Hodge Theory and Classical Algebraic Geometry

647 Hodge Theory and Classical Algebraic Geometry

Conference on Hodge Theory and Classical Algebraic Geometry May 13–15, 2013 The Ohio State University, Columbus, Ohio

Gary Kennedy Mirel Caibar˘ Ana-Maria Castravet Emanuele Macrì Editors

American Mathematical Society

Hodge Theory and Classical Algebraic Geometry

Conference on Hodge Theory and Classical Algebraic Geometry May 13–15, 2013 The Ohio State University, Columbus, Ohio

Gary Kennedy Mirel Caibar˘ Ana-Maria Castravet Emanuele Macrì Editors

647

Hodge Theory and Classical Algebraic Geometry

Conference on Hodge Theory and Classical Algebraic Geometry May 13–15, 2013 The Ohio State University, Columbus, Ohio

Gary Kennedy Mirel Caibar˘ Ana-Maria Castravet Emanuele Macrì Editors

American Mathematical Society Providence, Rhode Island

EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Martin J. Strauss

2010 Mathematics Subject Classiﬁcation. Primary 14C30, 14D07, 32G20, 58A14.

Library of Congress Cataloging-in-Publication Data

Hodge theory and classical algebraic geometry : a conference on Hodge theory and classical algebraic geometry : May 13-15, 2013, the Ohio State University, Columbus, Ohio / Gary Kennedy, Mirel Caib˘ar, Ana-Maria Castravet, Emanuele Macr`ı, editors. pages cm. – (Contemporary mathematics ; volume 647) Includes bibliographical references. ISBN 978-1-4704-0990-6 (alk. paper) 1. Geometry, Algebraic–Congresses. 2. Hodge theory–Congresses. I. Kennedy, Gary, 1950- II. Caib˘ar, Mirel, 1967- III. Castravet, Ana-Maria, 1980- IV. Macr`ı, Emanuele.

QA564.H55 2015 514.74–dc23 2015006623

Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online)

DOI: http://dx.doi.org/10.1090/conm/647

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Contents

Preface vii The stability manifolds of P1 and local P1 Aaron Bertram, Steffen Marcus, and Jie Wang 1 Reduced limit period mappings and orbits in Mumford-Tate varieties Mark Green and Phillip Griffiths 19 The primitive cohomology of theta divisors Elham Izadi and Jie Wang 79 Neighborhoods of subvarieties in homogeneous spaces Janos´ Kollar´ 91 Unconditional noncommutative motivic Galois groups Matilde Marcolli and Gonc¸alo Tabuada 109 Diﬀerential equations in Hilbert-Mumford Calculus Ziv Ran 117 Weak positivity via mixed Hodge modules Christian Schnell 129

Preface

This volume contains a selection of papers stemming from the conference Hodge Theory and Classical Algebraic Geometry, held on the campus of The Ohio State University in Columbus, Ohio, from May 13 to 15, 2013. The conference web page is still accessible at http://go.osu.edu/hodge. Most of the conference talks were captured on video, which may be viewed by following the appropriate links. A program and abstracts are also available there. In some instances the paper in this volume closely adheres to the conference lecture; in other instances there is a great difference. The idea of the conference was to offer young researchers a global view of recent developments and to have the speakers share their vision of the future. The papers in this proceeding follow essentially a similar idea; there are a few survey papers while others contain original research. The topics range from more classical aspects of Hodge theory to modern developments in compactifications of period domains, applications of Saito’s theory of mixed Hodge modules, and connections with derived category theory and non-commutative motives. The reader may note an odd feature: although there is no dedication on the title page, each paper in our volume is dedicated to Professor Herb Clemens. This is because our conference inadvertently used the venerable sales technique known as “bait and switch,” with Clemens as our bait. That is to say: we announced that he was retiring, and that we were organizing a conference in his honor. But it turned out after all that he was not in fact retiring, but instead beginning yet another chapter in his distinguished career. In view of this history, we tried to enforce the following rule at our conference: you are forbidden to say anything nice about Herb. Of course the rule was skirted repeatedly. One speaker tried to evade it by praising not Herb but rather his basement, where he had stayed as a houseguest while learning to ski in the mountains of Utah. Others broke the rule quite brazenly. For example, it would be violating this rule to remark how much energy he has brought to the Ohio State Mathematics Department and in particular to those who work in algebraic geometry. And again it would be against the rule to note the awe with which we seem to observe at least two or three different people doing full-time jobs, each one of them named Herb Clemens. One of us once had the opportunity to speak to Robert Moses, a civil rights pioneer and the founder of the Algebra Project, who told us of his admiration for Herb’s work in mathematics education, and Moses seemed genuinely astonished to learn that Herb was also famous for a conjecture named after him in pure mathematics. We are very glad to report that his efforts on behalf of mathematics research, education, and infrastructure continue unceasingly.

vii

viii PREFACE

The conference was supported in part by National Science Foundation Grant No. 1302880, with additional funding provided by The Ohio State University De- partment of Mathematics, including its Mathematics Research Institute. We used the lecture room and other facilities of the Mathematical Biosciences Institute. The conference organizers were the editors of the present volume, together with Elham Izadi and Christian Schnell, who likewise helped us in preparing the conference grant proposal. Paul Nylander created the striking image used for our poster and conference web pages.

Gary Kennedy Mirel Caib˘ar Ana-Maria Castravet Emanuele Macr`ı

Contemporary Mathematics Volume 647, 2015 http://dx.doi.org/10.1090/conm/647/12956

The stability manifolds of P1 and local P1

Aaron Bertram, Steﬀen Marcus, and Jie Wang

Dedicated to Herb Clemens

Abstract. In this expository paper, we review and compare the local homeomorphisms from the manifold of Bridgeland stability conditions to the space of central charges in the cases of both P1 and local P1.

1. Introduction The space of stability conditions on a triangulated category was introduced by Bridgeland in [3], following work on Π-stability in string theory due to Douglas [7]. A central result of [3] is that for an arbitrary triangulated category D the space Stab(D) of stability conditions is a topological manifold. Under nice conditions on D, the manifold Stab(D) is in fact a ﬁnite dimensional complex manifold. When working with an algebraic variety X, Bridgeland’s stability conditions allow one to abstract notions of slope stability for sheaves on projective varieties to the bounded derived category of coherent sheaves on X. Since [3], there have been many eﬀorts towards computing explicit examples in order to better understand the theory and its relation to enumerative geometry and mirror symmetry. For a smooth curve C of genus at least one, there is a rather uniform and complete description of Stab(D(C)). Macr`ı[11] and Bridgeland [3]provedthat + they are all isomorphic to the universal cover GL (2, R)ofGL+(2, R). The case of C = P1 turns out to be more subtle and involved due to the existence of full exceptional collections in D(P1). There exists “pathological” regions in Stab(D(P1)) ∼ where line bundles are unstable. Okada [12] proved that Stab(D(P1)) = C2. In [2], Bridgeland describes a connected component of Stab(D(X)) when X is the minimal resolution of a Kleinian quotient singularity C2/G. In particular, for G = Z/2Z this includes the local case of P1 embedded as the zero section in the total space of its cotangent bundle. The aim of this expository note is to provide a thorough review and comparison of the two cases of P1 and local P1, with emphasis on the local homeomorphisms p from the space of stability conditions to the space of central charges. Through these explicit examples, the authors hope to illustrate some general methods to study stability manifolds. Some of these techniques can be applied to study more complicated examples. Bayer and Macr`ı[6] gave a fairly explicit description of the stability manifold of local P2.

c 2015 American Mathematical Society 1

2 AARON BERTRAM, STEFFEN MARCUS, AND JIE WANG

There is a natural free action of the additive group C on the stability manifold and nothing interesting happens to the semistable objects under this action (c.f. Section 2). Therefore it is enough, and in fact more natural, to work with the quotient manifold and the map p descents to the quotient. The map p is a submersive surjection (but not proper) in the P1 case whereas it is a covering map in the local P1 case. Although we assume familiarity with stability conditions on a derived category throughout this paper, we provide a cursory review in Section 2 for the sake of read- ability. In Section 3 we recast Okada’s description of Stab(D(P1)) through the lens of geometric and non-geometric chambers in the stability manifold. Furthermore, we describe the local homeomorphism p explicitly in this case. Theorem 1.1. The stability manifold Stab(P1)/C is isomorphic to C and the natural map 1 1 ∼ 1 p :Stab(P )/C → P(HomZ(K(P ), C)) = P is a holomorphic surjective submersion (but not proper). In Section 4 we consider Bridgeland’s main component of the stability manifold Stab(D) of the triangulated subcategory of Db(T ∗P1) consisting of complexes with cohomology supported on the zero section. We provide the expected description of the local homeomorphism p in this setting.

∼ 1 Theorem 1.2. The image of p :Stab0(D)/C −→ P(HomZ(K(D), C)) = P is equal to C \ Z and p is a covering map. The case of local P1 is a restatement of a minor degenerate case of the main theorem in [2, Theorem 1.2]. We do not claim any originality in these results except the expository part and maybe the gluing argument in the proof of Theorem 1.1. We hope our approach provides a more geometric perspective on the structure of the stability manifold of P1 and local P1.

2. Preliminaries For a thorough introduction to stability conditions on triangulated categories, see [3]. We recall the salient deﬁnitions of the theory, beginning with that of a Bridgeland stability condition. Familiarity with the theory of triangulated categories, bounded t-structures, and tilting is assumed.

2.1. Stability conditions. Definition 2.1. A numerical pre-stability condition σ on a triangulated category D consists of a pair (Z, A),where √ 1 Z = −d + −1 r : K(D)num → C is a group homomorphism called central charge and A⊂Dis the heart of a t- structure, satisfying the following properties: (a) r(E) ≥ 0 for all E ∈A; (b) if r(E)=0and E ∈Anonzero, then d(E) > 0.

1Throughout this paper, we assume the numerical Grothendieck group of D is ﬁnite dimensional.

THE STABILITY MANIFOLDS OF P1 AND LOCAL P1 3

d(E) We can use them to define a notion of slope-stability via the slope μ(E)= r(E) . The slope μ takes value in (−∞, +∞]. Definition 2.2. An object E ∈Ais stable (resp. semistable) if for any subobject F of E in A, μ(F ) <μ(E)(resp.μ(F ) ≤ μ(E)). A pre-stability condition σ =(Z, A) is a stability condition if any nonzero object E ∈Aadmits a finite filtration:

0 E0 E1 ... En = E

Ei uniquely determined by the property that each Fi := is semistable and Ei−1

μ(F0) >μ(F1) > ... > μ(Fn). This property is called the Harder-Narasimhan property. We will also use the notation of slicing introduced by Bridgeland. For a stability condition σ and a real number φ ∈ (0, 1], we deﬁne a full abelian subcategory Pσ(φ), called a slicing, of D consisting of semistable objects of phase φ.Therelation between phase and slope is given by μ = − cot(πφ).

We then inductively deﬁne the category Pσ(φ) of semistable objects of arbitrary phase φ by enforcing Pσ(φ +1)=Pσ(φ)[1].

For any stability condition, HomD(P(φ1), P(φ2)) = 0 for φ1 >φ2. It follows that for each nonzero object E ∈Dthere is a unique sequence of real numbers φ1 >φ2 > ... > φn and a ﬁnite collection of triangles / / / / / EO 0 E1 ...... En−2 En−1 En = E } O w O tt }} ww tt } ww tt }} ww tt ~}} w{w ztt A1 An−1 An with Aj ∈Pσ(φj ) for all j. Finally denote Pσ(a, b] be the full subcategory consisting of objects whose semistable factors have phase in (a, b]. With this notation, the heart A = Pσ(0, 1]. Clearly (Z, Pσ) determines the stability condition σ =(Z, A) and vice versa. We will sometimes use the equivalent format (Z, P) of a stability condition. 2.2. Stability manifold. The set of locally ﬁnite2 stability conditions is denoted by Stab(D). Bridgeland proved that there exists suitable metric f on Stab(D) such that if f(σ, τ ) < 1andσ, τ have the same charge, then σ = τ.Themain result of [3] is the following. Theorem 2.3 (Theorem 1.2 and Corollary 1.3 of [3]). For each connected component Σ ⊂ Stab(D) there is an open set V (Σ) in a linear subspace of HomZ(K(D)num, C) and a local homeomorphism

(2.1) p :Σ−→ V (Σ) ⊂ HomZ(K(D)num, C)

2This condition is always satisﬁed for the categories considered in this paper.

4 AARON BERTRAM, STEFFEN MARCUS, AND JIE WANG mapping a stability condition σ =(Z, A) to its central charge Z. As a consequence, Stab(D) has a manifold structure induced from HomZ(K(D)num, C). Theorem 2.3 does not imply the local homeomorphism p is a covering map. If σ =(Z, P) ∈ Stab(D), and Z is another charge near Z,theremaynot exist σ =(Z, P)nearσ. Neverthelss, we have the following technical result which guarantees the existence of such σ. Theorem 2.4. [3, Theorem 7.1] Let σ =(Z, P) ∈ Stab(D),andchoose0 < <<1.IfZ ∈ HomZ(K(D), C) satisﬁes (2.2) |Z(E) − Z(E)| < sin(π)|Z(E)| for every σ-stable E in D, then there is a stability condition τ =(Z, P) ∈ Stab(D) with the distance f(σ, τ ) <. The stability manifold Stab(D) comes with natural left and right actions by Aut(D)andC respectively. An automorphism Φ ∈ Aut(D) induces an automor- D P phism ϕ of Knum( ). The action of Φ on a stability condition√ (Z, ) is deﬁned to be (Z ◦ ϕ−1, Φ(P)). The action of z ∈ C sends (Z, P(·)) to (e−π −1z · Z, P(Re(z)+·)). In particular, tilting a stability condition along μ ∈ R corresponds to the action of z = μ and the heart after tilting becomes P(μ, μ +1].

3. Stability manifold of P1 In this section we explicitly describe the local homeomorphism p in the case of P1. In this setting, the triangulated category under consideration is D = Db(Coh(P1)). We denote the stability manifold Stab(D)simplybyStab(P1). This space was studied by Okada and computed to be isomorphic to C2 in [12,Theo- rem 1.1]. Okada’s approach was to classify the various hearts and central charges that appear up to the action of Aut(D). We review this classiﬁcation from a slightly diﬀerent perspective. 3.1. Constructing stability conditions. We begin by constructing some interesting stability conditions in Stab(P1). In the next subsection, we will see these examples are essentially all the possibilities. 1 Start with the standard example of stability√ condition with heart A = Coh(P ) and central charge Z(E)=−deg(E)+ −1rk(E) as depicted in Figure 1. Since any vector bundle on P1 splits, the stable objects under this stability condition are all the line bundles and the skyscraper sheaf Cx.

Figure 1. The standard central charge.

THE STABILITY MANIFOLDS OF P1 AND LOCAL P1 5

Figure 2. Tilt along the angle μ.

Next tilt at an angle μ along a line between OP1 (d)andOP1 (d − 1) (Figure 2). Tilting does not change the set of stable objects but will result in a new heart Aμ (and of course, a rotated central charge). The stable objects in Aμ are OP1 (k) for k ≥ d and OP1 (k)[1] for k ≤ d − 1andCx. Since every stable object in Aμ can be written as successive extensions in terms of OP1 (d)andOP1 (d − 1)[1], Aμ = OP1 (d), OP1 (d − 1)[1] , the extension closure of the subcategory generated by OP1 (d)andOP1 (d − 1)[1]. Follow this by deforming the resulting central charge through a wall crossing (Figure 3).

Figure 3. A wall crossing.

The resulting heart does not change, remaining Aμ, but the collection of stable objects does change. For instance, the exact triangle

OP1 (d) −→ Cx −→ O P1 (d − 1)[1]

1 destabilizes Cx. Similarly, since HomD(OP1 (d), OP1 (d − 1)[1]) = 0, any objects in Aμ must be extensions of OP1 (d−1)[1] by OP1 (d). We conclude that every object in Aμ except OP1 (d), OP1 (d − 1)[1] and their respective direct sums is destabilized by OP1 (d). Thus the only stable objects are OP1 (d), OP1 (d − 1)[1] and their respective shifts. Next perform one more tilt along the vertical axis (Figure 4).

6 AARON BERTRAM, STEFFEN MARCUS, AND JIE WANG

Figure 4. Tilt along y-axis.

The new heart after this tilt is A = OP1 (d), OP1 (d − 1)[2] .Since 1 1 HomD(OP1 (d), OP1 (d − 1)[2]) = HomD(OP1 (d − 1)[2], OP1 (d)) = 0, the only objects in A are the two generators and their direct sums. We can then iterate the above process by deforming and then tilting. The resulting stability conditions could have OP1 (d), OP1 (d − 1)[l] for any l ≥ 2 as their heart.

3.2. Classiﬁcation of stability conditions. In this subsection we will show that all possible stability conditions on P1 areessentiallytheonesweconstructed in the previous subsection. The bounded derived category on a smooth curve has a relatively simple structure, this leads to the following observation: Proposition 3.1. Fix any stability condition σ ∈ Stab(P1).Thepotentially stable objects of D are either OP1 (n) or Cx or their shifts.

Proof. By Corollary 3.15 of [10], any object of D is isomorphic to a direct sum Ei[i]wheretheEi are themselves direct sums of line bundles and torsion sheaves. Any stable object must be indecomposable. Since any vector bundle on 1 P splits, E has to be either OP1 (n)orCx or their shifts.

The following lemma identiﬁes a collection of exact triangles in D involving the potentially stable objects. Lemma 3.2. ([9] Lemma 6.6) For any n, k ∈ Z, there exist exact triangles

⊕n−k ⊕n−k−1 (3.1) OP1 (k +1) −→ O P1 (n) −→ O P1 (k)[1] if kn,

(3.3) OP1 (k +1)−→ Cx −→ O P1 (k)[1]

Moreover, any exact triangle A → M → B with M either OP1 (n) or Cx and ≤0 HomD (A, B)=0is in one of the above form.

It is now possible to classify stability conditions based on the stability of OP1 (n) 1 and Cx.Letσ =(Z, P) ∈ Stab(P ) be a stability condition. There are two cases.

THE STABILITY MANIFOLDS OF P1 AND LOCAL P1 7

1 Case (a) OP1 (n) and Cx are σ-semistable for all n ∈ Z and x ∈ P . In this case, since there exists non-trivial exact triangles OP1 (n − 1) → OP1 (n) → Cx →OP1 (n − 1)[1], we have

(3.4) φ(OP1 (n − 1)) ≤ φ(OP1 (n)) ≤ φ(Cx) ≤ φ(OP1 (n)) + 1 1 for any n ∈ Z, x ∈ P . This also implies that φ(Cx)doesnotdependon 1 x ∈ P . This is because each Cx has the same class in K(D), so its phase has to differ by a multiple of 2 for a different point x. But the inequlity (3.4) implies this can not happen unless φ(Cx)doesnotdependonx. There are now two subcases. Case (a1) φ(OP1 ) <φ(Cx) <φ(OP1 )+1. Then ∼ Z : K(D) = Z[OP1 ] ⊕ Z[Cx] −→ C has rank 2 as a matrix with real coefficients, and√ it has the same orientation as the standard charge function − deg(E)+ −1 · rk(E). Therefore, up to the C-action, we can assume Z(E)=− deg(E)+w · rk(E) for w in the upper half plane and φ(Cx) = 1. It follows from (3.4) that OP1 (n) is in the heart for all n and the heart has to be Coh(P1). Therefore σ is the stability condition with heart Coh(P1) and charge − deg(E)+w·rk(E).

Case (a2) φ(Cx)=φ(OP1 ) or φ(Cx)=φ(OP1 )+1. Then Z has real rank 1, and we are on the “wall”. Up to rotation, we C 1 can assume that φ( x)=√ 2 .Letk + 1 be the smallest number such that O ∈ R · − O O 1 Z( P1 (k +1)) >0 1. Then φ( P1 (k +1))= φ( P1 (k)[1]) = 2 A P 1 O O O and the heart = ( 2 )= P1 (k +1), P1 (k)[1] .Since P1 (k +1)and OP1 (k)[1] are simple objects in the heart, they are stable. Everything else in the heart are strictly semistable.

1 Case (b) OP1 (n) or Cx is σ-unstable for some n ∈ Z or x ∈ P . We show that there exists an integer k such that OP1 (k), OP1 (k +1)and their shifts are the only stable objects and in which case the heart is OP1 (k +1), OP1 (k)[l] for l ≥ 1(seeFigures3and4). Suppose without loss of generality that OP1 (n) is unstable for some n (thecasethatCx is unstable can be proved similarly). Consider the ﬁrst exact triangle from the HN-ﬁltration of OP1 (n),

E1 −→ O P1 (n) −→ A0, ≤0 with A0 semistable. Since HomD (E1,A0) = 0, by Lemma 3.2, A0 is equal ⊕x0 to OP1 (k) up to shifts for some k ∈ Z and x0 ∈ Z+. Therefore OP1 (k) is semistable. Without loss of generality, let us assume k

⊕n−k ⊕n−k−1 (3.5) OP1 (k +1) −→ O P1 (n) −→ O P1 (k)[1] .

We claim that OP1 (k +1)isalsosemistableand(3.5)istheHN- ⊕n−k filtration of OP1 (n). If not, E1 = OP1 (k +1) is unstable. By Lemma 3.2 again, the first exact triangle in the HN-filtration of E1 has the form

E2 −→ E1 −→ A1

8 AARON BERTRAM, STEFFEN MARCUS, AND JIE WANG

⊕x1 ⊕y2 with A1 = OP1 (l)[j] semistable and E2 = OP1 (l +1)[j − 1] ,wherej is either 0 or 1.

If l

HomD(OP1 (l)[1], OP1 (k)[1]) = HomD(OP1 (l), OP1 (k)) =0 . If l>k+1,thenj =0and 1 HomD(OP1 (l), OP1 (k)[1]) = HomD(OP1 (l), OP1 (k)) =0 .

Either way, we find a contradiction with the fact that HomD(A1,A0)= 0. The claim is proved. As a consequence, φ(OP1 (k +1))>φ(OP1 (k)) + 1. Since HN-filtrations are unique, OP1 (l)andCx must be unstable for all l other than k, k + 1. Finally, since OP1 (k)andOP1 (k +1)aretheonly semistable line bundles and they have different phase, they are actually stable.

3.3. Coordinate description of p. We are now in a position to describe the stability manifold of P1 and the local homeomorphism to the space of charges up to C-action. 1 1 ∼ 1 p :Stab(P )/C −→ P(Hom(K(P ), C)) = P . For each k, consider the open subset 1 X˜k := {σ ∈ Stab(P ):OP1 (k) and OP1 (k + 1) are σ stable}.

Denote by Xk the quotient X˜k/C. We will abuse notation and denote by σ both the representative stability condition in X˜k and its image in the quotient. The content 1 of Section 3.2 is that {Xk | k ∈ Z} is an open covering of Stab(P )/C.

Lemma 3.3. The chart Xk is isomorphic to the upper half plane Hk and the map p restricted to Xk is given by 1 z → [− + k : −1]. exp(z) − 1

Proof. Fix σ ∈ X˜k.UptotheC-action, we can assume that Z(OP1 (k)[1]) = −1andφ(OP1 (k)[1]) = 1. Since there exists a non-zero morphism OP1 (k) → OP1 (k + 1), we know that φ(OP1 (k +1))>φ(OP1 (k)) = 0. On the other hand, we have already seen in section 3.1 that for any mass α>0 and any phase β>0, ∈ O there exists a unique stability condition σ(α, β) Xk such that φ( P1 (k +1))=√ β −1πβ and the charge Z := p(σ(α, β)) sends OP1 (k)[1] to −1andOP1 (k +1)to α · e . Switch to the standard basis of K(D), the charge is given by √ −1πβ Z([OP1 ]) = −kαe + k +1, √ −1πβ Z([Cx]) = αe − 1. Deﬁne a bijection ψ from H to X by k k k √ y z = x + −1y −→ σ ex, . π The composition

ψk / p / 1 Hk Xk P

THE STABILITY MANIFOLDS OF P1 AND LOCAL P1 9 is given by 1 z −→ [−kez + k +1:ez − 1] = − + k : −1 . ez − 1

It is straightforward to check that p ◦ ψk is a submersion with image 1 P \{k, k +1}. Since the complex structure of Xk is induced from p, the bijection ψk is an isomorphism.

3.4. Proof of Theorem 1.1. The remainder of the proof of Theorem 1.1 lies in gluing the above coordinate description of p. We give a chamber decomposition of each Xk in line with similar decompositions of stability manifolds into “algebraic” and “geometric” stability conditions. These correspond to the various cases in Section 3.2. √ The geometric chamber of Xk This is the region Uk = {x + −1y | 0 πcorresponding to the case (b) where only OP1 (k)[1] and OP1 (k + 1) are stable.

1 Lemma 3.4. Stab(P )/C is obtained by taking the disjoint union of Hk glued along Uk.

Proof. Points in Uk corresponds to the stability conditions σw with heart 1 Coh(P ) and charge function Zw(E)=− deg(E)+w · rk(E) for w in the upper half planeviathemap 1 (3.6) z −→ − + k. exp(z) − 1 The statements follows immediately.

To conclude the proof of Theorem 1.1, note that from the coordinate description, p is a surjective submersion. Now consider the action of a subgroup Z ⊂ Aut(D) generated by tensoring with line bundles. Clearly this action per- mutes the algebraic chambers and walls among Xk, but preserves the geometric chamber U ⊂ X . Thus we can ﬁnd a fundamental domain of this action in, say, ∼ k k X0 = H0. A fundamental domain of the action of Aut(D) on the geometric chamber U := {σw :Im(w) > 0} is given by {w ∈ C :0≤ Re(w) < 1, Im(w) > 0}.

Under the identiﬁcation (3.6) between U0 and U, we see that the region √ √ {z = x + −1y ∈ H : e|x| cos y ≤ 1, 0

10 AARON BERTRAM, STEFFEN MARCUS, AND JIE WANG bijectively to C \ R≥0. Moreover, this mapping extends continuously to the boundary and identiﬁes the two components of the boundary √ {z = x + −1y : e|x| cos y =1, 0 0} and √ {z = x + −1y : e|x| cos y =1, 0 0. Thus the quotient Z \ Stab(P )/C is conformal equivalent of C∗ (c.f [12] Lemma 4.4 for an explicit mapping). We conclude that Stab(P1)/C is isomorphic to the universal cover C of C∗.

4. Stability manifold of local P1 In this section we characterize the local homeomorphism p for the local model of P1 in a surface on a main component of the stability manifold. Denote the ∗ ∼ total space of T P1 by Y and let X = P1 be the zero section. We work with the triangulated subcategory D in Db(Y ) consisting of complexes whose cohomology is supported on X. This is equivalent, by the derived McKay correspondence (see [2]), to studying the bounded derived category of Z/2Z equivalent coherent sheaves on C2. Denote the stability manifold in question by Stab(D). In [2], Bridgeland identiﬁes the main component of the stability manifold for the minimal resolution of any arbitrary Kleinian quotient singularity and proves the analogue of Theorem 1.2. We denote this connected component by Stab0(D). The proof presented here follows the methods and techniques of [2], and is of interest mostly for a comparison with the P1 situation as described above.

4.1. Identifying the slope stable objects. Since Y has trivial canonical bundle, the category D is in fact a K3 category meaning the shift by 2 functor is a Serre functor, i.e. for any E,F ∈ D, ∼ ∨ 2 ∨ HomD(E, F) = HomD(F, E[2]) =HomD(F, E) . The standard t-structure on Db(Y ) induces a t-structure on D with heart A consisting of coherent sheaves on Y supported on X. Morphisms in A are as OY -modules. O n+1 O Denote the scheme Spec( Y /IX )byXn.Thenany Y -module supported O F∈A on X is an Xn -module for some n. It follows that any is a successive extensions of OX -modules, thus the natural map induced by the inclusion map i i∗ : K(Coh(X)) −→ K(A)=K(D) is surjective. Let π : Y −→ X denote the projection map of the cotangent bundle. Since there is no higher direct images of π for sheaves supported on X, the push- forward functor π∗ : A−→Coh(X) is exact. Therefore π∗ induces a map on the K-groups and π∗ is the inverse of i∗. ∼ We obtain an isomorphism K(A)=K(X) = Z[OX ] ⊕ Z[Cx]. Notice that F has the same class as π∗F in K(Aˆ). The following lemma gives a criterion for sheaves in A to be stable with respect to standard slope stability conditions, meaning stability conditions with central charges of the form Zw = − deg +w · rk

THE STABILITY MANIFOLDS OF P1 AND LOCAL P1 11 for w any ﬁxed point in the upper half plane. It is a statement similar in spirit to Proposition 3.1. Lemma 4.1. For any point w in the upper half plane U ={w : Im(w) > 0},there is a unique stability condition σw with heart Aˆ and charge Z(E)=− deg(π∗E)+ w · rk(π∗E).AsheafE ∈ Aˆ is σw-stable if and only if E is isomorphic to OX (n) or Cx. Proof. Since a sheaf E always has the same class as π∗E in K(A), Z(E) ∈ H ∪ R<0 for any nonzero E ∈ A. The existence of an HN-ﬁltration can be proved b 1 exactly as in the D (P ) case. Certainly OX (n)andCx are stable with respect to σw, and they are the only stable OX -modules. Conversely, assume E is stable. 0 Then End(E) = C · Id. Let V be the scheme theoretic support of E. H (V,OV ) 0 ∼ acts faithfully on E,soH (V,OV ) = C. Hence V must be contained in the scheme- 2 1 theoretic inverse image of the origin under the contraction Y → C /Z2, i.e V ⊂ P . Therefore E is in fact an OX -module and must be either OX (n)orCx.

4.2. Tilting slope stable√ objects. Now consider the stability condition σw constructed above with w = −1. We wish to understand the stability conditions resulting from certain elementary non-trivial tilts. We can tilt as in the 1 P case along a line between OX (1) and OX . By Lemma 4.1, the new heart A = OX , OX (−1)[1] after tilting is generated by extensions (in D)ofOX and 1 OX (−1)[1]. The diﬀerence to the P case is that for any stability condition with heart A, there is always a semistable object whose class in the Grothendieck group is [Cx]. We formalize this statement in the following lemma. Lemma 4.2. For each point Z in the region R := {Z ∈ HomZ(K(D), C):Im(Z(O )) > 0, X √ Im(Z(OX (−1)[1])) > 0,Z(Cx)= −1}

there is a unique stability condition σZ with heart A and central charge Z.More- over, there is always a semistable object for σZ whose class in K(D) is [Cx]. Proof. For the first half, it suffices to show the Harder-Narasimhan filtration exists. By construction, every object E ∈A has class [E]=m[OX ]+n[(OX (−1)[1]] for m, n ≥ 0. A subobject E of E in A has class [E ]=m [OX ]+n [(OX (−1)[1]] where m ≤ m and n ≤ n. Thus there do not exists infinite sequences of subobjects in A . For the second half, since OX and OX (−1)[1] are simple objects in A ,they are stable for any Z ∈ R. There are three cases

(a) φ(OX ) <φ(OX (−1)[1]). In this case, σZ is in the same C-orbit as the stability conditions in Lemma 4.1. Thus Cx is stable. (b) φ(OX )=φ(OX (−1)[1]). In this case, everything in A has the same phase, and therefore Cx is semistable. (c) φ(OX ) >φ(OX (−1)[1]). In this case, the exact sequence

OX −→ Cx −→ O X (−1)[1] in A destabilizes Cx. However, using the duality 1 O O − ∼ 1 O − O ∨ O − O ∼ C2 HomD( X , X ( 1)[1]) = HomD( X ( 1)[1], X ) =HomOY ( X( 1), X) = ,

12 AARON BERTRAM, STEFFEN MARCUS, AND JIE WANG

we see that there exists a non-trivial extension E ∈A: f / / (4.1) OX (−1)[1] E OX . Clearly [E]=[Cx] ∈ K(D). We claim that E is semistable. If not, let Ai for i =1, ..., n be the semistable factors of E in A . Since [Ai] ∈ Z≥0[OX ]⊕ Z≥0[OX (−1)[1]] and [E]=[OX ]+[OX (−1)[1]], the only possible HN- ﬁltration for E is of the form of the vertical triangle

f / / OX (−1)[1] E OX

g O(−1)[1]. The composition g ◦ f can not be zero since otherwise f would factor through a morphism from O(−1)[1] to OX , but HomD(O(−1)[1], OX )=0. Thus g ◦ f ∈ C∗ · Id by Schur’s lemma. We conclude that the extension (4.1) splits, a contradiction.

From now on we will abuse the notation by thinking of a point Z ∈ R also as the stability condition with heart A charge Z. 4.3. The covering property. Let Stab0(D) be the connected component of Stab(D) containing the set U. Lemma 4.3. Suppose E ∈ D is a stable object for some stability condition, then the class [E] in K(D)=Z[OX ] ⊕ Z[Cx] is either ±[OX (n)] or m[Cx] for some m, n ∈ Z. Proof. Consider the bilinear pairing χ on K(D)givenby − i i χ(E,F):= dimC( 1) HomD(E, F). i 0 ∼ 2 ∨ ∼ C If E is stable for some stability condition, HomD(E, E) = HomD(E, E) = .Thus ≤ 1 χ(E,E) 2 with equality holds when HomD(E, E) = 0. By Lemma 4.4, we see that if a stable object E has class [E]=a[OX ]+b[Cx], then χ(E,E) ≤ 2 if and only if −1 ≤ a ≤ 1. Therefore [E] is either ±[OX (n)] or m[Cx]forsomem, n ∈ Z.

Lemma 4.4. We have χ(OX , OX )=2and χ(OX , Cx)=χ(Cx, Cx)=0. Proof. Clearly we have 0 O O ∼ 2 O O ∨ ∼ C HomD( X, X) = HomD( X, X) = . 1 O O − O To compute HomD( X, X), apply the derived functor RHomD( , X )tothe exact sequence of OY -modules

0 −→ O Y (−X) −→ O Y −→ O X −→ 0.

THE STABILITY MANIFOLDS OF P1 AND LOCAL P1 13

Note that 1 O O 1 O O 1 O HomD( Y, X)=ExtOY ( Y, X)=H (X, X)=0, and 0 O − O O O HomD( Y( X), X)=HomOY ( Y, X(X)) O O − 0 O − =HomOY ( Y, X( 2)) = H (X, X( 2)) = 0. 1 O O O O We conclude that HomD( X, X)=0andχ( X , X )=2. − C 0 O C Similarly, apply the functor RHomD( , x), we obtain dimC HomD( X, x)= 1 O C 2 O C O C dimC HomD( X, x)=1andHomD( X, x)=0.Thusχ( X , x)=0. Finally apply the functor RHomD(−, Cx) to the exact sequence of OY -modules

0 −→ O X (−1) −→ O X −→ Cx −→ 0, 1 C C C C we obtain dimC HomD( x, x)=2andχ( x, x)=0. Lemma 4.5. The map p :Stab0(D) −→ HomZ(K(D), C) is a local homeomorphism onto an open subset of HomZ(K(D), C) containing B := {Z ∈ HomZ(K(D), C):Z([Cx]) =0 ,Z([OX (n)]) =0 for all n ∈ Z}. The restriction of p to p−1(B) is a covering map. Proof. By Theorem 2.3, p must be a local homeomorphism onto an open subset of Hom(K(D), C). By Lemma 4.2, this open subset contains R. Any charge Z ∈ B is in an GL+(2, R)-orbit of some charge in R. Since the universal cover + + GL (2; R)actsonStab0(D) with induced GL (2, R)-action on the charge space, we conclude that the image of p contains B. Fix a norm · on the vector space K(D)C.TakeanyZ ∈ B, and let δ>0be given. We can deﬁne open subsets Bδ(Z)={Z ∈ B : |(Z − Z )(v)| <δv for any v ∈ K(D)} in Hom(K(D), C). Similarly, for each σ ∈ p−1(Z) we can use the generalized metric f on Stab(D) in Section 2.2 to deﬁne open subsets 1 C (σ)= τ ∈ p−1(B (Z)) : f(σ, τ ) < ⊂ Stab (D). δ δ 2 0

For δ small enough the restriction of the map p to Cδ(σ) will be a homeomorphism onto Bδ(Z). To see that p gives a covering map, it is enough to see that we get the decomposition as a disjoint union −1 p (Bδ(Z)) = Cδ(σ). σ∈p−1(Z) The containment ⊃ is clear. For the other containment, consider a stability condition σ =(Z , P ) with Z ∈ Bδ(Z). Note that by construction of B, there exists a uniform constant η such that |Z (v)|≥ηv for any v =[OX (n)] or [Cx]. Thus by Lemma 4.3, the inequality (2.2) is satisﬁed for δ small enough. Theorem 2.4 then gives the existence of a stability condition σ with p(σ)=Z and f(σ, σ) <.Thus σ ∈ Cδ(σ). Remark. The same proof shows in the P1 case, that the map p :Stab(P1) → −1 HomZ(K(D), C) is also a covering map when restricted to p (B).

14 AARON BERTRAM, STEFFEN MARCUS, AND JIE WANG

4.4. The action of autoequivalences. Another diﬀerence between D and D is that OX and OX (−1)[1] are spherical objects in D. Recall that an object S ∈ D is spherical if C if k =0, 2; Homk (S, S)= D 0otherwise. Seidel and Thomas [14] showed that any spherical object determines an autoequivalence ΦS ∈ Aut(D) deﬁned via. the triangle

⊕kHomD(S[k],E) ⊗ S[k] −→ E −→ ΦS(E). Notice that ΦS(S)=S[−1], so ΦS has inﬁnite order in Aut(D). We denote by S1 and S2 the objects OX and OX (−1)[1] respectively. A similar O computation as in Lemma 4.4 shows that the deﬁning exact triangle for ΦS1 ( X (1)) is O⊕2 −→ O −→ O X X (1) ΦS1 ( X (1)). O O − Thus ΦS1 ( X (1)) = X ( 1)[1] by the Euler sequence.

Applying ΦS1 to the exact triangle

OX (1) −→ Cx −→ O X [1], we obtain a nontrivial exact triangle O − / C /O (4.2) X ( 1)[1] ΦS1 ( x) X . As a consequence, the induced map on the Grothendieck groups under the standard basis ∼ K(D) = Z[OX ] ⊕ Z[Cx] is given by the matrix −10 ϕ = . S1 01 O − Similarly, we compute ΦS2 (S1)= X ( 2)[1]. Applying ΦS2 to the exact triangle

OX −→ Cx −→ O X (−1)[1], we obtain O − −→ C −→ O − X ( 2)[1] ΦS2 ( x) X ( 1). Therefore the induced map on K(D)isgivenbythematrix −10 ϕ = . S2 21

For us to make use of the autoequivalences ΦS , it is necessary to show that i they preserve the connected component Stab0(D). This is given by the following two lemmas.

Lemma 4.6. Let σi =(Z, P) be a point in the boundary of R for which there is a unique Si ∈A with Im(Z(Si)) = 0. Assume further that Z(Si) ∈ R<0.Then the stability condition Φ−1(σ ) also lies in the boundary of R. Si i

THE STABILITY MANIFOLDS OF P1 AND LOCAL P1 15 Proof. Take a small open neighborhood V in Stab(D) containing σi and consider the open subset V− = {τ ∈ V | ImZτ (Si) < 0}. −1 We claim that if V is small enough, Φ (V−) ⊂ R. To this end, it suﬃces to show Si ∈ A that the heart of any τ V− is ΦSi ( ).

If i = 1, applying ΦS1 to the exact triangle / / Cx S2 S1[1] gives a nonsplit extension C / / (4.3) ΦS1 ( x) ΦS1 (S2) S1 . C ∈A ∈A By (4.2), ΦS1 ( x) , and therefore ΦS1 (S2) . The non-splittness of (4.3) implies that HomD(S1, ΦS1 (S2)) = 0 (since S1 is an spherical object). This means the semistable factors of ΦS1 (S2) with respect to σ1 have phases in the interval (0, 1). O − O O − If i = 2, using the fact that ΦS2 (S1)= X ( 2)[1] and φσ2 ( X ) <φσ2 ( X ( 1)[1]), we conclude that OX (−2)[1] ∈A is σ2-stable becasue σ2 is in the same C-orbit as one of the stability conditions in Lemma 4.1. A In either case, we have for i = j that the object ΦSi (Sj) lies in and has semistable factors with respect to σi in the interval (0, 1). Therefore ΦSi (Sj )is − in the heart of τ for any τ near σi.Wheni = j, the object ΦSi (Si)=Si[ 1] is σi-stable with phase 0, so we can assume Si[−1] is τ-stable with phase at most 1 for τ near σi.Forτ ∈ V−,wehave

0 <φτ (Si[−1]) ≤ 1. A ∈ We have proved that ΦSi ( ) is contained in the heart of τ for any τ V−, thus they are equal. The claim is proved. −1 It follows that ΦS (σi) lies in the closure of R. It can not lie in R because − i ΦSi (Si)=Si[ 1] lies on the positive real axis. The subgroup generated by these two autoequivalences plays an important role. D D We denote the subgroup of Aut( ) generated by ΦS1 and ΦS2 by A0( ). Lemma 4.6 shows that that the autoequivalence ΦSi reverses the orientation of the boundary of R, taking the side where ImZ(Si) < 0 to the side where ImZ(Si) > 0. As in [2], this implies the following: Lemma 4.7. For every stability condition σ =(Z, P) in Stab0(D),thecentral charge does not vanish on [Cx]. Furthermore, there is an autoequivalence Φ ∈ A0(D) such that up to C-action, Φ(σ) lies in the closure of R.

Proof. Assume Zσ([Cx]) = 0, we prove the second statement. We could take apathγ = σt in Stab0(D) connecting σ with a point in R. We can perturb the C C path a little such√ that Zt([ x]) =0forallt.Uptothe -action, we can also assume that Zt([Cx]) = −1 for all t. Each time γ passing through the boundary of R, Lemma 4.6 shows that there exists an autoequivalence Φ−1,fori = 1 or 2, that Si takes γ back to a stability condition√ in the closure of R. Since we are assuming Zt([Cx]) = Zt([S1]+[S2]) = −1 for all t, each time when Zt([Si]) cross the boundary of R, Zt([S3−i]) is far away from the boundary of R. Therefore after applying ﬁnitely many autoequivalences to σ, we are in the closure of R. Now suppose Z([Cx]) = 0, then there are no semistable objects in σ whose class is [Cx]. This is true in an open neighborhood of σ. But by the ﬁrst part, the stability

16 AARON BERTRAM, STEFFEN MARCUS, AND JIE WANG conditions near σ with Z([Cx]) =0canbemappedto R¯ by an autoequivalence Φ ∈ A0(D). Also notice that the induced map ϕ on the K-group has the property that ϕ([Cx]) = [Cx]. Lemma 4.2 then implies that the stability conditions near σ do have semistable objects in the class [Cx], a contradition. 4.5. Description of Stab0(D)/C. We now identify P(HomZ(K(D), C)) with 1 P by sending Z to [Z(OX ):Z(Cx)]. With this identiﬁcation, the charges (up to scalar) which vanish on [Cx] correspond to ∞ := [1 : 0] and the charges which vanish on [OX (n)] = [OX ]+n[Cx] correspond to [−n :1].ThustheopensetB in Lemma 4.5 modulo the C∗-action corresponds to C \ Z.

∼ 1 Lemma 4.8. The image of p :Stab0(D)/C −→ P(HomZ(K(D), C)) = P is equal to C \ Z. Proof. Lemma 4.5 implies that Im(p) ⊃ C \ Z.LetZ ∈ Im(p). By Lemma 4.7, we have Z(Cx) =0forany Z in the image of p. Therefore ∞ = [1 : 0] ∈/ Im(p). To show the image is equal to C \ Z, it suﬃces to show that O ∈ Z P Z( X (n)) =0foranyn . Suppose σ =(Z, ) is a stability√ condition such that Z(OX (n0)) = 0. Up to C-action, we can assume Z(Cx)= −1. By Lemma 4.7, C ∈ D ∈ ¯ up to -action, there exists an auto equivalence Φ A0( )√ such that Φ(σ) R −1 and with its corresponding charge Z ◦ φ sending [Cx]to −1. It is then clear −1 that Z ◦ φ does not vanish on [OX (l)] for any l ∈ Z by the construction of R¯. But the induced map ϕ([OX (n0)]) = ±[OX (l)] for some l, a contradiction. Proof of Theorem 1.2. This follows from Lemma 4.5 and Lemma 4.8. Proposition 4.9. The induced action of A0(D) on C \ Z is free. The quotient W has the free group in two generators F2 as its fundamental group and the composition of p with the quotient map to W q / / Stab0(D)/C C \ Z W. has A0(D) as the group of Deck transforms. Proof. The action on C \ Z by A0(D) is generated by precomposing a charge − − O Z by ϕ 1 and ϕ 1. Using complex coordinates z =[z :1]=[Z( X ) :1]forZ,then S1 S2 Z(Cx) the coordinates for Z ◦ ϕ−1 and Z ◦ ϕ−1 are −z and −z + 2 respectively. It is now S1 S2 routine to check that this action is free. A fundamental domain in C \ Z for this action is {z ∈ C :Im(z) > 0, −1 ≤ Re(z) < 1}∪{(−1, 0)}.

Denote W the quotient manifold. Clearly W has F2 as its fundamental group. By what was said above, A0(D) acts as Deck transforms for the covering map q / / Stab0(D)/C C \ Z W.

Conversely, if σ1, σ2 gets mapped to the same point in W , we need to show that there exists Φ ∈ A0(D) sending σ1 to σ2 up to C-action. By lemma 4.7, we can assume both σ1 and σ2 belongs to R¯.SinceR/C maps injectively to W , σ1 and ∗ σ2 have the same image in W implies that the have the same charge modulo C . Therefore they are in the same C-orbit.

THE STABILITY MANIFOLDS OF P1 AND LOCAL P1 17 Remark. By a result of Okada [13], Stab(D)=Stab0(D) is actually connected. It is proved in [1]thatStab0(D) is simply connected. This is equivalent to the fact that A0(D) is isomorphic to the free group F2. References

[1] A. Ishii, K. Ueda, and H. Uehara, Stability conditions on An-singularities, J. Diﬀerential Geom. 84 (2010), no. 1, 87–126. MR2629510 (2011f:14027) [2] T. Bridgeland, Stability conditions and Kleinian singularities, Int. Math. Res. Not. IMRN 21 (2009), 4142–4157, DOI 10.1093/imrn/rnp081. MR2549952 (2011b:14038) [3] T. Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), no. 2, 317–345, DOI 10.4007/annals.2007.166.317. MR2373143 (2009c:14026) [4] T. Bridgeland, Stability conditions on a non-compact Calabi-Yau threefold, Comm. Math. Phys. 266 (2006), no. 3, 715–733, DOI 10.1007/s00220-006-0048-7. MR2238896 (2007d:14075) [5] T. Bridgeland, Stability conditions on K3 surfaces, Duke Math. J. 141 (2008), no. 2, 241–291, DOI 10.1215/S0012-7094-08-14122-5. MR2376815 (2009b:14030) [6] A. Bayer and E. Macr`ı, The space of stability conditions on the local projective plane, Duke Math. J. 160 (2011), no. 2, 263–322, DOI 10.1215/00127094-1444249. MR2852118 (2012k:14019) [7] M. R. Douglas, Dirichlet branes, homological mirror symmetry, and stability, Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 395–408. MR1957548 (2004c:81200) [8] B. Dubrovin, Geometry and analytic theory of Frobenius manifolds, Proceedings of the In- ternational Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math. Extra Vol. II (1998), 315–326. MR1648082 (99j:32025) [9]A.L.Gorodentsev,S.A.Kuleshov,andA.N.Rudakov,t-stabilities and t-structures on triangulated categories (Russian, with Russian summary), Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), no. 4, 117–150, DOI 10.1070/IM2004v068n04ABEH000497; English transl., Izv. Math. 68 (2004), no. 4, 749–781. MR2084563 (2005j:18008) [10] D. Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2006. MR2244106 (2007f:14013) [11] E. Macr`ı, Stability conditions on curves, Math. Res. Lett. 14 (2007), no. 4, 657–672, DOI 10.4310/MRL.2007.v14.n4.a10. MR2335991 (2008k:18011) [12] S. Okada, Stability manifold of P1,J.AlgebraicGeom.15 (2006), no. 3, 487–505, DOI 10.1090/S1056-3911-06-00432-2. MR2219846 (2007b:14036) [13] S. Okada, On stability manifolds of Calabi-Yau surfaces,Int.Math.Res.Not.,postedon 2006, Art. ID 58743, 16, DOI 10.1155/IMRN/2006/58743. MR2276354 (2007j:14060) [14] P. Seidel and R. Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001), no. 1, 37–108, DOI 10.1215/S0012-7094-01-10812-0. MR1831820 (2002e:14030)

Department of Mathematics, University of Utah, 155 S 1300 E, Salt Lake City, Utah 84112 E-mail address: [email protected] Department of Mathematics, The college of New Jersey, PO Box 7718, Ewing, New Jersey 08628 E-mail address: [email protected] Department of Mathematics, University of California, San Diego, 9500 Gilman Dr., La Jolla, California 92093 E-mail address: [email protected]

Contemporary Mathematics Volume 647, 2015 http://dx.doi.org/10.1090/conm/647/12969

Reduced limit period mappings and orbits in Mumford-Tate varieties

Mark Green and Phillip Griﬃths For Herb Clemens, valued colleague and friend

Abstract. In this paper we use Lie group and Lie algebra methods to study properties of degenerating polarized Hodge structures. The central point is to assign to a limiting mixed Hodge structure a point in the boundary of the corresponding Mumford-Tate domain (reduced or naive limit period), and then use the analysis of the orbit structure of the boundary to infer results on the geometry of the limit mixed Hodge structure. As an application the description of the extremal limiting mixed Hodge structures for period domains is obtained. 0. Introduction I. GR-orbits A. Generalized flag varieties B. Open GR-orbits C. General GR-orbits II. Hodge structures, mixed Hodge structures and limiting mixed Hodge structures A. Polarized Hodge structures and Mumford-Tate domains B. Polarized Hodge structures in terms of grading elements C. Mixed Hodge structures D. Limiting mixed Hodge structures III. The reduced limit period mapping A. Definition and first properties B. The differential of the reduced limit period mapping C. Variations of the extension data in LMHS’s under the reduced limit period mapping D. Extremal degenerations of polarized Hodge structures

0. Introduction Period mappings, or equivalently variations of Hodge structure, (0.1) Φ : S → Γ\D arise by studying how the polarized Hodge structure in the cohomology varies in a family of smooth projective varieties. Here

2010 Mathematics Subject Classiﬁcation. Primary 32G20, 32S35; Secondary 22F30.

c 2015 American Mathematical Society 19

20 M. GREEN AND P. GRIFFITHS

• S is a complex analytic variety that is a Zariski open set in an analytic variety S¯; • D = GR/H is a Mumford-Tate domain embedded in its compact dual Dˇ = GC/Q; • Γ ⊂ G is a discrete subgroup that contains the monodromy group of the period mapping (0.1). It is of interest to study the behaviour of Φ(s)ass → S¯\S, which reﬂects how the Hodge structures degenerate as the varieties acquire singularities. This is the general context for this paper. We note that usually S is assumed to be smooth and S¯\S a normal crossing divisor, and we shall generally make this assumption even though in geometric examples the general case is frequently of ∼ ∗ interest. In addition we shall usually restrict to the local case where S = Δ k × Δl is a punctured polycylinder and S¯ =Δk × Δl. In this case the monodromy group Γ may be assumed to be generated by commuting unipotent transformations Ti = exp Ni around the punctures. Here the analysis of Φ(s)iscarriedoutin[CKS], which will be our fundamental reference. To set a context for this work, it lies at the conﬂuence between two classical and highly developed subjects:

(A) the study of the GR-orbits in Dˇ,forwhich[FHW] is the basic reference; (B) the study of completions, or partial compactifications Γ\D∗ of Γ\D. Regarding (A), the general theory in [FHW] is given without reference to the realization of an open GR-orbit D as a Mumford-Tate domain [GGK1]. In section I we shall summarize, and in some instances supplement, this theory in the form to be used where D has the additional Mumford-Tate domain structure. Here we shall generally follow the notations and conventions in [GGK2]. Regarding (B), a subject that has had an extensive and rich development, one may think of the theory as falling into two parts: (B1) compactifications of Γ\D when D is an Hermitian symmetric domain, or more generally when D is a non-compact Riemannian symmetric space (cf. [Sa], [BB], [AMRT]and[BJ]); (B2) completions of Γ\D when D is a general Mumford-Tate domain (cf. [CK] and [KU] for the period domain case and [KP1] for general Mumford- Tate domains). These two theories coincide in the very special case when a bounded symmetric domain has the additional structure of an unconstrained Mumford-Tate domain; i.e., one whose infinitesimal period relation is trivial. Deferring for the moment the discussion of (B2), under (B1) there are essentially two types of compactifications that are relevant for this work: (B.1M) These are the compactifications Γ\DM of Satake [Sa] and of Baily-Borel [BB]. They were in fact motivated by Hodge theory, viz. the classical case of polarized Hodge structures of weight n = 1 (abelian varieties). M Roughly speaking, D is obtained by attaching to D sets DP constructed from maximal parabolic subgroups P ofG, and then defining a topology M ∪ and analytic structure on D = D ( P DP ) on which the arithmetic group Γ operates analytically and properly discontinuously and such that M Γ\D is a compact analytic space. The DP are themselves Hermitian

REDUCED PERIOD MAPPING 21

symmetric domains which are acted on by the real Lie groups Pr,R associated to the reductive parts of the P ’s, and which are equivariantly embedded in the topological boundary ∂D. With considerable further work, Γ\DM is then proved to have the structure of a normal, projective algebraic variety. These compactifications are in a precise sense minimal (hence the “M”inDM ); in particular, period mappings (0.1) extend for any S and S¯, where the complement of S in S¯ is a reduced normal crossing divisor. (B.1T) These are the toroidal compactifications Γ\DT of Mumford et. al ([AMRT]). Among other things they are almost desingularizations of Γ\DM , meaning that they have at most quotient singularities. In contrast to the Γ\DM ’s, they are generally not canonical but rather depend on the choice of a fan Σ consisting of a set of nilpotent cones σ ⊂ gnilp that satisfy compatibility conditions. Turningto(B.2),weusethewordcompletions since the Γ\D∗’s are not in general compact but rather serve to complete period mappings (0.1) to Φ:Δ→ Γ\D∗. Thus, D∗ depends on the structure of D as a Mumford-Tate domain; it is obtained from D by attaching limits of holomorphic mappings that satisfy the infinitesimal period relation. The by now standard completions DKU are those based on [CKS] and constructed by Kato-Usui [KU]; they are constructed by adding to D the space of limiting mixed Hodge structures for a set of nilpotent cones constituting a fan, and thus are analogues of the toroidal compactifications in the weight n =1case. Here we should like to call attention to the papers [CCK]and[Ca]where the construction of the Satake-Baily-Borel and toroidal compactifications in the classical case are discussed from a purely Hodge theoretic perspective. In this situation there is no infinitesimal period relation and consequently the theory is much simpler; but it does suggest aspects of how the general case might proceed. An evident question is whether there is an analogue of the Satake-Baily-Borel construction for general Mumford-Tate domains. In fact, the paper [CK] discusses one such for period domains for weight n = 2 polarized Hodge structures. Here the critical role of the infinitesimal period relation comes into play, and the spaces constructed are completions relative to period mappings Φ : Δ∗ → Γ\D but they themselves are not compact. At this juncture one may ask what, if anything, do the above constructions 1 have to do with GR-orbits in the generalized flag varieties GC/Q? Their geometry, including the descriptions of the tangent and normal spaces and intrinsic Levi forms,2 is a subject of considerable interest in its own right irrespective of any Hodge theoretic considerations. Natural questions are KU (0.2) Do boundary components in the Kato-Usui spaces D map to GR-orbits in ∂D?

1Here we should like to call attention to [KP1]and[KP2], which overlap with and precede some of what is presented here. These papers will be further discussed at various points in this work. We also mention [He] where for period domains a number of the results given here may be found. 2The intrinsic Levi forms for GR-orbits in the ﬂag variety case of GC/B are brieﬂy discussed in the appendix to lecture 6 in [GGK2]. A more thorough treatment of them in the general case will appear in a future work. Important results in this area have been obtained by C. Robles.

22 M. GREEN AND P. GRIFFITHS

We shall see that the answer to this question is aﬃrmative. Then one may ask (0.3) Do the images of the Kato-Usui boundary components constitute boundary components for other completions? Again we shall see that, at least under very special circumstances, there is a positive answer to this question. In the classical weight n = 1 case it will turn out that the images of the Kato-Usui boundary components are isomorphic to those in the Satake-Baily-Borel compactiﬁcation.3 In the case of period domains for weight n =2theGR-orbits in the boundary of SO(2h, k)/U(k) ⊂ SO(2h + k, C)/B are used as a starting point for the completions in [CK]. A next question is (0.4) What Hodge theoretic information is retained under the mapping (0.2)? To explain this, here we shall restrict to 1-dimensional boundary components B(N) given for N ∈ gnilp as a set by • B(N)={limiting mixed Hodge structures (V,W•(N),F ), modulo rescaling F • ∼ eN F •}.4 • We denote by Gr(V,W•(N),F ) the direct sum of the polarized Hodge structures • on the primitive parts of (V,W•(N),F ). Then • (0.5) The mapping (0.2) captures all of Gr(V,W•(N),F ). When the weight n = 1 the answer to the question (0.4) may be phrased as saying • that Gr(V,W•(N),F ) is all that is captured by the mapping (0.2). However, we shall see that (0.6) When the weight n 2 some, but not all, of the extension data in the limiting mixed Hodge structure is captured by the mapping (0.2) A part of this work is devoted to quantifying and seeking to understand this phenomenon. A conceptual reason for (0.6) is the following: Denote by B(N)R ⊂ B(N) the equivalence classes of R-split limiting mixed Hodge structures having a given N, and recall the Deligne retraction map

B(N) → B(N)R. This mapping may be roughly thought of as taking the real part of some of the coordinates of the extension classes for points in B(N). Then in the following diagram the mapping in (0.2) factors (0.7) B(N) KKK KKΠ KKK K% ∂D9⊂ D.ˇ ss ss ss ss ss B(N)R

3This may be gleaned, e.g., from the discussion in pages 84ﬀ in [Ca]. It was alluded to above where we noted that the boundary components DP were Hermitian symmetric domains equivariantly embedded in ∂D ⊂ Dˇ. 4This is the same notation as in [KP1]. With one important exception, we have attempted to follow the terminology used there and in [KP2]. The exception is that we use reduced limit period mapping, rather than na¨ıve limit as in [KP2]. The reason for this terminology will be explained in footnote 7 below.

REDUCED PERIOD MAPPING 23

Thus (i) the mapping Π is holomorphic; (ii) we have the factorization (0.7) The extension data captured by Π is the maximum amount subject to (i) with the constraint (ii). Returning to the questions (0.2) and (0.3), one may additionally ask (0.8) Are the images of the Kato-Usui boundary components themselves orbits of real Lie subgroups of GR associated to Q-algebraic subgroups of G? Again in the special case considered in this paper, the answer is yes,andtheQ- 5 algebraic subgroups of G are just the centralizers ZG(N). In fact it will be seen that the diagram (0.7) is ZGR (N)-equivariant and we have

∼ B(N)R = ZGR (N)/M (N) ∩ ∂D wheretheLiealgebraofM(N)ism(N)=ker(adN) ∩ Im(ad N). This suggests, at least at the set level, how one might begin to seek to construct Satake-type extensions of the quotients Γ\D for general Mumford-Tate domains. The case of general boundary components B(σ)forconesσ is, however, considerably more complex and will require considerations well beyond those in these pages.6 A natural next question is: (0.9) What is the algebro-geometric significance of the information retained by the reduced limit period mapping (0.2)? To answer this it would seem that one needs to have a Kodaira-Spencer theory for deformations of a normal crossing variety, deformations that both smooth the normal crossing variety and that deform it to other normal crossing varieties (for the former see [F]). With this in hand one should then be able to compute the differential of the mapping to the Kato-Usui spaces and combine this with the differential of Π, which is computed below, to obtain an answer to (0.9). This will not be taken up here. Schematically this work may be pictured as ⎧ ⎫ ⎨⎪limiting mixed Hodge structures⎬⎪ GR-orbits in generalized (0.10) ←→ associated to nilpotent orbits flag varieties ⎩⎪ ⎭⎪ in Mumford-Tate varieties

5Cf. [He] for the case of period domains and [KP1] for the general case of Mumford-Tate domains. 6Among other things, the traditional compactiﬁcations are associated to Q-parabolic subgroups of G. For the induced monodromy weight ﬁltration W•(N)g,theW0(N)g’s give parabolic Lie sub-algebras that are related to, but not in as simple a way as in the classical weight n =1 case, to the centralizers Z(N) ⊂ g of N in g under the adjoint action. In the classical case the W0(N) are maximal parabolic subgroups; this is not true in the general case.

24 M. GREEN AND P. GRIFFITHS

Then this relationship is used to analyze the Hodge theoretic aspects of reduced limit period mappings7 KU (0.11) Π : D →{GR-orbits in Dˇ} inthecasewhenthesetB(N) of equivalence classes of single nilpotent orbits ezN ·F • is attached to D. The mapping (0.11) will be studied primarily by analyzing its differential using the Lie algebra and mixed Hodge structure descriptions of the tangent and normal spaces obtained in (0.10). This analysis will in part be done using the decomposition into irreducible sl2-modules for the natural sl2,R ⊂ gR that arises. In addition to an analysis of what information is retained under the reduced limit period mapping, we shall define and discuss the question (0.12) What are the minimal and maximal degenerations of a polarized Hodge structure? A refinement of this question is obtained by impossing the condition that the Mumford-Tate group of the polarized Hodge structures in the degeneration be contained in a given Q-algebraic group G; i.e., we require that the polarized Hodge structures have at least a given amount of symmetry. The definition of minimal and maximal is that the reduced limit period mapping should take the polarized limiting mixed Hodge structure to a codimension-1 GR-orbit (minimality) or to the unique closed GR-orbit (maximality). We are able to answer this question in the case of period domains, and some of these results will be presented here.8 The full story for a period domain, and some results for general Mumford-Tate domains, will appear elsewhere. The method used here builds on that for analyzing (0.6) above. Among other things, what must be understood is this: • Given an R-split limiting mixed Hodge structure (V,W•(N),F ),thereis • a corresponding R-split limiting mixed Hodge structure (g,W•(N)g,F gC) where g ⊂ Hom(V,V ). Describe, in a computationally useful way, the sl2- decomposition of gR in terms of that of VR. As will be seen below, the key phrase is “computationally useful” as that is what is needed to address the question (0.12). Finally there is the existence question that we shall only state very informally here: • (0.13) Given a mixed Hodge structure (V,W•(N),F ) that is a candidate to be a polarized limiting mixed Hodge structure, when is it one such? To be a candidate means at least that the weight filtration should be of the form

W−n ⊂···⊂W0 ⊂···⊂Wn W• where Grk is a polarized Hodge structure of weight k that for k 0 is isomorphic, W• up to Tate twists, to Gr−k . In some cases, including the one when D is a period W• domain and the Grk are of Hodge-Tate type, we shall formulate more precisely

7In [KP1]and[KP2] this mapping is referred to as the na¨ıve limit map.Weusethe terminology reduced limit period map as we think of it as being obtained from the full limit period mapping to the Kato-Usui spaces by reducing modulo the equivalence relation given by two mixed Hodge structures having the same R-split version. 8What is missing is a description of the polarized limiting mixed Hodge structures whose reduced limit period is in a closed orbit O whose Cauchy-Riemann tangent space T CRO is nonzero.

REDUCED PERIOD MAPPING 25 and give an answer to the above question. Although rather elementary, the result is perhaps of interest since it falls under the general rubric of being able to construct something in Hodge theory.9 We shall conclude this introduction with three comments. The first is that the results in this paper provide only some of the tools needed to define and address further questions such as: Is one is able to construct a Satake-Baily-Borel type M completion Γ\D by combining the GR-structure in ∂D together with suitable extensions of certain of the Q-parabolic subgroups of G associated to filtrations nilp W•(N)arisingfromN ∈ g and such that, for a variation of Hodge structure (0.2) where S is a projective algebraic variety, ωΓ\DM Φ(S) is ample? The second is that, as was noted above, this work was done largely indepen- dently but at a later time than the papers [KP1]and[KP2]. Although our perspective is somewhat different there is overlap in a number of the results presented here and which were first obtained in those works. We would like to thank those authors for valuable discussions about some of the material presented below. In addition, a number of questions on topics related to this work, including some that were mentioned above, are the subject of work in progress among Matt Kerr, Gregory Pearlstein, Colleen Robles and the two present authors. The third is to note both the paper [GGR] in which the results below on extremal degenerations of polarized Hodge structures have been extended, and the work in progress in a manuscript “Construction of limiting mixed Hodge structure” by Robles in which Proposition III.D.7 below has been significantly generalized.

I. GR-orbits I.A. Generalized flag varieties. Definition: A generalized flag variety is a rational, homogeneous projective algebraic variety. We shall use two notations for generalized flag varieties: • the usual one is

(I.A.1) X = GC/Q

where GC is a connected, complex semi-simple Lie group and Q is a parabolic subgroup;10 • sometimes we shall write X = Dˇ when we wish to emphasize that X is the compact dual Dˇ of a generalized flag domain D (defined below). The description (I.A.1) of rational homogeneous varieties is classical (cf. [FHW]). For our purposes, Q will be defined to be parabolic if its Lie algebra q is a parabolic sub-algebra, as defined below, of gC. The identification (I.A.1) depends on the choice of a reference point x0 whose isotropy group is Q. If we think of X as a rational, projective variety with a transitive action GC × X → X with

9Our approach to the question of which GR-orbit contains images of the reduced limit period mapping is complementary to that in [KP2]. There they assume the existence of an N to be able to deﬁne their condition of polarizability. Here we construct the N but do not a priori specify which GR-orbit in ∂D we are concerned with. 10We do not use P because we chose to use p in the Cartan decomposition gR = k ⊕ p so it is not available for the Lie algebra of P .

26 M. GREEN AND P. GRIFFITHS isotropy sub-groups Qx = {g ∈ GC : g · x = x where x ∈ X}, then a convenient alternate description to (I.A.1) is

(I.A.2) X = {qx =Adg · q where g ∈ GC} , the set of GC-conjugacy classes of the parabolic sub-algebra q ⊂ gC. We shall write the Levi decomposition of q as

q = qr ⊕ qn when qn is the unipotent radical of q and qr is the reductive complement. By definition, a flag variety isthecasewhenQ = B is a Borel subgroup of GC. For any Q there exist Borel subgroups B ⊂ Q and a fibration

GC/B → GC/Q whose fibres are flag varieties isomorphic to Qr/Qr ∩ B where Qr is the reductive sub-group of Q with Lie algebra qr. If Gu ⊂ GC is a compact real form of GC,thenGu acts transitively on X so that we have the identification

(I.A.3) X = Gu/Gu ∩ Q where Gu ∩ Q is a subgroup of Gu that contains a compact maximal torus T ,and in fact that may be described as ∩ Gu Q = ZGu (T ), the centralizer in Gu of a sub-torus T ⊂ T . The ﬂag variety case is when Gu ∩ Q = T . As usual, for computational purposes we shall use Lie algebra descriptions. Deﬁnition: A parabolic sub-algebra is a sub-algebra q ⊂ gC that contains a Borel sub-algebra b ⊂ q.

With respect to the compact real form Gu where T ⊂ Gu is a compact maximal torus, one choice of h is the complexification tC of the Lie algebra t of T . We shall use two descriptions of q in terms of h: • one is in terms of the root structure of (gC, h); • the other is in terms of the grading elements, which will be defined below. ∗ ∗ For the first we denote by Φ ⊂ h the set of roots and by Λrt =spanZ(Φ) ⊂ h the root lattice. Then we have α gC = h ⊕ ⊕ g α∈Φ where the α g = {X ∈ gC :[Y,X]=α(Y )X for Y ∈ h} are the 1-dimensional root spaces. We recall that • α ∈ Φ=⇒−α ∈ Φ; • [gα, gβ] ⊆ gα+β, with equality holding if α+β is a root and where [gα, g−α] ⊂ h; • the Weyl group W = NGC (h)/ZGC (h)actsonh, and this action is generated by the reflections in the root planes α⊥ = {Y ∈ h : α(Y )=0}. A choice of positive roots is given by a subset Φ+ ⊂ Φ such that • Φ+ is closed under addition; • Φ=Φ+ ∪ Φ− where Φ− = −Φ+ and Φ+, Φ− are disjoint.

REDUCED PERIOD MAPPING 27

Given Φ+, the choice of positive root systems are in 1-1 correspondence with the + + Weyl group via Φw =: w(Φ ). The Cartan-Killing form ( , ) is a non-degenerate, symmetric bilinear form on α −α α β gC that pairs g with g and satisfies (g , g )=0ifβ = −α. It is non-degenerate on h and serves to identify h with h∗. Given a choice Φ+ of positive roots, there is an associated Weyl chamber C ⊂ spanR(Φ) defined by C = {β :(α, β) 0 for all α ∈ Φ+}. Then • C is a fundamental domain for the action of W ; • the Weyl chambers are in 1-1 correspondence with the choices of positive roots. + + ⊂ + Given Φ , there is uniquely defined a minimal set Φs Φ of simple roots that generate Φ+ over Z+. ∈ + Definition: For a subset Σ Φs we set α β qΣ = h ⊕ ⊕ g ⊕ ⊕ g . α∈ Σ β∈Φ−\ Σ − q r qn Here, Σ ⊂Φ denotes the set of roots that are linear combinations of the roots in − − Σand Σ = Σ ∩Φ .ThenqΣ is a parabolic sub-algebra, and the decomposition

qΣ = qr ⊕ qn 11 is the Levi decomposition of qΣ into its reductive and nilpotent parts. There is a unique connected algebraic subgroup QΣ ⊂ GC whose Lie algebra is qΣ, and modulo the action of GC every parabolic sub-algebra arises in this way. Thus there is a surjective map ⎧ ⎫ ⎨⎪ data (h, Φ+, Σ) ⎬⎪ −→ parabolic subgroups ⎪ = ⎪ ⎩ ⎭ QΣ ⊂ GC subsets of the Dynkin diagram whose range is all parabolic subgroups QΣ ⊂ GC containing the Borel subgroup + BΦ+ determined by the choice of h and Φ . Fixing h, the map is finite-to-one with the fibre being the subgroup WΣ of W generated by reflections in the root planes corresponding to Σ . We shall sometimes also use the notations (cf. [FHW]) Φr = Σ − Φn =Φ−\ Σ where the role of Σ is to be understood.⎧ Then ⎨⎪ α qr = h ⊕ ⊕ g α∈Φr ⎪ β ⎩qn = ⊕ g . β∈Φn Turning to the second way we shall use to describe parabolic sub-algebras, we assume given a Cartan sub-algebra h ⊂ gC and have the

11 ˇ ∼ We have chosen to have qn a direct sum of negative spaces in order to have Tx0 D = gC/q a direct sum of positive root spaces. This is the opposite convention to that in [Ro].

28 M. GREEN AND P. GRIFFITHS

Deﬁnition: The set of grading elements is the lattice Hom(Λrt, Z)inh. + + { ··· } If we have chosen a set Φ of positive roots with the subset Φs = α1, ,αr of simple roots, then there is a dual basis L1,...,Lr for the set of grading elements where Li(αj )=δij. Any grading element is then expressed as L = niLi,ni ∈ Z. i If U is a gC-module it decomposes into weight spaces under the action of h, U = ⊕ U λ. λ∈Λwt For any grading element L we then have ⎧ ⎨ U = ⊕Ul λ ⎩Ul = ⊕ U . L(λ)=l Here l may be in Q, as the root lattice is generally of ﬁnite index in the weight lattice Λwt. Applying this to gC we obtain

gC = g−k ⊕···⊕g0 ⊕···⊕gk where α gk = ⊕ g . L(α)=k We note that

• gl and g−l pair non-degenerately under the Cartan-Killing form; • h ⊂ g0 and g0 is a reductive sub-algebra; • [gl, gm] ⊂ gl+m. Deﬁnition: Given a grading element L,weset

g− = g−k ⊕···⊕g−1 and deﬁne the parabolic sub-algebra

qL = g0 ⊕ g− =: qL,r ⊕ qL,n.

It follows from the above that qL is a parabolic sub-algebra with Levi decomposition as indicated by the equation just above. Given a grading element L, there are in general several choices Φ+ of positive roots such that for the corresponding Borel sub-algebra b = h ⊕ ⊕ g−α we α∈Φ+ have b ⊂ g0 ⊕ g−. For any such choice of Φ+ we deﬁne Σ ⊂ Φ+ by ! L s " ∈ + ΣL = α Φs : L(α)=0 . Then

qΣL = qL. Moreover, every Σ arises as a ΣL.Thusthemap {grading elements} →{parabolic sub-algebras containing a ﬁxed Borel b} is surjective.

REDUCED PERIOD MAPPING 29

We conclude this section with two remarks. The first is the identifications of tangent spaces # # ∼ α ∼ α Tx X = g = g 0 α∈Φ+ L(α)>0 α∈ Σ + to X = GC/Q in terms of the two descriptions of parabolic sub-algebras. In the flag variety case we have ∼ ⊕ α Tx0 X = g . α∈Φ+

If we think of X as the set bx of GC-conjugacy classes of Borel sub-algebras of ⊂ ⊂ ∗ gC, then the choice of a Cartan sub-algebra hx bx determines the set Φx hx + ⊂ of roots, and the above identiﬁcation uniquely singles out a subset Φx Φx of positive roots. In the general parabolic case, the choice of hx ⊂ qx only singles out + Φx up to the action of WΣ. For the second remark we have the

Definition: The enhanced generalized flag variety HQ is the total space in the fibration HQ → X whose fibre Hx over x ∈ X consists of the set of Cartan sub-algebras hx ⊂ qx. As noted above, the points of X = GC/Q may be identified with GC-conjugacy classes qx of the parabolic sub-algebra q. For computational purposes it is frequently convenient to choose a Cartan sub-algebra hx ⊂ qx; this choice is not unique and so it is sometimes useful to work up on H.

I.B. Open GR-orbits. We will consider the orbits in a generalized flag variety X = GC/Q of a non-compact and connected real form GR of GC. We assume that GR contains a compact maximal torus T ; then there is a unique maximal compact subgroup K ⊂ GR with T ⊂ K.Forx ∈ X the orbit GR · x will generally be denoted by Ox. In this section we are interested in the case when Ox is open in X, in which case we will use the notations D, D,... for the open orbits and will refer to them as generalized flag domains. The real form GR is uniquely determined by its Lie algebra gR ⊂ gC,whichis given as the fixed point set of a conjugation

σ : gC → gC 2 where σ = identity and σ(λX)=λσ¯ (X)forλ ∈ C, X ∈ gC.Wealsohavethe Cartan decomposition gR = k ⊕ p and Cartan involution → θ : gR gR − where θ k = identity and θ p = identity. Then τ = θ ◦ σ = σ ◦ θ is also a conjugation whose ﬁxed point set is the Lie algebra gu of the compact real form Gu of GC. The isotropy group in GR of x ∈ X will be denoted by

Sx = GR ∩ Qx.

30 M. GREEN AND P. GRIFFITHS

∈ For the reference point x0 = eQ X we set S = Sx0 and in order to have a Mumford-Tate domain shall assume that S is compact.

Then we may also assume that T ⊂ S and, as will be seen below, the GR-orbit of x0

D = GR/S will be open in X. In fact, a basic observation, which will be proved in the next section, is

(I.B.1) Ox is open in X ⇐⇒ Sx contains a compact maximal torus.

This does not imply that Sx is compact. For the generalized ﬂag domain D = GR/S where S is compact, since T ⊂ S we have S ⊂ K and thus there is a mapping of GR-homogeneous manifolds

(I.B.2) D = GR/S → GR/K.

Deﬁnition: We will say that D is classical if there is a GR-invariant complex structure on GR/K such that the mapping (I.B.2) is either holomorphic or anti- holomorphic. Otherwise, D is non-classical. In this work we are especially interested in the case when D is non-classical. We note that there may be open orbits D, D in X where D is non-classical, D is classical and where D and D have common GR-orbits in their boundaries; i.e., D ∩ D = ∅.Examplesofthisaregivenin[GGK2]. As Cartan sub-algebra h in q we may take the complexiﬁcation tC of the Lie algebra t of the compact maximal torus T .Then the roots are purely imaginary on t; i.e., Φ ⊂ it∗. In particular, we have g−α = gα where the conjugation is relative to σ.Sincet ⊂ k, the Cartan involution θ also acts on Φ and decomposing into ±1 eigenspaces we have

Φ=Φc ∪ Φnc where Φc are the compact roots⎧ and Φncare the non-compact roots. Thus ⎪ α ⎨kC = tC ⊕ ⊕ g α∈Φc ⎪ β ⎩pC = ⊕ g . β∈Φnc + ⊂ + Given a choice Φ of positive roots and a choice Σ Φs , we have, in order for S to be compact, Σ ⊂Φc; i.e., all the roots spanned by Σ are compact. For other open orbits of the action of GR on X, even though the isotopy group Sx contains a compact maximal torus ⊂ + Tx,itmaybethatSx is non-compact and therefore the Σx Φx corresponding to a choice of positive roots for (gC, tx,C) will have both compact and non-compact roots relative to the unique maximal compact subgroup Kx containing Tx.Ifwe have Σx ⊂ Φx,c, then we will say that Σx is of compact type; otherwise it is of non-compact type. We note the interesting point that if the group GR has a totally degenerate limit of discrete series, then a non-trivial Σx cannot be of compact type. This is because the existence of a TDLDS is equivalent to the non-existence of a compact simple root. Thus the connection between Hodge theory and TDLDS’s can

REDUCED PERIOD MAPPING 31 only occur when the Mumford-Tate domain is a ﬂag domain GR/T (cf. [GGK2] and the references cited there). Finally,wenote(cf.[FHW]) that the open GR-orbits in X are in 1-1 correspondence with the double coset space

WK \W/WΣ where the compact Weyl group WK is generated by the reﬂections in the root hyperplanes corresponding to the compact roots.

I.C. General GR-orbits. For a GR-orbit Ox = GR · x

GR · x = Ox = GR/Sx,Sx = GR ∩ Qx following [FHW] we shall describe the tangent space, CR-tangent space and normal space. The basic result needed to do this is

the parabolic sub-algebra qx contains a σ-stable Cartan sub- 12 algebra hx.

It follows that σ acts on the set Φx of roots of (gC, hx), and the description will be r ∩ r r ∩ n n ∩ r n ∩ n in terms of all of the intersections Φx σΦx,Φx σΦx, Φx σΦx and Φx Φx. Here we recall that • r ⊂ + Φx = Σx = roots spanned by the subset Σx Φx of simple roots corresponding to qx; • n { ∈ − ∈ r ∩ −} Φx = β Φx : β Φx Φx . + Although these descriptions seem to use a choice Φx of positive roots, this will not n + be the case since neither Σx nor Φx depend on the particular choice of Φx .Here ⊂ + we recall that for hx qx and any choice of Φx we have ∼ −β TxX = ⊕ g , ∈ n β Φx and α qx,r = hx ⊕ ⊕ g . α∈ Σx To avoid notational clutter we shall drop the subscript x, so that our orbit

O = GR/S, S = GR ∩ Q. For the real Lie algebra we have

sR =(q ∩ σq)R sR sR ⊕ sR = ,r ,n . reductive unipotent

Recalling that q = h ⊕ ⊕ gα ⊕ ⊕ gβ ∈ r ∈ n αΦ β Φ

qr qn

12 Recalling the enhanced ﬂag variety H → X = GC/Q,ifwedenotebyHσ ⊂ H the set of σ-stable Cartan sub-algebras, then this means that the mapping Hσ → X is surjective.

32 M. GREEN AND P. GRIFFITHS we have $ % ∩ ⊕ ⊕ α sR,r = qr σqr R = hR g ∈ r ∩ r $ % $ α Φ% σΦ$ R % ∩ ⊕ ∩ ⊕ ∩ sR,n = qn σqn R qr σqn R qn σqn R. The second equation may be expressed in terms of subsets of the roots as in the ﬁrst one, but we shall not need this. The basic idea in the above is $ % $ % ∩ ⊕ ⊕ sR = q σq R = hR root spaces R where the root spaces that appear in the second term are ⎧ ⎫ ⎪ r ∩ r ⎪ $ % $ % ⎨ union of Φ σΦ ⎬ r n r n Φ ∪ Φ ∩ σ Φ ∪ Φ = corresponding to sR plus . ⎩⎪ ,r ⎭⎪ the three corresponding to sR,n We will describe the tangent and CR-tangent spaces below for a special choice of point in the GR-orbit. Here we note that the complexiﬁcation of the projection ∼ ⊕ −β of the real normal space NOx/X,x in the complex tangent space TxX = g is β∈Φn given by ⊕ g−β. We write this as β∈Φn∩σΦn ⊕ −β NOx/X,x = g . β∈Φn∩σΦn R It follows that

n n n • codimR O =2#{α ∈ Φ ∩ σΦ : α = σα} +#{α ∈ Φ : α = σα} (I.C.1) x n n • Ox is open ⇐⇒ Φ ∩ σΦ = ∅.

In each GR-orbit there is a special point, arising from Matsuki duality, where qx contains a (σ, θ)-stable Cartan sub-algebra hx (cf. [FHW], [GGK2]andthe references cited there). The root system Φ of (gC, h) is acted on by both θ and σ. We have hR = t ⊕ A, where the Cartan subgroup HR ⊂ GR is a semi-direct product of a generally non- maximal compact torus T with Lie algebra t and a vector group A with Lie algebra A. For any root α we write

α = iαIm + αRe ∗ ∗ where αIm ∈ t and αRe ∈ A . Then by the (σ, θ)-stability of h σ(α)=−iαIm + αRe

θ(α)=iαIm − αRe are both roots. Since −α isalsoaroot

±iαIm ± αRe are roots The roots may then be divided into groups as follows:

• real roots,whereαIm =0;

REDUCED PERIOD MAPPING 33

compact imaginary α where g ⊂ kC • imaginary roots,whereαRe =0 PP PPP non-compact imaginary α where g ⊂ pC

• complex roots,whereαIm =0, αRe =0(quartets). For a choice Φ+ of positive roots the intersection Φ+ ∩ σΦ+ =Φ ∪ Φ where Φ = real roots in Φ+ Φ has one pair in Φ+ from each quartet. Thus #(Φ+ ∩ σΦ+) = # real roots + 2(# quartets). For the case of a ﬂag domain when Σ = ∅, this together with the above description of sR gives for the real tangent space ⎛ ⎞ ⎜ ⎟ # ⎜ # $ % ⎟ O ∼ α ⊕ ⎜ α σ(α) ⎟ (I.C.2) Tx x = gR/sR = gR ⎜ g + g R⎟ ⎝ ⎠ αIm=0 α= σα −α∈Φn −α∈Φn ασ∈Φn = CRO Tx x where the second summand is the Cauchy-Riemann tangent space with

JXα = ±Xσ(α) ∈ + ∈ + depending on whether α Φx or σ(α) Φx . We may now verify the assertion that D = GR/S is an open GR-orbit ⇐⇒ S contains a compact maximal torus.

Proof. In the GR-equivariant diagram −1 ⊂ π ⏐(D ) GC⏐/B - π- D = GR/S ⊂ GC/Q

−1 where B ⊂ Q is a Borel subgroup, the inverse image π (D )inGC/B is a union of ﬁnitely many GR-orbits, one of which must be open. Applying (I.C.2) to the case n − of the open GR-orbit in GC/B when Φ =Φ , we see that for a suitable choice of the Cartan sub-algebra in the isotropy subgroup of the open GR-orbit Φ− ∩ σ(Φ−)=∅ =⇒ all the roots are imaginary. This implies the isotropy group must be a compact maximal torus T ,andthen T ⊂ S is also a compact maximal torus.

34 M. GREEN AND P. GRIFFITHS

In general, there will also be non-open GR-orbits in GC/B that map onto D ⊂ GC/Q. These non-open orbits will not have a compact Cartan subgroup. Example: For SU(2, 1) one half of the mappings from the GR-orbits in GC/B to those in a GC/Q are illustrated, using the orbit pictures from [GGK2], as follows:

open → open

codimension-1 → open

codimension-1 → codimension-1

codimension-3 → codimension-1

In the ﬁrst picture the open orbit in GC/Q is SU(2, 1)/U(2), while in the next two pictures it is SU(2, 1)/U(1, 1). We note that the last two GR-orbits in GC/B both map to the closed, codimension-1 GR-orbit in GC/Q. The other half is given by following line instead of following the point. Finally we remark that in [GGK2] there are a number of low dimensional examples that appear throughout the text. Many of the general points listed here are illustrated in these examples. These examples deal mainly with the case of ﬂag domains and varieties. Here we shall illustrate the general parabolic case in two cases, following the notations from [GGK2].

REDUCED PERIOD MAPPING 35

SO(4, 1): There are two open GR-orbits in the ﬂag variety, each necessarily non- classical. One of these is speciﬁed by the root diagram + + rrr α rr+

+ rrr β Here we are following the notations and conventions in [GGK2]; the positive roots are marked with a + and the compact roots are denoted • . For this choice + ⊂ + Φ of positive roots there are two choices of proper subsets Σ Φs , depicted by Σ={α} and Σ = {β} above. For {α} the open orbit in the generalized ﬂag variety is SO(4, 1)/U(1, 1) and for {β} is is SO(4, 1)/U(2). Note that Σ = {α} leads to a non-compact S, and hence GR/S is not a Mumford-Tate domain. Sp(4): There are four open orbits in the ﬂag variety, two of which D and D are pictured below with D non-classical and D classical

r r α +

rr+ rr+ α1

rr+ rr+

rr + rr + β α2 r + r D D

+ For D there are the two choices {α1}, {α2} for Σ ⊂ Φ . In each case the open orbit in the generalized flag variety is Sp(4)/U(1, 1). For D the open orbit corresponding to {α} is again Sp(4)/U(1, 1), while for {β} it is Sp(4)/U(2). A standard way of generating non-open orbits in a flag variety is via Cayley transforms; cf. [Kn], [FHW]and[GGK2] for the approach we shall take here. Operationally the procedure is this: • start with a reference open orbit D = GR/T ; • select a necessarily imaginary non-compact root α; • the Cayley transform then is given by an element g ∈ GC such that Ad g(tC)=h is a (σ, θ)-stable Cartan sub-algebra with hR = t ⊕ A where Ad g(α) ∈ A∗; + + • we then select a set Φ of positive roots for (gC, h) with Ad g(α) ∈ Φ ; this specifies a Borel sub-algebra; i.e. a point x of Dˇ and the GR-orbit is GR · x.

36 M. GREEN AND P. GRIFFITHS

A standard example is SU(2, 1), where we begin with the non-classical complex structure D speciﬁed by the root diagram

rr α +

rr +

rr α +

Taking the Cayley transforms corresponding to α = α and α = α givesusthe two codimension-1 GR-orbits in ∂D. Ingeneralitiswellknown([Kn]) that any two GR-orbits in a ﬂag variety X are connected by a sequence of Cayley transforms. We will illustrate this for a 13 sample of the GR-orbits for SU(2, 1). For this we proceed as follows:

(i) starting from an open orbit D = GR/T we have the Cartan sub-algebra tC, and a Cayley transform is an element gβ of GC corresponding to a (necessarily imaginary) non-compact root β.ThenAdgβ(tC)=hβ is a (σ, θ)-stable Cartan sub-algebra whose root spaces are Ad gβ-transforms of the root spaces of tC.We shall use the same picture for the root diagram, with the understanding that in general the compact roots for tC may not be compact under the Cayley transform. Example: For SU(2, 1) with root diagram

β rr+

rr +

rr+ corresponding to the non-classical complex structure given as in [GGK2]bycon- ﬁgurations

s , the Cayley transform sends tC to a (σ, θ)-stable Cartan sub-algebra h with hR = t ⊕A where dim t =dimA = 1. The non-compact imaginary root β is transformed

13There is a more extensive discussion of Cayley transforms and illustrations of their use in [KP2].

REDUCED PERIOD MAPPING 37 to a real root. Pictorially we have for the new root diagram corresponding to h

rr A +

rr + ,

rr+ γ

and the GR-orbit is the set of conﬁgurations @s @ @ @ @ @ @ This is a codimension-1 orbit. (ii) Next we apply the element of the Weyl group corresponding to γ;thisis given by reﬂection in the dotted line above, and we obtain for the new root diagram

+rr A +

rr +

which corresponds to the conﬁgurations

This is the closed, codimension-3 orbit. To conclude this section we shall give a result that will be used later in this paper and in a further work in progress. (I.C.3) Proposition: For an open GR-orbit D = GR/S ⊂ Dˇ where S contains a compact maximal torus there is a one-to-one correspondence between the following:

38 M. GREEN AND P. GRIFFITHS

(i) codimension 1 GR-orbits in ∂D that contain images under the reduced • limit period mapping of nilpotent orbits (Fϕ,N) for a Mumford-Tate domain structure on D; (ii) simple non-compact positive roots β ∈ Φs\Σ and grading elements L where L(β)=−2 and N = (constant) Xβ,whereXβ is the root vector corresponding to the real root under the Cayley transform associated to β.14 The argument will proceed in several steps. Step one: We will show that for γ ∈ Σ ,α∈ Φ+\ Σ ⇐⇒α + γ ∈ Φ+\ Σ . Here Φ+ is a (non-unique unless Q = B) choice of positive roots giving a complex structure on the open orbit. + { } Proof. If Φs = α1,...αk,γ1,...γl where Σ = γ1,...,γl , then by Proposition 2.49 on page 155 in [Kn] α =Σaiαi +Σbj γj where αi 0, bj 0andsomeai > 0. If γ =Σcj αj ,thenifα + γ is a root we have for all j that bj + cj 0. Step two: We have − α, α ∈ Φ\ Σ and α ∈ Φ+ ⇐⇒ α ∈ Φ α, α ∈ Σ =⇒ α = −α.

+ Proof. The ﬁrst is because if α, α ∈ Φ \ Σ ,thengR∩qx is too large, contradicting the assumption that Ox has codimension one. The second is because Σ consists of compact roots. + Step three: We now deﬁne a new choice Φnew of positive roots giving the same complex structure on D: • ∈ \ ∈ + ⇐⇒ ∈ + for α Φ Σ , α Φnew α Φ ; • ∈ ∈ + ⇐⇒ ∈ − for γ Σ , γ Φnew γ Φ . Then a small computation gives ∈ + ⇒ ∈ + α, β Φnew = α + β Φnew, ∈ + ⇐⇒ − ∈ − α Φnew α Φnew. + \ \ Since Φnew Σ =Φ Σ we do not change the complex structure on D.

Step four: As we saw above, the codimension-1 orbits Ox with x a Matsuki point and with Φ = Φx and Σ = Σx as above correspond to • dim A =1; • one pair of real roots ±β with β ∈ Σ ; • corresponding to β there is one Cayley transform from the open orbit D to Ox. Then we have ∈ + ⇐⇒ ∈ − for all α = β,α Φnew α Φnew.

14This result is similar to and preceded by one given in [KP2].

REDUCED PERIOD MAPPING 39

+ { } ∈ ∈ + Step ﬁve: If Φnew,s = δ1,...,δm for β Σaiδi,,whereai 0, if β Φnew,s we have ∈ − β = β =Σaiδi Φnew, ∈ + which is a contradiction. So β Φnew,s. + { } ∈ \ ∈ Step six: If now Φnew,s = β,α1,...,αk,γ1,...,γl where αi Φ Σ , γj Σ , then we deﬁne • L(β)=−2; • L(γj)=0; L(αi) ≡ 0(mod4) ifαi is compact in D • L(αi) < 0and L(αi) ≡ 2(mod4) ifαi is non-compact in D. With this choice of L we complete the proof of (I.C.3). As a consequence of the proposition we have the (I.C.4) Corollary: Under the circumstances of the proposition, we have

−1,−1 gR =spanR N.

−1,−1 Here, gR is deﬁned in terms of the canonical R-split limiting mixed Hodge structure associated to the nilpotent orbit (cf. [CKS] and just below (II.D.9) below).

II. Hodge structures, mixed Hodge structures and limiting mixed Hodge structures II.A. Polarized Hodge structures and Mumford-Tate domains. We assume given a Q-vector space V and a non-degenerate form Q : V ⊗Q V → Q where Q(u, v)=(−1)nQ(v, u). We will denote the circle by S1 = {e2πiθ}. Definition: A polarized Hodge structure (PHS) (V,Q,ϕ)ofweightn is given by a homomorphism 1 ϕ : S → Aut(VR,Q) where the normalized eigenspace decomposition ⎧ ⎨ p,q VC = ⊕ V p+q=n ⎩ p,q 2πiθ 2πi(p−q)θ V = {v ∈ VC : ϕ(e )=e v} = V q,p gives a polarized Hodge structure in the usual sense. This means that the Hodge-Riemann bilinear relations (HRI) Q(V p,q,Vp ,q )=0forp = n − p (HRII) ip−qQ(V p,q, V p,q) > 0 should be satisfied. For the Hodge structures arising from the primitive cohomology n H (M,Q)prim of a smooth projective variety, the weight n 0andp, q 0, but we shall not require this in general. We shall usually describe the polarized Hodge structure (V,Q,ϕ) by the associated Hodge filtration p ⊕ p,n−p Fϕ = V . pp

40 M. GREEN AND P. GRIFFITHS

Then (HRI) and (HRII) translate into p n−p+1 Q(Fϕ,Fϕ )=0 Q(Cv,v¯) > 0forv =0 where C = ϕ(i) is the Weil operator. The Tate Hodge structure Q(1)of weight −2isgivenby V = Q,Q(1, 1) = 1 −1,−1 VC = V (we omit the usual 2πi factor). Polarized Hodge structures admit the standard operations ⊕, ⊗, Hom(•, •)of linear algebra. We follow the usual notation Q(n)=⊗nQ(1) and V (n)=V ⊗Q(n). APHS(V,Q,ϕ) induces one of weight zero on End(V )where ! " i,−i p,q p+i,q−i End(V ) = X ∈ End(VC):X(V ) ⊂ V for all p, q . For a PHS (V,Q,ϕ)ofevenweightn =2p the Hodge classes are p,p Hg(Vϕ)={v ∈ V : ϕ(z)v = v} = V ∩ V . k l In the tensor algebra ⊗•,•V = ⊕ V ⊗ ⊗ V ∗⊗ we have the algebra of Hodge k,l •,• − ≡ tensors Hg(Vϕ ) given by the Hodge classes in the summands where k l 0(2).

Deﬁnition: The Mumford-Tate group Gϕ is the subgroup of GL(V )thatﬁxes •,• Hg(Vϕ ).

It is known that Gϕ is a reductive Q-algebraic group whose associated real Lie group Gϕ,R contains a compact maximal torus [GGK1]. The isotropy group p,q p,q Sϕ = {A ∈ Gϕ,R : A(V )=V } 1 = ZGϕ,R (ϕ(S )) is compact and the quotient

Dϕ = Gϕ,R/Sϕ is a generalized ﬂag domain of the type considered above. The compact dual

Dˇ ϕ = Gϕ,C /Qϕ where ! " ∈ p p Qϕ = g Gϕ,C : g(Fϕ)=Fϕ is a generalized flag variety in which Dϕ is an open Gϕ,R orbit. We emphasize that realizing Dϕ as a set of flags in VC leads to a realization of Dˇϕ as a subvariety of • n n−1 0 flags F = {F ⊂ F ⊂···⊂F = VC}. We will be interested in the Hodge theoretic interpretations of general Gϕ,R- orbits in Dˇϕ. For this we write points of Dˇϕ as x, with x0 ∈ Dϕ being the reference point corresponding to ϕ.Thenforx = g(x0) ∈ Dϕ where g ∈ Gϕ,R

ϕx =Adg(ϕ) gives a polarized Hodge structure (V,Q,ϕx). We may think of Dϕ as the set of p,q p,q polarized Hodge structure’s of weight n with given Hodge numbers h =dimVϕ and whose generic member has Gϕ as its Mumford-Tate group. For the other open orbits, we have the

REDUCED PERIOD MAPPING 41

Theorem: p ∈ ˇ For Fx Dϕ we set p,q p ∩ q q,p Vx = Fx Fx = Vx .

Then x is an open Gϕ,R orbit if, and only if, ⊕ p,q VC = Vx p+q=n gives a Hodge structure on V . Moreover, this Hodge structure satisﬁes (HRI),and the Hermitian forms in (HRII) are non-singular but may not be positive. p We may think of Fx as giving an indeﬁnitely polarized Hodge structure (IPHS). The isotropy group Sx ⊂ GR will not in general be compact but it will contain a compact maximal torus. Informally we may say that

the open Gϕ,R-orbits in Dˇϕ correspond to Mumford-Tate domains for IPHS’s.

For the proof of the theorem, if the orbit GR · x = GR/Sx is open then by (I.B.1) there is a compact maximal torus T ⊂ Sx. The eigenvalues of the action of T on VC are purely imaginary and occur in conjugate pairs, which then leads to p,q the above Hodge decomposition of VC. Moreover, the subspaces Vx are naturally orthogonal relative to the Hermitian form inQ(v, w¯), while Q is non-degenerate. Thus the forms in (HRII) are non-singular, although they may be indeﬁnite. For the opposite implication, we note that for x ∈ Dˇ having the decomposition ⊕ p ∩ q VC = Fx F x is an open, GR-invariant condition. p+q=n

II.B. Polarized Hodge structures in terms of grading elements. A convenient way to describe PHS’s is in terms of grading elements (cf. [Ro]).15 For this we begin with a Q-algebraic group G ⊂ Aut(V,Q), which for simplicity of exposition we assume to be semi-simple, and whose associated real Lie group GR contains a compact maximal torus T .WearethinkingofG as the Mumford-Tate group of a polarized Hodge structure (V,Q,ϕ). Up to a factor of 2πi, the grading 1 element Lϕ associated to (V,Q,ϕ) is the diﬀerential of the circle ϕ : S → T . The derivative at the identity

dϕe(∂/∂θ) ∈ t acts by bracket on g, and the root vectors Xα,α∈ Φ are eigenvectors. We have

[dϕe(∂/∂θ),Xα]=2πiLϕ(α)Xα for α ∈ Φ.

Deﬁnition: The grading element Lϕ ∈ Hom(Λrt, Z) is deﬁned by the above formula. Alternatively, we may think of 1 L = dϕ (∂/∂θ) ∈ t. ϕ 2πi e If h = tC is the Cartan sub-algebra of gC associated to the compact maximal torus T ,then Lϕ ∈ h and Lϕ = −Lϕ p,q − − Lϕ acts on Vϕ with weight (p q)=(2p n).

15As noted earlier, our positive/negative indexing is the opposite of that in [Ro].

42 M. GREEN AND P. GRIFFITHS

Conversely, given n ∈ Z, L ∈ h with Lϕ = −Lϕ,andwhereLϕ acts on VC with integral weights that are in n +2Z, we obtain a Hodge structure p,n−p − n−p,p VC = ⊕ V , V p,n p = V p of weight n on V .Moreover,sinceG preserves Q, the spaces V p,q and V p ,q are or- p,q thogonal under Q unless p = q (= n−p), and the Hermitian forms ip−qQ(V p,q, V ) are non-singular. Thus we obtain an IPHS, which is polarized when these Hermitian forms are positive. When V = g, B is the Cartan-Killing form, and n = 0, a grading element gives aHodgestructure p,−p gC = ⊕ g = ⊕ g2p p where the second equality is the notation that we used earlier for the action of a grading element on gC.Inparticular, 0,0 qr = g = g0 p,−p qn = ⊕ g = g−. p>0 Each gp,−p is a direct sum of root spaces, and the condition to have a polarized Hodge structure is 2i+1,−2i−1 g ⊂ pC (= ⊕ (non-compact root spaces)) 2i,−2i g ⊂ kC (= h ⊕ (compact root spaces)); i.e., Lϕ(α) ≡ 0(mod4)ifα ∈ Φc

Lϕ(α) ≡ 2(mod4)ifα ∈ Φnc.

If x = gx0 ∈ D where g ∈ GR, then a grading element Lx ∈ hx is given by

Lx =Adg(Lϕ).

If, however x = gx0 ∈ Dˇ but is not in D, e.g., if x ∈ ∂D,theng ∈ GC and so − • Lx =Adg(Lϕ)doesnotgenerallysatisfyLx = Lx. Equivalently the filtration Fx does not satisfy − p ⊕ n p+1 −∼→ Fx F x VC p q for all p. A more subtle description of how the Fx and F x interact is required; this is given in [KP2] and will be discussed below. Given a gC-module VC and a grading element L,wemayextendL to Hom(Λwt, Q)and we can decompose (l) VC = ⊕V , V (l) = ⊕(weight spaces where L acts by l). • This induces a filtration FL on VC. More specifically, we index the l’s that appear above as ln >ln−1 > ···>l1 >l0 and set p ⊕ (li) FLVC = V . ip We shall refer to this as the filtration associated to the grading element L. For the Lϕ’s above we obtain the Hodge filtration.

REDUCED PERIOD MAPPING 43

II.C. Mixed Hodge structures. We recall that a mixed Hodge structure • (MHS) (V,W•,F )isgivenbyaQ-vector space V together with

• a finite increasing filtration ···⊂Wk ⊂ Wk+1 ⊂··· of V , p p+1 • a decreasing filtration ···⊃F ⊃ F ⊃··· of VC such that the induced filtration p F ∩ Wk/Wk−1

W• on Grk = Wk/Wk−1 induces a pure Hodge structure of weight k. Herewereally p should write F ∩ Wk,C + Wk−1,C/Wk−1,C, but we will omit the C to help avoid notation clutter. Mixed Hodge structures admit the usual operations ⊕, ⊗, Hom, dual,... oflinearalgebra. A morphism of weight r • • ψ :(V,W•,F ) → (V ,W•,F ) is given by ψ : V → V satisfying → ψ : Wk Wk+2r ψ : F p → F p+r. Any such ψ is strict in the sense that ∩ p+r ∩ Im ψ (F Wk+r)=Im ψ p . F ∩Wk This implies that the set of MHS’s constitutes an abelian category. In particular, k • • the groups ExtMHS( , ) are deﬁned. p,q A bigrading of a MHS is given by a decomposition VC = ⊕J such that ⎧ ⎪ p,q ⎨Wk = ⊕ J p+qk ⎪ ⎩ p,q q,p J ≡ J mod Wp+q−1.

Any bigrading gives both a grading LW associated to the weight ﬁltration and a grading LF associated to the Hodge ﬁltration, where p,q LW = p + q on J p,q LF = p on J .

p,q According to Deligne there is a canonical bi-grading VC = ⊕I such that

p,q q,p I ≡ I mod Wp+q−2. For any morphism of weight r as above, we have

ψ : Ip,q → I p+r,q+r. The construction of the Ip,q is given by − p,q p q q 1 I =(F ∩ Wp+q) ∩ (F ∩ Wp+q + F ∩ Wp+q−2 + ···). The MHS is r-split, or split over R,if q,p Ip,q = I . R • W•,R Thus over ,(VR,W•,R,F ) is a direct sum of the pure Hodge structures Grk,R .

44 M. GREEN AND P. GRIFFITHS

• There is also canonically associated to any MHS (V,W•,F )anR-split MHS .• (V,W•, F ) with the same weight filtration and where the Hodge filtrations are related by (II.C.1) F.• = e−iδF where δ ∈ L−1,−1 =: ⊕ Ip,q. p,q<0 For the construction of δ, setting L = LW we have L ∈ L + L−1,−1. Then a little argument shows that L = eZ L −1,−1 −1,−1 where Z ∈ L ,andfromZ = −Z we may write Z = −2iδ where δ ∈ LR . Symbolically we may write F.p = F p + δF p + δ2F p + ··· . In case D is a Mumford-Tate domain of adjoint type, meaning that it arises p i,−i from a PHS (g,B,ϕ)ofweightn = 0 given by a filtration F gC = ⊕ g ,thenfor ip ∈ ˇ p any x D there is a filtration Fx on gC.Onemayask:What is the Hodge-theoretic meaning of this filtration? Forthiswehavethe

Proposition: Given a choice of σ-stable Cartan sub-algebra hx ⊂ qx,thereexists an R-split mixed Hodge structure ⎧ p,q p,q q,p ⎪ gC= ⊕g , g = g ⎪ #x x x ⎪ p,q ⎨WkgC= gx p+qk ⎪ # ⎪ p p ,q ⎪F gC = g . ⎩⎪ x x pp q p,q Moreover, the spaces gx are mutually orthogonal, except for p,q −q,−p B(gx , gx ) which is non-singular. Remarks: This result, in a slightly different and stronger form, is given in [KP2], p,q where they show in addition that the dim gx do not depend on the choice of hx. We note that the weight filtration is only defined over R.Wemaythinkofthe Cartan-Killing form B as defining an indefinitely polarized mixed Hodge structure.16 Proof. The argument will give us a chance to introduce some useful points. Let ∈ Z ⊂ ∗ L Hom(Λrt, 2 ) hx ⊂ + ⊂ be a grading element. Here, Λrt Φx are the roots of (gC, hx), and Φx Φx are the positive roots relative to a choice of Borel sub-algebra bx ⊂ qx as discussed above. If we have ∈ + L(α) 0if,andonlyif,α Φx L(α)=0 if,andonlyif,α ∈ Σx

16For the deﬁnition of a polarized mixed Hodge structure, cf. [He]aswellas[KP2].

REDUCED PERIOD MAPPING 45 then we will say that Z is a compatible grading at x.17 The associated filtration is ⎧ ⎨ ⊕ α gx for p>0 p L(α)2p FL = ⎩ ⊕ α hx gx for p 0. We may always define such compatible gradings. Given one such L,bythe σ-stability of hx for α ∈ Φx we have σ(α) ∈ Φx,andthen L(α)=2p, L(σ(α)) = 2q for some p, q ∈ Z. Noting that gσ(α) = gα, we set α Wk = ⊕ g L(α)+L(σ(α))k to obtain a filtration W• that is defined over R and with ∼ ⊕ α Wk/Wk−1 = gx . L(α)+L(σ(α))=2k

We then deﬁne # p,q α gx = gx . L(α=2p L(σ(α))=2q In this way we obtain an R-split mixed Hodge structure which, using the standard properties of the Cartan-Killing form, has an indeﬁnite polarization. It remains to choose L so that

p p FL = Fx .

One choice may be obtained as follows: Let L0 ∈ tC be the grading element associated to the polarized Hodge structure at the reference point x0.Letg ∈ GC satisfy g · x0 = x and Ad g(tC)=hx.ThenAdg(L0) ∈ hx deﬁnes a grading element with p p FL = Fx .

Remark: Although L0 = −L0, it is generally not the case that L = −L,andwe set L + L Y = (= p + q on gp,q) 2 x L − L L = (= p − q on gp,q). ϕ 2 x

Then the pair {Y,Lϕ} deﬁnes a bi-grading on gC which deﬁnes the above mixed Hodge structure.18

17The signs are a little confusing because on the one hand for generalized ﬂag domains the tangent space corresponds to positive roots, while on the other hand for Mumford-Tate domains the tangent space corresponds to gp,q’s where p is negative; this is the reason for the sign in L(α) 0. 18 The notation Lϕ is chosen so that in the case of an open orbit we obtain the grading element associated to ϕ : S1 → Aut(gR,B).

46 M. GREEN AND P. GRIFFITHS

The above result is relatively crude in the following sense: If we have x ∈ ∂D ˇ p ∈ where x is limit in D of a family Fz D,Imz 0, arising from a variation of Hodge structure Φ:Δ∗ →{T k}\D, $ % ∗ { | | } 1 where Δ = 0 < t < 1 and, z = 2πi log t and T is the monodromy, then p p limIm z→∞ Fz = F∞ has a ﬁner structure than that given by the theorem. This will be explained at the beginning of the next section.

II.D. Limiting mixed Hodge structures. Let D = GR/S baMumford- Tate parametrizing polarized Hodge structures of weight n on (V,Q) as described p above. Denote by F0 the Hodge ﬁltration on VC corresponding to the identity coset eS. As noted above, on g ⊂ End(V ) there is an induced PHS; we assume that the form Q induces on g the Cartan-Killing form. We will think of the PHS on V p ∈ corresponding to F0 as given by a grading element L0 h = tC; the action of L0 p on VC has integral eigenvalues and the corresponding ﬁltration of VC is F . Via the adjoint action of L0 on gC ∈ End(VC) there is induced a grading element in Hom(Λrt, Z) giving a decomposition

gC = g−k ⊕···⊕g0 ⊕···⊕gk where g0 = sC

gl = g−l. The subspace −1,1 g1 = g , p p−1 which in Hom(VC,VC) may be thought of as {X ∈ gC : X(F ) ⊂ F }, deﬁnes a homogeneous sub-bundle I ⊂ T Dˇ; explicitly

I = GC ×S g1. For our purposes, a variation of Hodge structure over the punctured disc Δ∗ = {0 < |t| < 1} is given by a locally liftable holomorphic mapping (II.D.1) Φ : Δ∗ →{T m}\D that will be described by its lift Φ:. H → D to the upper half plane H = {z ∈ C :Imz>0}, t = e2πiz. The lift will then satisfy (i) the functional equation Φ(. z +1)=eN · Φ(. z) where N ∈ gnilp is the logarithm of the unipotent monodromy element T ∈ G,and (ii) the inﬁnitesimal period relation (IPR) . (II.D.2) Φ∗ : T H → I. Associated to (II.D.1) are two limit points in Dˇ: (i) Setting t = e2πiz and Θ(t):=e−zN · Φ(. z)

REDUCED PERIOD MAPPING 47

gives a well-deﬁned map Θ : Δ∗ → Dˇ that extends across t =0,andwe set • ∈ ˇ (II.D.3) Flim =Θ(0) D. The IPR (II.D.2) implies that p ⊂ p−1 N(Flim) Flim . (ii) The limit of Φ(. z) ∈ D ⊂ Dˇ exists in Dˇ as Im z →∞and we set • . F∞ = lim Φ(z). Im z→∞ The existence of both of these limits is the result of Schmid’s nilpotent orbit theorem • [Sc], which says that Flim exists and that . zN · • →∞ Φ(z)ande Flim are exponentially close as Im z . Assuming N =0wehave • ∈ • ∈ 19 F∞ ∂D whereas in general Flim D.

• . On the face of it, it would seem that F∞ depends on the particular lift ΦofΦ. k Any two lifts diﬀer by an element T ∈ GR, and one may show that • • (II.D.4) TF∞ = F∞. In fact, (II.D.4) is a consequence of the stronger statement

(II.D.5) the vector field ξN on Dˇ induced by N ∈ gC vanishes nd • to 2 order at F∞ ∈ ∂D. This result is in the proof of lemma (3.12) in [CKS]. An argument also appears in [GGK2].$ The% model is the vector field ∂/∂z on C ⊂ P1,whichforw =1/z is − 1 given by w2 ∂/∂w. We shall use the notation zN · • ∈ ˇ Π(z)=e Flim D for the nilpotent orbit. Then in addition to (II.D.3) we have Π(z +1)=T · Π(z) (II.D.6) Π(z) ∈ D for Im z 0. We note that a change of coordinates 2 t = λ(t + a1t + ···) in the disc Δ induces the change λN (λ+z)N · • Π (z)=e Π(z)=e Flim so that the nilpotent orbit is well defined up to scaling by the original VHS.20 The second of Schmid’s results associates to the VHS (II.D.1), via the data N • ∈ and Flim, a special kind of MHS. Namely, a nilpotent transformation N End(V )

19 • These matters are discussed in detail in §3of[CKS]. In [KP2] F∞ is called the na¨ıve limit of the VHS (II.D.1). As was explained in the introduction and will be recalled below, we shall • • refer to the map (V,W•(N),F ) → F∞ as the reduced limit period mapping. 20 π −1 For a family of algebraic varieties X −→ ΔwithXt = π (t) smooth for t =0and X0 a n reduced normal crossing divisor, there is a nilpotent orbit associated to the H (Xt)prim and to a tangent vector ξ ∈ T0Δ.

48 M. GREEN AND P. GRIFFITHS

n n+1 where N = 0 but N = 0 deﬁnes a unique monodromy weight ﬁltration W•(N), centered at zero, and pictured as

(0) ⊂ W−n(N) ⊂···⊂W0(N) ⊂···⊂Wn(N)=V, and which satisfies the defining relations N : Wk(N) → Wk−2(N) ∼ N k :Grk −→ Gr−k ,k 1. W•(N) W•(N) • Schmid’s theorems (loc. cit.) imply that (V,W•(N),Flim) defines a polarized mixed Hodge structure (PMHS). We will explain the polarization conditions as needed below. His results lead to the following Definition: A limiting mixed Hodge structure (LMHS) is given by the data N ∈ gnilp,F• ∈ Dˇ where • (a) (V,W•(N),F )isaMHS; (b) N(F p) ⊂ F p−1. Thus a LMHS is a MHS of a very special sort. The polarization conditions, which are essential in the proof of Schmid’s result but which will play a secondary role here, may be remembered by the analogy ⊕ W•(N) ←→ ⊕ n−k Q Grk H (X, ) L ←→ Lefshetz operator L where X is a smooth, n-dimensional projective algebraic variety and L ∈ H2(X, Q) is the Chern class of an ample line bundle. The decomposition of Hn−k(X, Q), n−k k+1 k 0, into the primitive parts H (X, Q)prim =kerL , and the polarizations induced on these by the pairings

n−k n−k / H (X, Q)prim ⊗ H (X, Q)prim Q ∈ ∈ α ⊗ β /(Lkα ∧ β)[X], induce, using the analogy ↔, on the mixed Hodge structure the polarization conditions for a LMHS. All of our limiting mixed Hodge structures will be understood to be polarized, and we shall not use the more precise acronym PLMHS. Returning to the general discussion from [CKS], there is an equivalence between the data • nilpotent orbits {LMHS’s (V,W•(N),F )}⇐⇒ . ezN · F, modulo rescaling Originally, this result applied to the case of period domains, but as noted in [KP1] this equivalence applies equally to general Mumford-Tate domains. Deﬁnition: We set ! " B.(N)= F • ∈ Dˇ : N(F p) ⊂ F p−1 and ezN F • ∈ D for Im z 0 B(N)=B.(N)/{rescalings F • → eλN F •}.

REDUCED PERIOD MAPPING 49

Then B(N)=C\B.(N)whereC acts as indicated. There is a well-deﬁned map

(II.D.7) Π∞ : B(N) → ∂D where • zN • (II.D.8) Π∞(F ) = lim e · F . Im z→∞ Deﬁnition: The mapping (II.D.8) is the reduced limit period mapping associated to the VHS (II.D.1). Thinking of B(N) as the set of LMHS’s associated to the monodromy transfor- N mation T = e , the reduced limit period mapping Π∞ will capture some, but not all, of the information in the LMHS. This is the reason for the adjective “reduced.” The analysis and interpretation of this mapping will be done in the next sections. The conceptual framework is that LMHS’s associated D ∪ B(N)=D ∪ to N ∈ gnilp is a building block for the Kato-Usui spaces [KU]. These latter are universal for period mappings, playing a role for general VHS’s similar to that played by the toroidal compactiﬁcations of Mumford et al. for the case of PHS’s of weight n =1 (polarized abelian varieties). The mapping (II.D.7) will capture some, but not all, of the information in the mappings the the Kato-Usui spaces.21 • To a LMHS (V,W•(N),F )thereiscanonically associated an R-split MHS .• (V,W•(N), F ). As noted above F.• =¯e iδF •, andasprovedin[CKS] (II.D.9) [δ, N]=0. From this it follows that .• (V,W•(N), F ) is a limiting mixed Hodge structure. Applying this to the case V = g we have a canonical grading element 0,0 Y ∈ gR where Y = p + q on gp,q. In addition to (II.D.9) another somewhat subtle point is this: The point F.• ∈ Dˇ but is not a point in D.Infact,F.• is a point x ∈ ∂D.If # .0 p,q qx = F gC = g , p0 q we have for the reductive part 0,0 qx,r = g .

Then qx contains a σ-stable Cartan subalgebra hx,andthepointis 0,0 0,0 (II.D.10) hx is contained in g , and in fact we may assume that Y ∈ gR .

21One may reasonably ask: Why study a situation in which information is lost? This question was brieﬂy touched on in the introduction and will be further discussed at the end of section III.D.

50 M. GREEN AND P. GRIFFITHS

In addition to this property which implies that Y splits the ﬁltration W•(N), we also have the commutation relation

[Y,N]=−2N,

+ from which it follows that we may uniquely complete Y,N to an sl2-triple {N,Y,N }, meaning that ⎧ ⎨⎪[Y,N+]=2N + [Y,N]=−2N ⎩⎪ [N +,N]=Y.

We thus have an SL2(R) ⊂ GR and the above leads to an equivariant VHS

Ξ:H → D $ % 01 −10 00 + ∈ R ⊂ H .• ∈ where ( 00) corresponds to N, 01 to Y ,(10)toN ,0 ∂ to F ∂D and the mapping is given by

(II.D.11) Ξ(z)=ezN · F.•.

This SL2-orbit approximates the nilpotent orbit, and is related in a subtle way to the SL2-orbit in Schmid’s original work; cf. [CKS]. Assuming that we have a (σ, θ)-stable Cartan sub-algebra, the relation between the root structure and the gp,q decomposition is • the real root spaces are in the gp,p’s; • the imaginary root spaces are in the gp,−p’s; • the quartets are in gp,q gq,p g−p,−q g−q,−p.

In general these are inclusions and not equalities; e.g., for SU(2, 1) there are complex roots in gr,r and gr,−r.

III. The reduced limit period mapping III.A. Deﬁnition and ﬁrst properties. We are studying the reduced limit period mapping (II.D.7)

Π∞ : B(N) → ∂D. It will be more convenient to study the lift . . Π∞ : B(N) → ∂D.

The reason is that B.(N) ⊂ Dˇ and we will have a good description of the tangent . spaces TF • B(N) ⊂ TF • Dˇ. . • Thinking of B(N)asthesetofLMHS’s(V,W•(N),F ), as discussed above . there is canonically associated to each LMHS the R-split LMHS (V,W•(N), F•).

REDUCED PERIOD MAPPING 51

• .• (III.A.1) Theorem: Π∞(F )=Π∞(F ). Thus, if we denote by B(N)R the R-split LMHS’s, the mapping Π∞ factors in a diagram B(N)H HH HHΠ∞ HH H# δ v;∂D vv vv vv vv Π∞,R B(N)R where the vertical map F • → e−iδF • = F.• is Deligne’s canonical R-splitting map. Here we note two points: (i) the mapping Π∞ is holomorphic; (ii) the mapping δ is not a holomorphic mapping to B(N)R ⊂ Dˇ. We have the sense that Π∞ is maximal subject to the constraint (i) and above factorization. We observe that, using the notation introduced just above, the map Ξ given by (II.D.11) extends to a mapping Ξ · P1 → Dˇ where Ξ(0) = F.• .• Ξ(∞)=Π∞(F ). .• .• .• In fact, Ξ maps ∂H to ∂D with ∞→Π∞(F ), and the roles of F and Π∞(F )are symmetric.$ % The vector ﬁeld generated by exp(tN)is∂/∂t, which as noted above is − 1/w2 ∂/∂w where w =1/z. Example: The following example will illustrate the mechanism in the mappings in the above diagram and the points (i) and (ii). Setting l(t)=(1/2πi)logt,the normalized period matrix Ω(t) of a family of genus 2 curves Ct degenerating to a nodal one over the disc Δ = {0 < |t| < 1} is ⎛ ⎞ 10δ∗ ⎜ ⎟ 1 ⎜ ⎟ ∗ ⎜ 01⎟ δ2 Ω(t)=⎜ ⎟ ∗ ⎝l(t)+a(t) b(t)⎠ γ1 ∗ b(t) c(t) γ2 where the picture is

γ γ1 2

δ1 δ2

52 M. GREEN AND P. GRIFFITHS and where a(t),b(t),c(t) are holomorphic with Im c(t) > 0. The Hodge ﬁltration 1 1 F (t)is spanned by the columns of Ω(t) relative to the basis of H (Ct, Z)givenby • the duals of cycles in the ﬁgure. Setting F = Flim the nilpotent orbit (V,W•(N),F ) has

⎛ ⎞ 0000 ⎜ ⎟ ⎜ ⎟ ⎜0000⎟ N = ⎜ ⎟ ⎝1000⎠ 0000 ⎛ ⎞ ⎛ ⎞ 0 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜0⎟ ⎜∗⎟ ⎜ ⎟ = W0 ⊂ W1 = ⎜ ⎟ ⎝∗⎠ ⎝∗⎠ 0 ∗ ⎛ ⎞ 10 ⎜ ⎟ ⎜ ⎟ 1 ⎜01⎟ F = ⎜ ⎟ a = a(0),b= b(0),c= c(0). ⎝ab⎠ bc

By rescaling we may assume that a =0.Then

⎛ ⎞ 1 ⎜ ⎟ ⎜ −2i(Im b/ Im c) ⎟ 1,1 1 1 ⎜ ⎟ I = F ∩ (F + W0)=spanC ⎜ ⎟ . ⎝ −2i(Im b/ Im c)b ⎠ b − 2i(Im b/ Im c)c

Thus

I1,1 = I1,1 ⇐⇒ Im b =0.

The map F • → F.• is

⎛ ⎞ ⎛ ⎞ 10 10 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜01⎟ ⎜ 01⎟ ⎜ ⎟ −→ ⎜ ⎟ , ⎝0 b⎠ ⎝ 0Reb⎠ bc Re bc

REDUCED PERIOD MAPPING 53 which retracts the extension b in the LMHS onto Re⎛b.⎞ We also have 0 ⎜ ⎟ ⎜ ⎟ 1,0 1 ⎜1⎟ I = F ∩ W1 =spanC ⎜ ⎟ ⎝b⎠ c ⎛ ⎞ 0 ⎜ ⎟ ⎜ ⎟ 0,1 ⎜1⎟ 1,0 I =spanC ⎜ ⎟ = I ⎝¯b⎠ c¯ ⎛ ⎞ 0 ⎜ ⎟ ⎜ ⎟ 0,0) ⎜0⎟ I =spanC ⎜ ⎟ . ⎝1⎠ 0 The I.p,q are given by the same expressions with b replaced by Re b.Thus ⎛ ⎞ ⎛ ⎞ 00 00 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ p,0 ⎜10⎟ ⎜ 10⎟ .p,0 ⊕ I =spanC ⎜ ⎟ =spanC ⎜ ⎟ = ⊕ I . p ⎝b 1⎠ ⎝Re b 1⎠ p c 0 c 0 From this we see that the non-holomorphic part of the map F → F. disappears • .• .• • under the composite map F → F → F∞ = F∞. We see below (cf. III.A.3) that this composite map only captures the GrW•(N)-part of the LMHS, which will be a general phenomenon when the weight n = 1. We will see that the situation is quite diﬀerent in the non-classical case n 2. Returning to the general discussion, regarding notations we shall set • • .• (III.A.2) F∞ =Π∞(F )=Π∞(F ). • .• The LMHS’s (V,W•(N),F )and(V,W•(N), F ) each have their Deligne bi-grading Ip,q and I.p,q. However, to keep the notation simple because of the theorem we shall simply use Ip,q and, in the adjoint case when V = g, we shall use gp,q. • (III.A.3) Theorem: For (V,W•(N),F ) p r,s F∞ = ⊕ I , sn−p • and for (g,W•(N)g,Fg ) # p r,s Fg,∞ = g . s−p r Roughly speaking p n−p+1 F∞ = (VC/Flim ), and since F n−p−1 varies holomorphically with F •, the double-conjugation means p • that F∞ varies anti- anti-holomorphically = holomorphically with F .

54 M. GREEN AND P. GRIFFITHS

Both of these theorems are consequences of the analysis of LMHS’s that results ∗ from the much more precise approximations to a VHS over Δ given by the SL2- orbit theorem. This analysis is explained in detail in [CKS] where the above result is given in the proof of the lemma (3.12) there. The above consequences of the analysis are also stated in the appendix to lecture 10 in [GGK2] and in [KP1]. Here we shall ﬁrst give the idea in the case of weight n = 2. For this we assume that the LMHS may be pictured as a Hodge diamond with the action of N given by the vertical arrows

F 2 F 1 •

•• ? • • • ? ? • • ? •

2 F∞ Recalling that 2 zN 2 zN 2,q F∞ = lim e · F = ⊕ lim e · I Im z→∞ q Im z→∞ 2 22 we see that F∞ is given by the places marked • in the above diagram. Proof of the theorem. • We denote by (W•(N)g and Fg the weight and R • Hodge ﬁltrations on g and gC.Forthe -split LMHS (g,W•(N)g,Fg ) there is a canonical grading element 0,0 Y ∈ gR which satisﬁes Y = p + q on gp,q, [Y,N]=−2N, and as noted above we may then uniquely complete N,Y to be an sl2-triple + {N,Y,N } in gR. We may then decompose VR into a direct sum of sl2(R)rep- resentations and thus reduce to the case of an irreducible one. We shall give the argument for the case V = g. A similar one works in general. The key step is the d Lemma: If V is the irreducible representation of sl2(R) of dimension d +1,represented by V d =span(xd,xd−1y, ···yd) ∂ ∂ ∂ ∂ N = y ,N+ = x ,Y = x − y ∂x ∂y ∂x ∂y F pV d =span{xiyd−i | p i},

W•(N) 22 • If Gr2,prim = (0), there is an additional in the middle spot in the Hodge diamond.

REDUCED PERIOD MAPPING 55 then lim etN (F pV d)=span{xd−iyi | i p}. t→∞ Proof. etN (xiyd−i)=(x + ty)iyd−i so etN F pV d =span{(x + ty)iyd−i | i p} x i =span + y yd−i | i p t x p = + y · span{xd−p,xd−p−1y,...,yd−p}. t Thus lim etN F pV d = yp span{xd−p,xd−p−1y,...,yd−p} t→∞ =span{xd−iyi | i p} concluding the proof of the lemma. If for an N-string, xd ∈ ga,b then F a−r ∩ N-string = span{xd,...,xd−ryr},r 0 so $ % lim etN N-string) ∩ F a−r =span{xryd−r,...,yd} t→∞ $ % = (N-string ∩⊕g−b+j,−a+j. 0jr Aggregating over N-strings # lim etN F p = gr,s. t→N s−p r

The method of decomposing a general limiting mixed Hodge structure into irreducible sl2-submodules in both VR and in gR and examining how the resulting (p, q) decompositions interact with these strings is perhaps the basic tool in the ﬁne analysis of limiting mixed Hodge structures and the reduced limit period mapping. As an example of this we have the (III.A.4) Theorem: Assuming we have chosen the Cartan sub-algebra h so that −1,−1 Y ∈ h,inthe(p, q) plane, in which under the action of N ∈ gR sl2-strings run from the upper right to the lower left, we have • strings cross (0, 0) on h or Σ ; • strings cross p =0,q =0 only at roots in Σ ; • the picture is symmetric under p →−p, q →−q and under p → q, q → p. What this means is that gp,q is a direct sum of root spaces, and the g0,0, g0,q, gp,0 where crossings occur are a direct sum of root spaces as indicated. The picture

56 M. GREEN AND P. GRIFFITHS is

r r r r r r r r r r r r r r r r r r r r r

A special case occurs when Q = B and X is a ﬂag variety, Σ = ∅ and we see that the only non-zero gp,q in the ﬁrst and third quadrants are the gi,i’s. The picture in this case is r r r r r r r r r r r r r r

The proof of the above theorem is very similar to that given in [GGK2]when −1,−1 Q = B. The essential observation is that since N ∈ gR its action on a string moves down by p → p − 1andq → q − 1; thus a string cannot jump over an axis. We shall not write out the details here. • O • · For the orbit F∞ = GR F∞ we have the identiﬁcations

O • = GR/S • ⊂ GC/Q • F∞ # F∞ F∞ • p,q QF∞ = g q0 # • ˇ • ∼ p,q TF∞ D = GC/QF∞ = g q>0 all p R T • OF • =Im(gR → gC/qF • ) F∞ ∞ # ∞ # ∼ p,q q,p p,q q,p = (g ⊕ g ) ∩ gR ⊕ (g ⊕ g ) ∩ gR q>0 qp>0 p0 # # ∼ p,q p,q q,p = ResC/R g ⊕ (g ⊕ g ) ∩ gR q>0 q p>0 # p 0 ∼ p,q q,p ∩ NO • ˇ = i(g + g ) gR F∞ /D qp>0

REDUCED PERIOD MAPPING 57 # CR ∼ p,q • O • TF∞,R F∞ = g . q>0 p0 The picture of the (p, q) plane of the various Lie algebras are

• ←→ qF∞

• • ∩ • ←→ sF∞,C = qF∞ qF ∞

and for the tangent spaces

C T • O • ←→ F∞,C F∞

CR T • O • ←→ F∞,C F∞

Note: In general, given a complex manifold X and a real submanifold Y ⊂ X,at apointp ∈ X with the complex structure given by

J : TpX → TpX then CR ∩ Tp,R Y = Tp,RY JTp,RY, and there is an isomorphism CR −→J Tp,RY/Tp,R Y NY/X,p. The intrinsic Levi form is an Hermitian form CR CR CR L : TR Y ⊗ TR Y → TRY/TR Y. Using the previous map, the intrinsic Levi form is sometimes thought of as having values in the normal bundle NX/Y . O • ⊂ ˇ Inthecaseof F∞ D,wehave

CR 1,0 0,1 T O • C = T O • ⊕ T O • F∞, / F∞ / F∞ - - II IV

58 M. GREEN AND P. GRIFFITHS where the notation means the gp,q’s in the second and fourth quadrants, II and IV, respectively. The instrinsic Levi form is given by

1,0 0,1 CR T O • ⊗ T • → (T O • /T O • ) ⊗ C F∞ F∞ F∞ F∞ and using the above identiﬁcation it is II ⊗ IV → I,

23 which denotes the bracket in gC followed projection to the quadrant I. Since the mapping B(N) −Π−∞→ Dˇ

O • is holomorphic, the image is an integral manifold of the distribution in T F∞ given CRO • by the sub-bundle T F∞ . In fact, this is clear from the above description of Π∞,∗. Interesting work on whether the maximal integral manifolds of this distribution are given in this way, therefore arising from Hodge theory, has been done by C. Robles.

III.B. The diﬀerential of the reduced limit period mapping. Recall • • that B(N) consists of the equivalence classes of LMHS’s (V,W•(N),F )asF varies in Dˇ subject to the restrictions N(F p) ⊆ F p−1 and that ezN · F • ∈ D for Im z 0, which is an open condition. Denoting by [F •] the equivalence class of F • under rescaling F • → eλN · F •, we want to compute the diﬀerential

• → ˇ (III.B.1) Π∞,∗ : T[F ]B(N) TF∞ D. • ∈ ˇ At a point Fx D, we always have ˇ ⊂⊕ p p TxD Hom(Fx ,VC/Fx ). p Moreover, if we have a grading that gives the direct sum decompositions ⎧ ⎨ VC = ⊕Vj p ⊕ ⎩Fx = Vi ip then we will have an inclusion

TxDˇ ⊂⊕ Hom(Vi,Vi+j). i j>0 • At our R-split LMHS we have gradings that deﬁne both W•(N)andF .Thisgives an identiﬁcation ⎧ p,q ⎨TF • Dˇ = ⊕ g (III.B.2) p<0 ⎩ • ˇ ⊕ r,s TF∞ D = g . s>0

23The root-theoretic formula for the intrinsic Levi form in the ﬂag domain case is given in [GGK2]. The argument given there extends to the general GC/Q case.

REDUCED PERIOD MAPPING 59

p,q (III.B.3) Theorem: Denoting by g0 the bottom ends of sl2-strings and using the identifications (III.B.2), the differential (III.B.1) is induced by the inclusion # # p,q → p,q g0 g . p<0 p<0 q>0 . Proof. Here, the LHS is identified with TF • B(N) where the quotient T[F •]B(N) is obtained by factoring out by the action of CN, and as in (III.B.2) the • ˇ RHS is a subspace of TF∞ D. An essential step in the proof is the identification • . Z ∩ • ˇ (III.B.4) TF B(N)= (N) TF D. This is obtained by differentiating the relation p ⊂ p−1 [N,Ft ] Ft as in [KP1] and in [GGK2], proof of the proposition on page 258, where essen- −1,−1 . tial use is made of the fact that N ∈ g . As noted, passing from TF • B(N)to T[F •]B(N) is reflected by passing to the quotient by CN. For the formal argument, if v ∈ V ,thenforX ∈ Z(N)/F 0 ∩ Z(N), lim etN Xv = X lim etN v t→∞ t→∞ tN and thus the action of X commutes with limt→∞ e . • 0 The induced action on F∞ requires that we map X to gC/F∞gC,sofor X ∈ gp,q,p<0 the map we get is 0 if q 0 and the identity if q>0.

Denoting by and by the regions correponding respectively to • • ˇ T[F ]B(N)andTF∞ D, the mapping in the theorem is pictorially represented by the double shaded region in the (p, q)-plane

q 6

- p

st Thus the diﬀerential Π∞,∗ has cokernel in the 1 quadrant and kernel the ends of k sl2-strings in the third quadrant (except for N terms which are modded out). Some conclusions that may be drawn from the theorem are

(1) The identity component of Z(N)R acts transitively on the components of B(N)(cf.[He] for the case of period domains and [KP1] for the general case);

60 M. GREEN AND P. GRIFFITHS

(2) Using the second identiﬁcation in (III.B.2), as previously noted the real • normal space to the orbit GR · F∞ is p,q q,p 24 ⊕ i (g + g )R . qp>0

Z ∩ ⊕ p,q q,p To obtain (1) we observe that TeZ(N)R = (N) gR is equal to (g0 + g0 )R, p,q and any sl2-string appears here for some p<0, except for the N-strings of length one in g0,0. This identiﬁcation was pictured in the previous section. • From (2), if x = F∞ ∈ ∂D and O = GR · x is the orbit, we have x R O R p,q q,p ∩ codimDˇ x = dim (g + g ) gR qp>0 C C =2 dim (gp,q)+ dim gp,p. q>p>0 p>0

The algebro-geometric interpretation of the diﬀerential Π∞,∗ separates naturally into two steps. (A) Kodaira-Spencer maps to Kato-Usui spaces. This means the following: Let π X0 be a normal crossing variety and X −→ Δ a deformation of X0 where X is smooth. Essentially there are two kinds of such deformations: −1 (i) when the Xt = π (t) are normal crossing varieties; (ii) when the Xt for t = 0 are smooth.

Incase(i)wehaveinmindthattheXt are themselves smoothable to a family Xs,t where X0,t = Xt. Thus in both of the above situations there is a monodromy • n weight filtration W•(N) and Hodge filtration F on a vector space V ⊆ H (smooth n variety) such that H (X0) fits into a Clemens-Schmid exact sequence. Denoting by DN = D ∪ B(N) the Kato-Usui space, where the notation means that the set of LMHS’s associated to W•(N) is attached to the Mumford-Tate domain D as in [KU], there will be a Kodaira-Spencer map {∂/∂s,∂/∂t}→TF • DN . This map has been analyzed in Friedman [F] in case (ii), and (i) appears in various places in the literature as the deformation theory for a pair (Y,Z)whereY is a compact, complex manifold and Z ⊂ Y is a reduced normal crossing divisor. In this case we will have

(III.B.5) ∂/∂t → T[F •]B(N),

24Here we recall the convention that for a real vector space WR with a complex structure given by J : WR → WR satisfying J2 = −I, writing as usual WC = W 1,0 ⊕ W 0,1 we may identify WR with W 1,0 as real vector spaces, written 1,0 WR =ResC/R W , and where J and the LHS corresponds to the “i” on the RHS. If WR is the real tangent space Tx,RM to a complex manifold M at a point x ∈ M,thenwealsohave 1,0 Tx,RM = TxM where the RHS is understood to be the complex or holomorphic tangent space in the usual sense. The summation here is over all dots in the third quadrant. The index range p q<0isto prevent double counting.

REDUCED PERIOD MAPPING 61 which will describe how the LMHS’s associated to the degenerations Xt,s → Xt,0 vary to ﬁrst order in t. (B) The mapping (III.B.5) may then be composed with

• → • ˇ Π∞,∗ : T[F ]B(N) TF∞ D to describe how much of the LHS’s in the family Xs,t are captured by the reduced limit period mapping. Acknowledging that both (A) and (B) are necessary for the full story, it is (B) that we are concerned with here, leaving aside (A) for future consideration. A ﬁrst observation relates to the mapping 0 (III.B.6) Gr : B(N) → Di i where the Di are the period domains that appear in the polarized Hodge structures W • . Gr• associated to the PLMHS’s (V,W•(N),F ) ∈ B(N). (III.B.7) Proposition: The diﬀerential of the map (III.B.6) factors 0 Gr∗ / TB(NF) TDi FF < FF xx i FF xx FF xx Π∞,∗ F" xx

• ˇ TF∞ D. Thus, at least variationally the reduced limit period mapping captures the W• • PHS’s given by the Gr (V,W•(N),F )’s. The proof is a consequence of k 0 −p,p T Di ⊂⊕g , p>0 i from which it follows that Gr∗ picks out the intersection Z ∩ ⊕ −p,p ⊕ −p,p (N) g = g0 . p>0 p>0

W•(N) We will see below that for weight n = 1 it is only Gr∗ that is captured by Π∞,∗.Forn 2 some, but not all, of the extension data in the LMHS’s will be detected by Π∞,∗. This will be analyzed in the next section. III.C. Variations of the extension data in LMHS’s under the reduced limit period mapping. We have seen in proposition (III.B.7) that the reduced limit period mapping captures the variation of the associated graded PHS’s in a family of LMHS’s. A more subtle question is the extent to which the extension data in the LMHS is picked up by the reduced limit period mapping. We shall now discuss one answer to this question, where we restrict to the behavior of the 1st order variation. For this we begin with an informal discussion of how one might picture the extension data in a MHS. • For this one may think of a MHS (V,W•,F ) with weight ﬁltration

0 ⊂ W0 ⊂···⊂Wn−1 ⊂ Wn = V in the following way: • W• the top level data are the Gri = Wi/Wi−1 =: Vi which are pure Hodge structures of weight i;

62 M. GREEN AND P. GRIFFITHS

• the next level of data are given by the extension classes

∈ 1 ei ExtMHS(Vi,Vi−1);

• 1 then the next level of data below the ExtMHS’s, which are well deﬁned when the ei = 0, are what one might call the “extensions upon extensions.”

The induced structure on W0(gC) ⊂ End(VC) may be pictured as

Vn Vn−1 Vn−2 ··· V0 ⎛ ⎞ Vn ⎜ ⎟ ⎜ ⎟ Vn−1 ⎜ ⎟ ⎜ ⎟ Vn−2 ⎜ ⎟ ⎜ ⎟ (III.C.1) · ⎜ · · · ⎟ · ⎝ · · · · ⎠

V0 · ↑!"

W−ngC W− gC W gC 1 0 . W−2gC W−1gC

Here, the arrows denote the entire string of boxes running from upper left to lower right. For the tangent spaces (III.C.2)⎧ ∼ W gC ∼ − ⎪ T (associated graded) 0 ⊕ g p,p; ⎪ = W− gC+(F 0gC∩W gC) = ⎪ 1 0 p>0 ⎪ ⎨⎪ T (1st set of extensions)∼ W−1gC ∼ ⊕ g−p,−1+p; = W− gC+(F 0gC∩W− gC) = 2 1 p>0 ⎪ ⎪ ∼ W−2gC ∼ ⊕ −p,−2+p ⎪T (extensions upon extensions)= 0 ∩ = g ⎪ W−3gC+(F gC W−2gC) ⎪ p>0 ⎩⎪ . .

This means that W0(gC)actsonVC preserving the weight ﬁltration W•(VC), and the graded pieces of W0(gC) act on the graded pieces of W•(VC) by the indicated blocks, where the isomorphisms on the RHS assume we are at an R-split MHS. The following will illustrate, in explicit matrix terms, the extension classes in the case of period domains for n = 2 and Hodge type (2, 2, 2) and where the polarized limiting mixed Hodge structure is assumed to be of the form

0 N 0 N 0 0 ∼ H (−2) −−→ H (−1) −−→ H ,H= Q ⊕ Q.

We will describe this situation by using period matrices, by which we mean choosing bases for F 2 and F 1 and expressing these in terms of the natural bases for the H0(−i)’s. The bases for the H0(−i)’s correspond to the rows and the bases for the

REDUCED PERIOD MAPPING 63

F p correspond to the columns. In the above situation we obtain ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ I 00 000 00I ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝AI0⎠ N = ⎝I 00⎠ Q = ⎝0 −I 0⎠ B tAI 0 I 0 I 00 F2 F 1 where the matrices are all 2 × 2andthefirstoneistheperiodmatrix.Thefirst Hodge-Riemann bilinear relation gives B + tB − A tA =0. Thus A is free as is the anti-symmetric part of B, where free means that the entries can be arbitrarily specified. We note that the condition [N,F2] ⊂ F 1 holds. ∼ For the adjoint representation where g = so(4, 2) the N-strings are

Λ2H0(−2) −−N→ Λ2H0(−1) −−N→ Λ2H0 −−N→ Λ2H0(1) −−N→ Λ2H0(2)

S2H0(−1) −−N→ S2H0 −−N→ S2H0(1) Λ2H0

Using the abbreviation Wi = Wi(N)wehave

W−1 ∼ 2 0 ⊕ 2 0 0 = Λ H (1) S H (1), W−2 + F ∩ W−1 A where the notation means that the RHS corresponds to the matrix A decomposed into its anti-symmetric and symmetric parts and

W−2 ∼ 2 0 0 = Λ H (2) . F ∩ W−2 B

Since ⎡ ⎛ ⎞⎤ ⎛ ⎞ I 00 000 ⎢ ⎜ ⎟⎥ ⎜ ⎟ ⎣N,⎝AI0⎠⎦ = ⎝ 000⎠ B tAI A − tA 00 this gives ⎧⎛ ⎞ ⎫ ⎨⎪ I 00 ⎬⎪ ∼ ⎜ ⎟ t t t ZG(N) = ⎝AI0⎠ : B + B = A A, A = A ; ⎩⎪ ⎭⎪ B tAI

A gives the S2H0(1) above and the free part of B is the Λ2H0(2); i.e., the ends of the N-strings in g.

64 M. GREEN AND P. GRIFFITHS

Remark: If we have an integral structure then the action of W−1(GZ)onthe period matrix is ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ I 00 I 00 I 00 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝CI0⎠ ⎝AI0⎠ = ⎝ A + CI0⎠ ; D tCI B tA 0 B + tCA + D tA + tCI i.e., A → A + C B → B + tCA + D. Thefreeanti-symmetricpartofB transforms by B − tB → B − tB + tCA − tAC + D − tD. 1 This behaves like an iterated integral on a punctured P .Namely,ifω1 is a diﬀer- rd ential of the 3 kind with integral residues at the punctures and ω2 is holomorphic P1\{ } on punctures ,for

f(p)= ω1 γ

g(p)= f(p)ω2 γ

∗ γ λ p0

→ under a transformation γ γλ we have

f → f + ω1 λ

g → g + ω2 ω1. γ λ We note that A ∈ Hom(Z2, Z2) ⊗ C/Z (extension) 2 2 B ∈ Homa(Z , Z ) ⊗ C/Z when A = 0 (extension upon an extension) The above is but the tip of a very interesting iceberg relating the extensions and extensions of extensions to iterated integrals; cf. [Ha]. This paper also contains an excellent exposition of the general theory of degenerations of polarized Hodge structures, including Schmid’s work [Sc].

Remark: The groups corresponding to the Lie algebra Z(N)andtheWi(g)have appeared for period domains in [He] and for general Mumford-Tate domains in [KP1], [KP2]. As noted above, there it is proved that Z(N)R acts transitively on B(N)R so that ∼ B(N)R = Z(N)R/M (N)R wheretheLiealgebraofM(N)ism(N)=ker(adN) ∩ im(ad N).

REDUCED PERIOD MAPPING 65

If one wants a group that acts transitively on B(N), then Z(N)R must be enlarged to take into account the complex extension data in the LMHS. For this it is convenient to assume that we have a Q-split LMHS, an assumption that is always possible in B(N)R (cf. [KP2]). Then there is an sl2 ⊂ g deﬁned over Q,say + sl2 = {N,Y,N }.UsingY we may split the weight ﬁltration WiZ(N)=:Z(N)i on Z(N)tohave

Z(N)=Z(N)r ⊕ Z(N)−1 where Z(N)−1 is the unipotent radical and Z(N)r ⊂ Z(N)isareductiveLevi complement to Z(N)−1.Then

Z(N)C = Z(N)r,C × Z(N)−1,C is a semi-direct product. Moreover, the group

Z(N)r,R × Z(N)−1,C acts transitively on B(N). The details of this action in the period domain case are in [He] and in the general case in [KP1]. As regards W0(g), its unipotent radical is W−1(g). In the classical case when D is an Hermitian symmetric domain, W0(g) is a maximal parabolic subalgebra of g (cf. [Sa]and[BB]). The real Lie group W0(N)r,R ⊂ GR corresponding to the reductive Levi complement of W−1(g) acts on an Hermitian symmetric domain D that is equivariantly embedded in ∂D and the reduced limit period mapping takes nilpotent orbits to D. We may think of D as having as its tangent spaces the • associated graded to the limiting mixed Hodge structures (V,W•(N),F ). In the higher weight case no such construction seems to be known, although the paper [CK] takes steps in that direction in the weight n = 2 case and for period domains. The issue is that the reductive group W0(N)r,R does not act naturally on the extension data captured by the limit period mapping. What seems to be needed is a larger group that acts on a “period-like domain” that parametrizes the potential images of reduced limit period mappings. When we have a PLMHS the weight ﬁltration is

⊂···⊂ ⊂··· W −n W0 Wn .

The associated graded terms over the brackets are isomorphic, up to Tate twists, via the operator N k, but the extension data will involve all the terms. A convenient way to picture the structure is using the sl2-strings. For n = 2 the picture is H0(−2) H0(−1) H0 r - rr-

H1(−1) H1 (III.C.3) r - r

H2 r

66 M. GREEN AND P. GRIFFITHS

The horizontal arrows are the action of N.Inthe(p, q) plane the picture is

re r r

r © r © r er

r © r © re

W• Here the circled terms are the primitive part of Gr2 (V ) which correspond to the H2 in the previous picture. The boxed terms • and diagonally boxed terms • are as indicated. For a general n, the picture is

H0(−n) −−N→ H0(−(n − 1)) −−N→···−−N→ H0 H1(−(n − 1)) −−N→···−−N→ H1 . . Hn−1(−1) −−N→ Hn−1 Hn where Hj is a polarized pure Hodge structure of weight j. The general line looks like

Hn−k(−k) −−N→ Hn−k(−(k − 1)) −−N→···−−N→ Hn−k.

• • ˇ nd The issue is this: The image of T[F ]B(N)inTF∞ D lies in the 2 quadrant and is given by the

p,q Z ∩ p,q g0 = (N) g where p<0,q >0.

On the other hand, by (III.C.2) the respective tangents to the associated graded, the 1st set of extensions, the 2nd set of extensions of extensions, . . . also lie in the 2nd quadrant. We need to determine which of these are in Z(N). For the period domain, the corresponding picture for the adjoint representation has the following N-strings: (i) For 0 a

Hom(Hn−a,Hn−b)(−b) −−N→···−−N→ Hom(Hn−a,Hn−b)(a) Hom(Hn−a,Hn−b)(−(b − 1)) −−N→···−−N→ Hom(Hn−a,Hn−b)(a − 1) . . Hom(Hn−a,Hn−b)(−(b − a)) −−N→···−−N→ Hom(Hn−a,Hn−b).

REDUCED PERIOD MAPPING 67

(ii) For 0 a n, there are two cases, depending on the parity of n.Forn odd, we get N-strings

n−a n−a N N n−a n−a Homsym(H ,H )(−a) −−→···−−→ Homsym(H H )(a) n−a n−a N N n−a n−a Homalt(H ,H )(−(a − 1)) −−→···−−→ Homalt(H ,H )(a − 1) . . n−a n−a Homa/s(H ,H ) where a/s is sym if a is even and alt if a is odd. For n even, the only change is that the roles of alt and sym are reversed — we start with Homalt and then alternate with Homsym in successive lines. With this set-up, Z(N) acting on gC has as graded pieces the rightmost terms of each N-string. We thus have for 0 a

This does not contribute to the image of Π∞,∗ as we never have p<0, q>0. We have in (III.C.5) 0 0 0 0 1 1 Homsym(H ,H )(1), Homalt(H ,H ), Homsym(H ,H ). The ﬁrst two are equal to their respective F 0’s, so they do not contribute. The W• latter gives T Gr1 . Case n =2: (III.C.4) has Hom(H2,H1), Hom(H2,H0), Hom(H1,H0), Hom(H1,H0)(1). By analyzing the position in the N-strings Hom(H2,H1) ⊆ g−2,1 ⊕ g−1,0 ⊕ g0,−1 ⊕ g1,−2 −2,1 and g contributes to the image of Π∞,∗; Hom(H2,H0) ⊆ g−2,0 ⊕ g−1,−1 ⊕ g0,−2 which does not give anything; Hom(H1,H0) ⊆ g−1,0 ⊕ g0,−1

68 M. GREEN AND P. GRIFFITHS so here there is also no contribution. Finally, Hom(H1,H0)(1) ⊆ g−2,−1 ⊕ g−1,−2 which again, gives nothing. Next, (III.C.5) has 0 0 −2,−2 Homalt(H ,H )(2) ⊆ g , 0 0 −1,−1 Homsym(H ,H )(1) ⊆ g , 0 0 0,0 Homalt(H ,H ) ⊆ g , 1 1 −2,0 −1,−1 0,−2 Homalt(H ,H )(1) ⊆ g ⊕ g ⊕ g , 1 1 −1,1 0,0 1,−1 Homsym(H ,H ) ⊆ g ⊕ g ⊕ g , 2 2 −2,2 −1,1 0,0 1,−1 2,−2 Homalt(H ,H ) ⊆ g ⊕ g ⊕ g ⊕ g ⊕ g .

From this we conclude that the only contributions to the image of Π∞,∗ are 1 1 2 2 W• Homsym(H ,H )andHomalt(H ,H ), which are in T Gr∗ . −2,1 We then ﬁnally have g as the only contribution to the image of Π∞,∗ beyond W• T Gr∗ .

III.D. Extremal degenerations of polarized Hodge structures. An interesting algebro-geometric question is Given a smooth projective variety X, what are the minimal and maximal degenerations of X to a singular variety X? Of course, as it stands this question is not well defined and we shall not try to do so here. The Hodge-theoretic version of the problem is this: π −1 Given X as above, consider families X −→ Δ where X = π (t0) for some t0 =0 and where X0 may be singular. What are the minimally and maximally degenerate LMHS’s associated to such families? More precisely, we look for the minimally and maximally degenerate LMHS’s n (V,Q,N) associated to such families where V = H (X, Q)prim. And here again one must define minimally and maximally degenerate LMHS’s; we shall do so below. This leads to the purely Hodge-theoretic question (III.D.1) What are the minimal and maximal degenerations of a polarized Hodge structure (V,Q,F•)? This question may be refined to requiring that degeneration take place in a family of polarized Hodge structures whose Mumford-Tate groups are contained in a given G; this is the form we shall discuss here. Let D be a Mumford-Tate domain and recall the 1-1 correspondence ⎧ ⎫ ⎧ ⎫ ⎨⎪equivalence classes⎬⎪ ⎨⎪limiting mixed Hodge structures⎬⎪ • of nilpotent orbits ←→ (V,W•(N),F ) . ⎩⎪ ⎭⎪ ⎩⎪ ⎭⎪ (F •,N)inD where F • ∈ Dˇ and N ∈ gnilp To each element on the left where N = 0 the reduced limit period mapping gives • • Π∞(F ,N)=F∞ ∈ ∂D. We shall think of this as giving a degeneration of the polarized Hodge structure corresponding to a point in D.

REDUCED PERIOD MAPPING 69

• Deﬁnitions: The degeneration is minimal if F∞ belongs to a codimension 1 GR- • orbit in ∂D. The degeneration is maximal if F∞ belongs to the closed orbit in ∂D. Here we recall ([FHW], page 30) that there is a unique closed GR-orbit in ∂D. Restricting to the case when D is a period domain we shall prove the following:

(III.D.2) Minimal degenerations: (i) For odd weight n =2m +1, assuming hm+1,m =0 there is a unique codimension 1 orbit. For this we have N 2 =0and the limiting mixed Hodge structure will be explicitly described. (ii) For n =2there is again a unique codimension 1 orbit. If h1,1 =1then N 2 =0 ,whileN 2 =0 if h1,1 2. In both cases the limiting mixed Hodge structure can be explicitly described. Turning to maximal degenerations, we first have a general result. We recall that the Q-split case is when the subgroup Q ⊂ GC is defined over R,inwhichcase the closed orbit is O = GR/QR where QR = Q∩GR. In fact we will see that the following are equivalent conditions for the closed orbit O: (i) we are in the split case O = GR/QR; (ii) dimR O =dimC Dˇ; (iii) O is totally real, in the sense that T CRO =0. This will be proved below. In general we shall write LMHS → O to mean that for the corresponding nilpotent orbit • • Π∞(F ,N)=F∞ ∈ O; i.e., the reduced limit period mapping sends the limiting mixed Hodge structure • (V,W•(N),F )toapointinO. Finally we shall say that a mixed Hodge structure is of Hodge-Tate type if the associated graded consists only of Q(−k)’s. (III.D.3) Theorem: For general Mumford-Tate domains the above conditions (i), (ii), (iii) are all equivalent. If LMHS → O then the LMHS is of Hodge-Tate type if, and only if, Q is R-split. Finally, in the case of period domains • • (V,W•(N),F ) is of ⇐⇒ (g,W•(N)g,Fg is Hodge-Tate type of Hodge-Tate type The proof of this theorem will be given below. We shall first discuss the conditions on the Hodge numbers to have a maximal degeneration in the period domain case. (III.D.4) If n =2then in the split case a necessary condition to have a maximal degeneration is h2,0 h1,1.

70 M. GREEN AND P. GRIFFITHS

If n =3, which is automatically in the split case, a necessary condition to have a maximal degeneration is h3,0 h2,1. The general result is • (III.D.5) If (V,W•(N),F ) is of Hodge-Tate type, then hn,0 hn−1,1 ··· hn−[n/2],[n/2].25 The proof of this result consists in analyzing the relation between the numbers f p =dimF p and dim Ip,q when all of the N-strings are of Hodge-Tate type. We now turn to minimal degenerations, including the proof of (III.D.2). For this we shall ﬁrst do the n = 1 case, as this illustrates in a familiar case the basic idea behind the arguments. We shall show For the period domain and for polarized Hodge structures of weight n =1, the codimension-1 GR-orbits include those of the form

H0(−1) −−N→ H0 dim H0 =1

H1.

It may be that any codimension-1 GR-orbit is of this form, but we have not been able to show this.

Proof. A general limiting mixed Hodge structure in the n =1caseis

H0(−1) −−N→ H0

H1 where dim H0 = h0 is arbitrary. The associated limiting polarized mixed Hodge structure on g =sp(2g)is

0 0 N 0 0 N 0 0 Homs(H ,H )(−1) −−→ Homs(H ,H ) −→ Homs(H ,H )(1) 0 0 Homa(H ,H ) Hom(H1,H0)(−1) −−N→ Hom(H1,H0)

1 1 Homs(H ,H ). Here the subscripts s and a refer respectively to symmetric and alternating. p,q 0 0 For the g corresponding to Homs(H ,H )(1) we have p = −1, p + q = −2. Thus 0 0 −1,−1 Homs(H ,H )(1) ⊂ g . The GR-orbit O then satisﬁes h0 +1 codim O dim g−1,−1 . 2

25An interesting algebro-geometric question is whether the conditions (III.D.5) are suﬃcient for there to be a degenerating family of algebraic varieties whose polarized Hodge structure has a limiting mixed Hodge structure one of Hodge-Tate type.

REDUCED PERIOD MAPPING 71

Thus, for codimension 1 we must have h0 = 1. Assuming this to be the case, the limiting mixed Hodge structure for g is then dim g1,1 dim g0,0 dim g−1,−1 ⊕Q(−1) −−N→⊕Q −→ ⊕ Q(1) 0

∗ g0,1 ⊕ g1,0 ⊃ H1(−1)∗ −−N→ H1 ⊂ g−1,0 ⊕ g0,−1

1 1 −1,1 0,0 1,−1 Homs(H ,H ) ⊂ g ⊕ g ⊕ g . O p,q Since codim = pq<0 dim g , we see that the polarized limiting mixed Hodge structures with h0 = 1 map a codimension-1 orbit. A similar argument gives for the general odd weight n =2m+1 where hm+1,m = 0 that the limiting mixed Hodge structures that map to a codimension-1 orbit are H0(−m − 1) −−N→ H0(−m)dimH0 =1

H2m+1; in terms of the Hodge numbers hp,q for polarized Hodge structures in D, H2m+1 has Hodge numbers (h2m+1,0,...,hm+2,m−1,hm+1,m − 1). Algebro-geometrically one thinks of the middle intermediate Jacobians of X where dim X =2m +1and where the specialization X0 has an ordinary double point. There is also an existence issue here: Are there in fact polarized limiting mixed Hodge structures of the above type? It is a general result (cf. (I.C.3) above and [KP2]) that for any codimension-1 orbit O ⊂ ∂D where D is a generalized flag domain with a compact isotropy group, there exists a Mumford-Tate domain structure on D having a nilpotent orbit (F •,N) such that • 26 (III.D.6) (V,W•(N),F ) → O. Here, we are given the Mumford-Tate domain D, which in this case is a period domain, and we are asking for the existence of a polarized limiting mixed Hodge structure for this Mumford-Tate domain structure for which (III.D.6) is satisfied. Under the assumption that (III.D.5) is satisfied, this is true and the proof of a stronger result will be given below. We now turn to minimal degenerations in the weight n =2casefortheperiod domain with Hodge numbers (a, b, a). In this case GR =SO(2a, b) and a polarized limiting mixed Hodge structure must look like H0(−2) −−N→ H0(−1) −−N→ H0 H1(−1) −−N→ H1 H2.

26 −1,−1 The point is that codimension-1 implies that dim gR = 1, so that once we know that − − gR 1, 1 is deﬁned over Q we know where to ﬁnd the N. Then there turn out to be many Mumford- • • Tate domain structures on D for which there is an F such that (g,W•(N),F ) gives a polarized limiting mixed Hodge structure.

72 M. GREEN AND P. GRIFFITHS

For so(2a, b)oneN-chain is ∧2H0(−2) /∧2H0(−1) /∧2H0 /∧2H0(1) /∧2H0(2)

∩∩∩∩∩

g2,2 g1,1 g0,0 g−1,−1 g−2,−2. If h0 2weobtaindimg−1,−1 1, dim g−2,−2 1 forcing the codimension of the orbit O to be 2. Thus codim O =1 =⇒ h0 1. Another N-strand is 1 1 N 1 1 N 1 1 Homa(H ,H )(−1) −−→ Homa(H ,H ) −−→ Homa(H ,H )(1). Then 1 1 1,0 1,0 Homa(H ,H )(1) ⊇ Hom(H ,H )(1) and since this terms has p + q = −2, Hom(H1,0,H1,1)(1) ⊆ g−1,−1 which gives h1,0 1 if the codimension of O is equal to 1; i.e.

codim O =1 =⇒ h1,0 1. We next have a strand Hom(H1,H0)(−2) → Hom(H1,H0)(−1) → Hom(H1,H0) → Hom(H1,H0)(1). Then Hom(H1,0,H0)(1) ⊂ g−2,−1 Hom(H0,1,H0)(1) ⊂ g−1,−2 each term of which contributes h1,0h0 to the codimension. Thus codim O =1 =⇒ h0 =0 or h1,0 =0. Finally we have a strand Hom(H2,H0)(−2) −−N→ Hom(H2,H0)(−1) −−N→ Hom(H2,H0), and from Hom(H1,1,H0) ⊂ g−1,−1 we infer O ⇒ 0 1,1 codim =1 = h =0 or hprim =0. No other strand has p + q −2, which is necessary for p<0, q<0, and gives new conditions beyond those listed above. We may then conclude that the possible polarized limiting mixed Hodge structures are Q(−2) → Q(−1) → Q b =1

REDUCED PERIOD MAPPING 73 where H2 has Hodge numbers (a − 1, 0,a− 1), and H1(−1) → H2 b>1

H2 where H1 has Hodge numbers (1, 1) and H2 has Hodge numbers (a−1,b−2,a−1). Proof of Theorem (III.D.3). We will now give the proof of Theorem (III.D.3), R • beginning with an -split limiting mixed Hodge structure (g,W•(N)g,Fg ) corresponding to a nilpotent orbit (F •,N) where the limit period mapping • Π∞(F ,N) ∈ O = GR/QR.

In the case when for the (σ, θ)-stable Cartan subaglebra hx at a suitably chosen point x ∈ O we have hx = Ax; 0,0 p,q i.e., every root is real; the argument is particularly simple: Since hx ⊂ g ,theg α p,q are direct sums of root spaces. For Xα ∈ g ⊂ g we have q,p Xα = Xα ∈ g =⇒ p = q, which gives gp,q =(0)forp = q. The proof in the general case will be given below. We next will show • If (g,W•(N)g,Fg ) is of Hodge-Tate type, then in the period domain case • (V,W•(N),F ) is also of Hodge-Tate type. Proof. The polarized mixed Hodge structure looks like H0(−n) → H0(−(n − 1)) →···→H0(−1) → H0 H1(−(n − 1)) →···→H1(−1) → H1 . . Hn where Hk is a polarized Hodge structure of weight k.27 For any k the strings Hn−k(−k) −−N→···−→Hn−k for n odd produce strings in g (thisiswhereweusethatweareintheperioddomain case) n−k n−k n−k n−k Homs(H ,H )(−k) →···→Homs(H ,H )(k), n−k n−k n−k n−k Homa(H ,H )(−(k − 1)) →···→Homa(H ,H )(k − 1), length 2k −1. For n even there are the same strings with Homs and Homa reversed. We observe that Hn−k not of Hodge-Tate type n−k n−k =⇒ Homs(H ,H ) not of Hodge-Tate type.

27This includes the case where Hk = H k−l(k)withH k−l a polarized Hodge structure of weight k − l with hk−l,0 =0 .

74 M. GREEN AND P. GRIFFITHS

n−k n−k For k 1, since 2k − 1 1wehaveHoms(H ,H ) appearing in g.Thus,g of Hodge-Tate type implies that Hn−k is of Hodge-Tate type for all k 1. If for some k 1wehaveHn−k =0,thenHom( Hn,Hn−k)appearsing.SinceHn−k is of Hodge-Tate type, we see that Hom(Hn,Hn−k) of Hodge-Tate type =⇒ Hn is of Hodge-Tate type. To complete the proof of Theorem (III.D.3) we have the further • Theorem: For an R-split LMHS (V,W•(N),F ) mapping to x ∈ O, the following are equivalent: • (i) (V,W•(N),F ) is of Hodge-Tate type; (ii) the induced LMHS on g is of Hodge Tate type; (iii) gp,q =0unless p = q; (iv) T CRO =0; (v) qx ∩ gR is a real form of qx, i.e., qx = qx. Proof. (i) ⇐⇒ (ii): This is by deﬁnition. ⇐⇒ W•(N) ∼ ⊕ p,q (ii) (iii): Grk g = g , and these are all of Hodge-Tate type if p+q=k and only if gp,q = 0 unless p = q. (iii) =⇒ (iv): This follows from the formula # CR p,q TO = g . q>0 p0 (iv) =⇒ (iii): T CRO =0fromgp,q =0ifq>0,p 0 and thus also if p>0, q 0. Now any N-string that starts in ga,b goes through (a−1,b−1),...,(−b, −a). If a<0, this goes through (0,b − a), (−1,b − a − 1),... .Butb>aimplies b − a − 1 0, so p<0, q 0, which is forbidden. Thus a = b, as desired. p,q (iv) ⇐⇒ (v): We have qx = ⊕ g .So q0 ⇐⇒ p,q qx = qx g =0 for p>0,q 0. But the latter condition is equivalent to (iv). Finally we shall deal with an existence question. The issue is to construct a nilpotent orbit or, equivalently, a limiting mixed Hodge structure. For this we recall that we are representing a general limiting mixed Hodge structure as a direct sum of basic limiting mixed Hodge structures of the form Hn−k(−k) −−N→···−−N→ Hn−k where Hn−k is a polarized Hodge structure of weight n − k. (III.D.7) Proposition: A Hodge structure that is a direct sum of polarized Hodge structures of the form (III.D.8) Hn−k(−k) →···→Hn−k is a limiting mixed Hodge structure. Proof. We have to construct • the vector space V , • the polarizing form Q, • the Hodge ﬁltration F •, • the endomorphism N ∈ HomQ(V,V ).

REDUCED PERIOD MAPPING 75

The vector space V will be the direct sum of the vector spaces appearing in (III.D.8). The polarizing form on (III.D.8) will be the anti-diagonal block matrix ⎛ ⎞ 0 Q n−k ⎜ H ⎟ ⎜ −Q n−k ⎟ Q = ⎝ · H ⎠ · · 0

n−k where QHn−k is the polarizing form on H . The Hodge ﬁltration will be the direct sum of the Hodge ﬁltrations in (III.D.8). Finally the endomorphism N will be ⎛ ⎞ 0 ⎜ ⎟ ⎜ ⎟ ⎜I 0 ⎟ ⎜ ⎟ N = ⎜ I 0 ⎟ . ⎜ . ⎟ ⎝ .. ⎠ I 0 Essentially by construction this gives a limiting mixed Hodge structure with the required properties.

As an application, given a set of Hodge numbers hp,q,p+ q = n, satisfying hn,0 hn−1,1 ··· we may construct a direct sum of Hodge-Tate structures of the form (III.D.8) where Hn−k = H0(−(n − k)), the F p with dim F p = f p are as constructed in the proposition, and where f n = hn,0 f n−1 = hn,0 + hn−1,1 . . This provides a converse to the above result on the necessary conditions on the Hodge numbers for a polarized Hodge structure of weight n to have a Hodge-Tate degeneration. It suggests a complementary approach to the question addressed in [KP2].

For D an open GR-orbit in GC/Q,forwhichGR-orbits O in ∂D is there a Mumford-Tate domain structure on D and a limiting mixed Hodge structure • (V,W•(N),F ) with • (V,W•(N),F ) → O? The complementary approach would be to use the root-theoretic knowledge of the GR-orbits in ∂D and from these single out those that contain homogeneous submanifolds D = GR/S where GR ⊂ GR is a semi-simple subgroup where S contains a compact maximal torus in GR.TheseD ⊂ ∂D are then Mumford-Tate domains and one may try to use arguments such as that just given to realize points of D as the right endpoint of N-strings associated to the reduced limit period mappings for a Mumford-Tate domain structure on D.TheSU(2, 1) example discussed in [GGK2] show that this is not always possible.

76 M. GREEN AND P. GRIFFITHS

References [AMRT] A. Ash, D. Mumford, M. Rapoport, and Y. Tai, Smooth compactification of locally symmetric varieties, Math. Sci. Press, Brookline, Mass., 1975. Lie Groups: History, Frontiers and Applications, Vol. IV. MR0457437 (56 #15642) [BB] W. L. Baily Jr. and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 84 (1966), 442–528. MR0216035 (35 #6870) [BJ] A. Borel and L. Ji, Compactifications of locally symmetric spaces, J. Differential Geom. 73 (2006), no. 2, 263–317. MR2226955 (2007d:22031) [Ca] E. H. Cattani, Mixed Hodge structures, compactifications and monodromy weight filtration, Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982), Ann. of Math. Stud., vol. 106, Princeton Univ. Press, Princeton, NJ, 1984, pp. 75–100. MR756847 [CCK] J.A.Carlson,E.H.Cattani,andA.G.Kaplan,Mixed Hodge structures and compactifications of Siegel’s space (preliminary report),Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980, pp. 77–105. MR605337 (82i:32053) [CK] E. H. Cattani and A. G. Kaplan, Extension of period mappings for Hodge structures of weight two, Duke Math. J. 44 (1977), no. 1, 1–43. MR0432925 (55 #5904) [CKS] E.Cattani,A.Kaplan,andW.Schmid,Degeneration of Hodge structures, Ann. of Math. (2) 123 (1986), no. 3, 457–535, DOI 10.2307/1971333. MR840721 (88a:32029) [FHW] G. Fels, A. Huckleberry, and J. A. Wolf, Cycle spaces of flag domains, Progress in Mathematics, vol. 245, Birkhäuser Boston, Inc., Boston, MA, 2006. A complex geometric viewpoint. MR2188135 (2006h:32018) [F] R. Friedman, The period map at the boundary of moduli, Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982), Ann. of Math. Stud., vol. 106, Princeton Univ. Press, Princeton, NJ, 1984, pp. 183–208. MR756852 [GGK1] M. Green, P. Griffiths, and M. Kerr, Mumford-Tate groups and domains, Annals of Mathematics Studies, vol. 183, Princeton University Press, Princeton, NJ, 2012. Their geometry and arithmetic. MR2918237 [GGK2] M. Green, P. Griffiths, and M. Kerr, Hodge theory, complex geometry, and representation theory, CBMS Regional Conference Series in Mathematics, vol. 118, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2013. MR3115136 [GGR] M. Green, P. Griffiths, and C. Robles, Extremal degenerations of polarized Hodge structures, to appear. [Ha] R. Hain, Periods of limit mixed Hodge structures, Current developments in mathematics, 2002, Int. Press, Somerville, MA, 2003, pp. 113–133. MR2059020 (2005e:14015) [He] C. Hertling, Classifying spaces for polarized mixed Hodge structures and for Brieskorn lattices, Compositio Math. 116 (1999), no. 1, 1–37, DOI 10.1023/A:1000638508890. MR1669448 (2000d:32048) [KU] K. Kato and S. Usui, Classifying spaces of degenerating polarized Hodge structures, Annals of Mathematics Studies, vol. 169, Princeton University Press, Princeton, NJ, 2009. MR2465224 (2009m:14012) [KP1] M. Kerr and G. Pearlstein, Boundary components of Mumford-Tate domains,preprint, arXiv:1210.5301. [KP2] M. Kerr and G. Pearlstein, Naive boundary strata and nilpotent orbits,preprint, http://arxiv.org/abs/1307.7945. [Kn] A. W. Knapp, Lie groups beyond an introduction, 2nd ed., Progress in Mathematics, vol. 140, Birkhäuser Boston, Inc., Boston, MA, 2002. MR1920389 (2003c:22001) [Ro] C. Robles, Schubert varieties as variations of Hodge structure, Selecta Math. (N.S.) 20 (2014), no. 3, 719–768, DOI 10.1007/s00029-014-0148-8. MR3217458 [Sa] I. Satake, On compactifications of the quotient spaces for arithmetically defined discon- tinuous groups, Ann. of Math. (2) 72 (1960), 555–580. MR0170356 (30 #594) [Sc] W. Schmid, Variation of Hodge structure: the singularities of the period mapping,Invent. Math. 22 (1973), 211–319. MR0382272 (52 #3157)

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Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095 E-mail address: [email protected] Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540 E-mail address: [email protected]

Contemporary Mathematics Volume 647, 2015 http://dx.doi.org/10.1090/conm/647/12981

The primitive cohomology of theta divisors

Elham Izadi and Jie Wang Dedicated to Herb Clemens

Abstract. The primitive cohomology of the theta divisor of a principally polarized abelian variety of dimension g is a Hodge structure of level g − 3. The Hodge conjecture predicts that it is contained in the image, under the Abel-Jacobi map, of the cohomology of a family of curves in the theta divisor. We survey some of the results known about this primitive cohomology, prove a few general facts and mention some interesting open problems.

Contents Introduction 1. General considerations 2. Useful facts about Prym varieties 3. The n-gonal construction 4. The case g =4 5. The case g =5 6. Higher dimensional cases 7. Open problems References

Introduction Let A be an Abelian variety of dimension g and let Θ ⊂ A be a theta divisor. In other words, Θ is an ample divisor such that h0(A, Θ) = 1. We call the pair (A, Θ) a principally polarized abelian variety or ppav, with Θ uniquely determined up to translation. In this paper we assume Θ is smooth. The primitive cohomology K of Θ can be deﬁned as the kernel of Gysin push- forward Hg−1(Θ, Z) → Hg+1(A, Z) (see Section 1 below). We shall see that K inherits an integral Hodge structure of level g − 3 from the cohomology of Θ.

2010 Mathematics Subject Classification. Primary 14C30; Secondary 14D06, 14K12, 14H40. Key words and phrases. Theta divisor, Abelian variety, primitive cohomology, primal cohomology, Hodge conjecture, curves, limit mixed Hodge structure. The first author was partially supported by the National Science Foundation. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF.

c 2015 American Mathematical Society 79

80 ELHAM IZADI AND JIE WANG

Recall that the level of an integral or rational Hodge structure H is deﬁned to be l(H):=max{|p − q| : Hp,q =0 } p,q where H is the (p, q) component of the Hodge decomposition of HC. Alternatively, we deﬁne the coniveau of H to be γ(H):=min{q : Hp,q =0 }. Thus l(H)+2γ(H)=weightofH. We will always designate a Hodge structure by its lattice or rational vector p,q space H, the splitting HC := H ⊗ C = ⊕p+q=mH being implicit. The general Hodge conjecture says that KQ is contained in the image of the cohomology of a proper algebraic subset of Θ. More precisely, let X be a smooth projective algebraic variety and m and p two positive integers with 2p ≤ m. Grothendieck’s version of the general Hodge conjecture [Gro69] can be stated as Conjecture 1. GHC(X,m,p): For every rational sub-Hodge structure V of Hm(X, Q) with level ≤ m − 2p, there exists a closed algebraic subset Z of X of pure codimension p such that V ⊂ Ker{Hm(X, Q) → Hm(X \ Z, Q)}.

In the case of KQ,wehaveX =Θ,m = g − 1andp = 1. We are therefore looking for a divisor in Θ. The general Hodge conjecture for KQ can be answered positively for g ≤ 5. Here we survey these results and the tools used to obtain them. We also say a few words about higher dimensional cases and mention other interesting problems related to the primitive cohomology group K. There are relatively few examples of lower level sub-Hodge structures of the cohomology of algebraic varieties that are not already contained in the images of the cohomology groups of subvarieties for trivial reasons. Some of the most interesting such examples are provided by abelian varieties, such as abelian varieties of Weil type (see, e.g., [Iza10]). For a smooth hypersuface Y of degree d in Pg, the primitive cohomology g−1 H (Y,Q)0 is a sub-Hodge structure of coniveau at least 1 if and only if Y is g−1,0 ∼ 0 Fano, since H (Y ) = H (Y,KY ). Thus, if Y is of general type or Calabi-Yau, 0 i.e. d ≥ g +1,H (Y,Q)0 is of coniveau 0. On the other hand, the general Hodge conjecture is true for Fano hypersurfaces with coniveau 1 primitive cohomology group (see, e.g.,[Voi11]).

1. General considerations There is a strong relation between the cohomology of A and Θ. For instance, one has the Lefschetz hyperplane theorem: Theorem 1.1. Let j :Θ→ A be the inclusion. Then k k+2 j∗ : Hk(Θ, Z) −→ Hk(A, Z) j! : H (Θ, Z) −→ H (A, Z) ∗ k k ! j : H (A, Z) −→ H (Θ, Z) j : Hk+2(A, Z) −→ Hk(Θ, Z) are isomorphisms for kg− 1. ∗ ! Also j∗ and j! are surjective for k = g − 1, j and j are injective for k = g − 1. The ! maps j and j! are deﬁned to be ! ∗ −1 −1 j := PΘ.j .PA ,j! := PA .j∗.PΘ,

THE PRIMITIVE COHOMOLOGY OF THETA DIVISORS 81

k k where PΘ : H (Θ, Z) −→ H2g−2−k(Θ, Z), PA : H (A, Z) −→ H2g−k(A, Z)arethe Poincar´e duality maps. We also have ∗ ∪θ = j!.j , where ∪θ is the cup product with the fundamental class of Θ. Proof. See, e.g., [AF59]. It is well-known (see, e.g., [IS95] Proposition 1.1) that the integral cohomology and homology groups of Θ and A are torsion-free. The cohomology of Θ is therefore determined by that of A except in degree g − 1. Following [IS95]and[ITW], deﬁne g−1 g+1 K := ker{j! : H (Θ, Z)−→→ H (A, Z)} so that its dual lattice is (see [IS95] Proposition 1.3) K∗ =coker{j∗ : Hg−1(A, Z) → Hg−1(Θ, Z)}. Lemma . K 1.2 The rank of is 1 2g rank(K)=g! − . g +1 g Proof. It follows easily from Theorem 1.1 that the rank of K is g+1 g g−1 rank(K)=h (A) − h (A)+(−1) (χtop(Θ) − χtop(A)). Using the exact sequence

0 −→ TΘ −→ TA|Θ −→ O Θ(Θ) −→ 0, we see that the total Chern class c(TΘ) of the tangent bundle of Θ satisfies the identity c(TΘ)(1 + θ|Θ)=1. Therefore − g−1 $ % χtop(Θ) = deg$ %cg−1(TΘ)=( 1) g!. As hg+1(A)= 2g and hg(A)= 2g , we find g+1 g 1 2g rank(K)=g! − g +1 g as claimed. The integral lattices K and K∗ inherit Hodge structures from Hg−1(Θ, Z). One can use Griffiths’ residue calculus [Gri69] to compute all the Hodge summands of K as follows. Put U := A \ Θ and let i : U → A be the natural inclusion. Also, for an integer k ∈{0,...,g},let g−k k+1 g−k g+k+2 H (A, Q)prim := ker{∪θ : H (A, Q) −→ H (A, Q)} be the (g − k)-th primitive cohomology group of A. The Gysin exact sequence (see [Voi03, p. 159]) j ∗ j −→ Hg−2(Θ, Q) −→! Hg(A, Q) −→i Hg(U, Q) −→Res Hg−1(Θ, Q) −→! Hg+1(A, Q) −→ induces a short exact sequence of mixed Hodge structures

g g Res 0 −→ H (A, Q)prim −→ H (U, Q) −→ KQ −→ 0.

82 ELHAM IZADI AND JIE WANG

Thus for 0 ≤ p ≤ g, the induced sequence on the Hodge ﬁltration F g−pHg(U) (1.1) 0 −→ Hg−p,p(A) −→ −→ Kg−p−1,p −→ 0 prim F g−p+1Hg(U) is exact. Griﬃths’ residue calculus implies that there is an exact sequence 0 g−1 −→d 0 −→αp g−p g −→ (1.2) H (ΩA (pΘ)) H (ωA((p + 1)Θ)) F H (U) 0 where the leftmost map is the exterior derivative and the middle map sends a rational g-form on A with a pole of order ≤ p + 1 on Θ to its De-Rham class in U (c.f. [Voi03, pp. 160-162]). Denote 0 g−p g 0 O ∼ H (ωA((p + 1)Θ)) −→ F H (U) αp : H (Θ, Θ((p + 1)Θ)) = 0 g−p+1 g H (ωA(pΘ)) F H (U) the induced map.

g−1,0 g−2,1 Lemma 1.3. TheHodgestructureonK satisﬁes K =0anddimC K = g − − g(g+1) K K∗ − ≥ ≥ 2 1 2 .Thus and have level g 3(g 3). For p 2, we have an exact sequence

0 0 H (Θ, OΘ(Θ)) ⊗ H (Θ, OΘ(pΘ)) −→ g−p g αp F H (U) H0(Θ, O ((p + 1)Θ)) −→ −→ 0, Θ F g−p+1Hg(U)

F g−pHg (U) i.e. F g−p+1Hg (U) is isomorphic to the Koszul cohomology group ∼ K0,p+1(Θ, OΘ(Θ)) = K0,p+1(Θ,KΘ) (see, e.g., [Gre84a]). Proof. When p =0, 1, the image of the exterior derivative in (1.2) is zero. 0 ∼ g g ∼ We conclude from (1.2) that H (ωA(Θ)) = F H (U) = C and g−1 g 0 ∼F H (U) H (Θ, O (2Θ))= . Θ F gHg(U) g−1,0 0 g Therefore, by (1.1), K =0.Sinceh ((Θ, OΘ(2Θ)) = 2 − 1 by Riemann-Roch g−p,p g(g+1) and h (A)prim = 2 , we obtain from (1.1) that

g−2,1 g g(g +1) dimC K =2 − 1 − . 2 ∈ 0 O Qp+1 ∧ ∧ ∈ g−p+1 g Let Qp+1 H ( A((p+1)Θ)) be such that θp+1 dz1 ... dzg F H (U) { } 0 1 ∈ where dz1, ..., dzg form a basis of H (A, ΩA). Thus, by (1.2), there exists Qp 0 H (OA(pΘ)) such that the rational form Q − θQ ( p+1 p )dz ∧ ... ∧ dz = dγ θp+1 1 g g si ∧ ∧ ˆ ∧ ∧ ∈ 0 O for some γ = i=1( θp )dz1 ... dzi ... dzg, with si H ( A(pΘ)). We directly compute g ∂si ∂θ θ +(−p)si dγ = (−1)i ∂zi ∂zi dz ∧ ... ∧ dz . θp+1 1 g i=1

THE PRIMITIVE COHOMOLOGY OF THETA DIVISORS 83

Comparingthetwosides,weseethat g ∂s ∂θ Q − θQ = (−1)i i θ +(−p)s . p+1 p ∂z i ∂z i=1 i i Restricting the above equality to Θ, we obtain g ∂θ Q | =(−p) (−1)i s | ∈ H0(Θ, O((p + 1)Θ)). p+1 Θ i ∂z Θ i=1 i ∂θ 0 Since { : i =1, ..., g} form a basis of H (Θ, OΘ(Θ)) (see, e.g, [Gre84b, p. 92]), ∂zi we conclude our proof. For g ≤ 2, K =0.Forg = 3, the lattice K has rank 1 and level 0, i.e., it is generated by a Hodge class of degree 2. By the Lefschetz (1, 1)-theorem, this is a rational linear combination of classes of algebraic cycles. In fact, in this case, one can write an explicit cycle generating K as follows. The abelian variety (A, Θ) = (JC,ΘC ) is the Jacobian of a curve of genus 3. The theta divisor is isomorphic to the second symmetric power C(2) of C and K is generated by the class θ − 2η where η is the cohomology class of the image of C in C(2) via addition of a point p of C: C→ C(2) t → t + p. Forhighervaluesofg, the following equivalent formulation of the Hodge conjecture has been useful (see e.g. [Iza10]). Conjecture 2. There exists a nonsingular projective family of curves in Θ q Z −→ Θ ↓r S whose base is a (possibly reducible) nonsingular projective variety S of dimension g−3 ∗ g − 3 such that the image of H (S, Q) by the Abel-Jacobi map q!r of the family contains KQ. For g =4, 5, a positive answer was given to the above conjecture by using the “largest” sub-Hodge structure H of coniveau 1 of Hg−1(Θ, Z) deﬁned as follows (see [IS95]and[ITW]). First consider the image of Hg−3(Θ, Z)=j∗Hg−3(A, Z) under cup product with the cohomolgy class θ of Θ. This is also a sub-Hodge structure of level g − 3 and satisﬁes the Hodge conjecture since it is contained in the image, for instance, of the cohomology of an intersection of a translate of Θ with Θ. Put g−1 g−1 2 g−1 g+3 P := H (A, Z)prim =ker{∪θ : H (A, Z) −→ H (A, Z)}. 1 Choosing a symplectic basis {α1,...,αg,β1,...,βg} of H (A, Z), it is immediately ∧ ∧ Z g−1 seen that the wedge products γi1 ... γig−1 form a -basis of P where γi = αi or βi and i1 <...

84 ELHAM IZADI AND JIE WANG after identifying Hg+1(A, Z) with the dual of Hg−1(A, Z) using the intersection pairing. The Hodge structure H can then be deﬁned as the kernel of the composition

j! H := ker{Hg−1(Θ, Z) −→→ Hg+1(A, Z)−→→ (P g−1)∗}. It follows from the results of [Haz94]that,for(A, Θ) generic (i.e., outside a count- able union of Zariski closed subsets of the moduli space Ag), any rational sub-Hodge structure of Hg−1(A, Q) of coniveau 1 or more is contained in θ ∪ Hg−3(A, Q). Therefore any rational sub-Hodge structure of Hg−1(Θ, Q) of coniveau 1 or more is contained in HQ = H ⊗ Q. g−3 Note that HQ = KQ ⊕ θ ∪ H (A, Q). Therefore the Hodge conjecture for HQ is equivalent to the Hodge conjecture for KQ. We discuss the cases g = 4 and 5 in Sections 4 and 5. In Sections 2 and 3 below we review two of the main tools used in the proofs for g =4, 5: Prym varieties and n-gonal constructions.

2. Useful facts about Prym varieties

Let Rg+1 be the coarse moduli space of admissible (in the sense of [Bea77]) double covers of stable curves of genus g + 1. The moduli space Rg+1 is a partial compactiﬁcation of the moduli space of ´etale double covers of smooth curves. Beauville [Bea77] showed that the Prym map Pg : Rg+1 −→ A g is proper. The . prym map Pg associates to each admissible double cover (π : X −→ X)ofastable curve X of genus g + 1 its Prym variety P (X,X. ):=Im(1 − σ∗ : J(X.) −→ J(X.)) = Ker0(ν : J(X.) −→ J(X)) where σ is the involution interchanging the two sheets of π, ν : Pic(X.) −→ Pic(X) is the norm map and by Ker0(ν) we mean the component of the identity in the kernel of ν. For general background on the Prym construction we refer to [Bea77] and [Mum74]. The Prym maps P4 and P5 aresurjective[Bea77].

There is a useful parametrization of the Prym variety of a covering. Consider the following subvarieties of Pic2g(X.)

+ 2g . ∼ 0 A := {D ∈ Pic (X):ν(D) = ωX ,h (D)even}

− 2g . ∼ 0 A := {D ∈ Pic (X):ν(D) = ωX ,h (D)odd} Both are principal homogeneous spaces over A. The divisor Θ is a translate of Θ+ = {L ∈ A+ : h0(L) > 0}. For each D ∈ A− we have an embedding (see [Iza95, p. 97])

. + . φD : X −→ A ⊂ J(X); x → D(Lx − σ(Lx)) . where Lx is an eﬀective Cartier divisor of degree 1 on X with support x. The im- . . age XD of such a morphism is called a Prym-embedding of X or a Prym-embedded curve.

THE PRIMITIVE COHOMOLOGY OF THETA DIVISORS 85

. + 0 Note that XD ⊂ Θ if and only if h (D) ≥ 3. The set of Prym-embeddings of X. in Θ+ is therefore parametrized by λ(X.):={D ∈ A− : h0(D) > 1}.

The involution σ : X. → X. induces an involution, also denoted σ: σ : λ(X.) −→ λ(X.); D −→ σ∗D. We put λ(X):=λ(X.)/σ. Note that σ has finitely many fixed points in A−, hence at most finitely many fixed points in λ(X.).

3. The n-gonal construction Suppose given an étale double cover κ : X. → X of a smooth curve X of genus g + 1. Suppose also given a non-constant map X → P1 of degree n. Sending a point of P1 to the sum of the points of X above it, allows us to think of P1 as a subscheme of X(n),whereX(n) is the n-th symmetric power of X.Let C. ⊂ X. n be the curve defined by the fiber product diagram

C. → X. (n) (3.1) ↓↓κ(n) P1 → X(n). In other words, the curve C. parametrizes the 2n points lifting the same point of P1. The involution σ also induces an involution on C., still denoted σ.Thecurve . . . C has two connected components C1 and C2 which are exchanged under σ if n is odd. If n is even, σ leaves each component globally invariant (see e.g. [Bea82]).

4. The case g =4 A R P−1 Since dim( 4) = 10 and dim( 5) = 12, the fiber 4 (A)forA generic in A4 isasmoothsurface.WhenΘissmooth,thefiberisalwaysasurfaceandthe generic elements of any component of the fiber are double covers of smooth curves (see [Iza95] pages 111, 119 and 125). If A is neither decomposable nor the Jacobian of a hyperelliptic curve, then λ(X.)isacurveandthePrymvarietyofthedoublecoverλ(X.) → λ(X)isisomor- phic to (A, Θ) (see [Iza95] p. 119). Sending λ :(X,X. )to(λ(X.),λ(X)) defines an involution λ acting on the fibers of the Prym map P4. To (A, Θ) ∈A4 with smooth Θ, one can associate a smooth cubic threefold T P−1 ([Iza95], [Don92]). The quotient of the fiber 4 (A) by the involution λ can be identified with the Fano surface F of lines on T . Let F be the scheme parametrizing the family of Prym–embedded curves in- side Θ. It follows that the fiber of the natural projection F−→P−1 4 (A)

86 ELHAM IZADI AND JIE WANG

. ∈P−1 . over the point (X,X) 4 (A)isthecurveλ(X). In particular, the dimension of F is three. In general, F might be singular, but for A general F is smooth, see [IS95, Section 3]. Let C→Fbe the tautological family over F with the natural map to Θ+:

q C /Θ+

r F.

Theorem 4.1. ([IS95]) For (A, Θ) general in A4, the image of the Abel-Jacobi ∗ 5 3 + map q!r : H (F, Q) → H (Θ , Q)isequaltoHQ. It is in fact proved in [IS95] that for any (A, Θ) with Θ smooth, the image of F in the intermediate Jacobian of Θ generates the abelian subvariety associated to H.

5. The case g =5

The spaces A5 and R6 both have dimension 15 and P5 is surjective. So P5 is generically finite and its degree was computed in [DS81] to be 27. In [ITW], we use the 5-gonal construction to construct a family of curves in Θ as follows. Let X be a smooth curve of genus 6 with an étale double cover X. of genus 11. For a pencil M of degree 5 on X consider the curve BM defined by the pull-back diagram . (5) BM ⊂ X ↓↓ P1 = |M|⊂X(5).

By [Bea82, p. 360] the curve BM has two isomorphic connected components, say 1 2 | − | ∈ ⊂ . (5) BM and BM .PutM = KX M . Then, for any D BM X and any . (5) D ∈ BM ⊂ X , the push-forward to X of D + D is a canonical divisor on X. Hence the image of 10 . BM × BM −→ Pic X −→ O (D, D ) X (D + D ) + − is contained in A ∪ A . If we have labeled the connected components of BM and 1 × 1 + 2 × 2 BM in such a way that BM BM maps into A ,thenBM BM also maps into + 1 × 2 2 × 1 − A while BM BM and BM BM map into A . By construction, the images of 1 × 1 2 × 2 + BM BM and BM BM lie in Θ . To obtain a family of curves in Θ+, we globalize the above construction. 1 The scheme G5(X) parametrizing linear systems of degree 5 and dimension at least 1 on X has a determinantal structure which is a smooth surface for X 1 suﬃciently general (see, e.g., [ACGH85, Chapter V]). The universal family P5 of 1 P1 1 divisors of the elements of G5 is a bundle over G5 with natural maps 1 −→ (5) P5 X ↓ 1 G5

THE PRIMITIVE COHOMOLOGY OF THETA DIVISORS 87 . whose pull-back via X → X givesusthefamilyofthecurvesBM as M varies: B −→ X. (5) ↓↓ 1 −→ (5) P5 X ↓ 1 G5.

The parameter space of the connected components of the curves BM is an ´etale .1 1 double cover G5 of G5. If we make a base change, / B1 ∪ B2 B

.1 / 1 G5 G5 , →r .1 the family of curves B splitsintotocomponentsB1 and B2,whereB1 G5 is the | | i ∈ .1 tautological family, i.e, the fiber of B1 over a point ( M ,BM ) G5 is exactly the i curve BM . The family of curves F is then defined to be the fiber product / F B1

ι◦r r /.1 B1 G5, .1 | | i | | i where ι is the involution on G5 sending ( M ,BM )to(M ,BM ). For X suﬃciently general, we obtain a family F of smooth curves of genus 25 + + ∼ . over a smooth threefold B1 in the theta divisor Θ of A = P (X,X): q F /Θ+

r B1. The main result of [ITW]is Theorem 5.1. For a general Prym variety P (X,X. ), the image of the Abel- ∗ 4 4 + Jacobi map q!r : H (B1, Q) → H (Θ , Q)isequaltoHQ. 4 2 Note that H (B1, Q) is a level 2 Hodge structure isomorphic to H (B1, Q) under the Lefschetz isomorphism. Combining Theorem 5.1 with the main result of [Haz94], we obtain Corollary 5.2. For (A, Θ) in the complement of countably many proper Zariski closed subsets of A5, the general Hodge conjecture holds for Θ. As far as we are aware, the primitive cohomology of the theta divisor of an abelian fivefold is the first nontrivial case of a proof of the Hodge conjecture for a family of fourfolds of general type. The proof was considerably more difficult than the case of of the theta divisor of the abelian fourfold worked out in [IS95]and required a difficult degeneration to the case of a Jacobian. The computation of the Abel-Jacobi map was broken into computations on different graded pieces of the limit mixed Hodge structures of F and Θ, see [ITW]forthefulldetails.

88 ELHAM IZADI AND JIE WANG

6. Higher dimensional cases As is often the case with deep conjectures such as the Hodge conjecture, the level of difficulty goes up exponentially with the dimension of the varieties concerned or, perhaps more accurately, with their Kodaira dimension. In higher dimensions a general principally polarized abelian variety is no longer a Prym variety. It is however, a Prym-Tyurin variety [Wel87]. This is, up to now, the most promising generalization of Prym varieties. For any curve C generating the abelian variety A as a group, pull-back on the first cohomology induces a map A → JC which has finite kernel. Assume given a curve C and a symmetric correspondence D ⊂ C × C.DenoteΘC a Riemann theta divisor on JC, i.e., a g−1 translate of the variety Wg−1 ⊂ Pic C of effective divisor classes. Also denote by D the endomorphism JC → JC induced by D.Wehavethefollowing Definition 6.1. We say that (A, Θ) is a Prym-Tyurin variety for (C, D)if there exists a positive integer m such that D satisfies the equation (D − 1)(D + m − 1) = 0 ∼ and there is an isomorphism A = im(D − 1) inducing an algebraic equivalence Θ ≡ mΘC |im(D−1). The integer m is called the index of the Prym-Tyurin variety. To find a family of curves in Θ that would give an answer to the Hodge conjecture for HQ or KQ (as in the cases g =4, 5), we need an explicit Prym-Tyurin structure on (A, Θ). In particular, we need to know at least one value of the index m. In general, there is very little known about the indices of ppav. In dimension 6 however, we have the following (see [ADFIO]). Theorem 6.2. For (A, Θ) general of dimension 6, there is a Prym-Tyurin structure (C, D) of index 6 on (A, Θ). Furthermore, there is a morphism π : C → P1 of degree 27 such that the 1 Galois group of the associated Galois cover X → P is the Weyl group W (E6). The morphism π has 24 branch points and above each branch point there are 6 simple 1 ramification points in C.IfP ∈ P is not a branch point of π, the action of W (E6) on π−1(P ) gives an identification of π−1(P ) with the set of lines on a smooth cubic surface such that the restriction of the correspondence D to π−1(P ) × π−1(P )can be identified with the incidence correspondence of lines on a smooth cubic surface. Prym-Tyurin structures for correspondences obtained from covers with monodromy group Weyl groups of Lie algebras were constructed by Kanev [Kan95] (also see [LR08]) who also proved irreducibility results for some of the Hurwitz schemes parametrizing such covers [Kan06]. In particular, Kanev proved that the Hurwitz scheme parametrizing covers as in the above theorem is irreducible.

7. Open problems Irreducibility: It would be interesting to know whether the Hodge structure K is irreducible. This is trivially true in dimensions up to 3 and follows from the results of [IS95] in dimension 4. In dimension 5 this would simplify the computation of the Abel-Jacobi map hence shorten the proof of [ITW].

THE PRIMITIVE COHOMOLOGY OF THETA DIVISORS 89

E6 structure when g =5: The monodromy group of the Prym map R6 → A5 is the Weyl group W (E6) of the exceptional Lie algebra E6 (see [Don92, Theorem 4.2]). Also, the lattice K has rank 78 for g =5whichis equal to the dimension of E6. So one might wonder whether it is possible to define a natural isomorphism between KC := K ⊗ C and E6. Generalization of the n-gonal construction: As we saw the 5-gonal (or pentagonal) construction is used in the construction of the family of curves in dimension 5 and the 4-gonal (or tetragonal) construction is important for understanding the family of curves in dimension 4. Therefore, one can ask whether there is a good generalization of the n-gonal construction for correspondences (in analogy with double covers) that would allow one to construct a good family of curves in higher dimensions. Catalan numbers: The g-th Catalan number can be directly defined as 1 2g C := g g +1 g and is the solution to many different counting problems (see, e.g., [Kos08]). For instance, Cg is the number of permutations of g letters that avoid the pattern 1, 2, 3. This means that, if we represent a permu- tation σ by the sequence s(σ):=(σ(1),σ(2),...,σ(g)), then the sequence s(σ) does not contain any strictly increasing subsequence of length 3. Or, g!−Cg = dim(KQ) is the number of permutations of g letters that contain the pattern 1, 2, 3 (i.e., s(σ) does contain a strictly increasing subsequence of length 3). An interesting question would be to find degenerations of Θ, i.e., K, that illustrate some of these counting problems. For instance, a degeneration of Θ and K that would exhibit a natural basis of K indexed by the permutations of g letters that contain the pattern 1, 2, 3.

References [ACGH85] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Princi- ples of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR770932 (86h:14019) [ADFIO] V. Alexeev, R. Donagi, G. Farkas, E. Izadi, and A. Ortega. A uniformization of the moduli space of abelian sixfolds. In preparation. [AF59] A. Andreotti and T. Frankel, The Lefschetz theorem on hyperplane sections, Ann. of Math. (2) 69 (1959), 713–717. MR0177422 (31 #1685) [Bea77] A. Beauville, Prym varieties and the Schottky problem, Invent. Math. 41 (1977), no. 2, 149–196. MR0572974 (58 #27995) [Bea82] A. Beauville, Sous-variétés spéciales des variétés de Prym (French), Compositio Math. 45 (1982), no. 3, 357–383. MR656611 (83f:14025) [Don92] R. Donagi, The fibers of the Prym map, Curves, Jacobians, and abelian varieties (Amherst, MA, 1990), Contemp. Math., vol. 136, Amer. Math. Soc., Providence, RI, 1992, pp. 55–125, DOI 10.1090/conm/136/1188194. MR1188194 (94e:14037) [DS81] R. Donagi and R. C. Smith, The structure of the Prym map,ActaMath.146 (1981), no. 1-2, 25–102, DOI 10.1007/BF02392458. MR594627 (82k:14030b) [Gre84a] M. L. Green, Koszul cohomology and the geometry of projective varieties, J. Differential Geom. 19 (1984), no. 1, 125–171. MR739785 (85e:14022) [Gre84b] M. L. Green, Quadrics of rank four in the ideal of a canonical curve, Invent. Math. 75 (1984), no. 1, 85–104, DOI 10.1007/BF01403092. MR728141 (85f:14028) [Gri69] P. A. Griffiths, On the periods of certain rational integrals. I, II, Ann. of Math. (2) 90 (1969), 460-495; ibid. (2) 90 (1969), 496–541. MR0260733 (41 #5357)

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[Gro69] A. Grothendieck, Hodge’s general conjecture is false for trivial reasons, Topology 8 (1969), 299–303. MR0252404 (40 #5624) [Haz94] F. Hazama, The generalized Hodge conjecture for stably nondegenerate abelian varieties, Compositio Math. 93 (1994), no. 2, 129–137. MR1287693 (95d:14011) [IS95] E. Izadi and D. van Straten, The intermediate Jacobians of the theta divisors of four- dimensional principally polarized abelian varieties, J. Algebraic Geom. 4 (1995), no. 3, 557–590. MR1325792 (96e:14053) [ITW] E. Izadi, Cs. Tamas, and J. Wang. The primitive cohomology of the theta divisor of an abelian ﬁvefold. archive 1311.6212. [Iza95] E. Izadi, The geometric structure of A4, the structure of the Prym map, double solids and Γ00-divisors,J.ReineAngew.Math.462 (1995), 93–158, DOI 10.1515/crll.1995.462.93. MR1329904 (96d:14042) [Iza10] E. Izadi, Some remarks on the Hodge conjecture for abelian varieties, Ann. Mat. Pura Appl. (4) 189 (2010), no. 3, 487–495, DOI 10.1007/s10231-009-0119-4. MR2657421 (2011f:14073) [Kan95] V. Kanev, Spectral curves and Prym-Tjurin varieties. I, Abelian varieties (Egloﬀstein, 1993), de Gruyter, Berlin, 1995, pp. 151–198. MR1336606 (96d:14024) [Kan06] V. Kanev, Hurwitz spaces of Galois coverings of P1, whose Galois groups are Weyl groups,J.Algebra305 (2006), no. 1, 442–456, DOI 10.1016/j.jalgebra.2006.01.008. MR2264138 (2007g:14032) [Kos08] T. Koshy, Catalan numbers with applications, Oxford University Press, Oxford, 2009. MR2526440 (2010g:05008) [LR08] H. Lange and A. M. Rojas, A Galois-theoretic approach to Kanev’s correspondence, Manuscripta Math. 125 (2008), no. 2, 225–240, DOI 10.1007/s00229-007-0143-x. MR2373083 (2008m:14085) [Mum74] D. Mumford, Prym varieties. I, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 325–350. MR0379510 (52 #415) [Voi03] C. Voisin, Hodge theory and complex algebraic geometry. II, Cambridge Studies in Advanced Mathematics, vol. 77, Cambridge University Press, Cambridge, 2003. Trans- lated from the French by Leila Schneps. MR1997577 (2005c:32024b) [Voi11] C. Voisin, Lectures on the Hodge and Grothendieck-Hodge conjectures, Rend. Semin. Mat. Univ. Politec. Torino 69 (2011), no. 2, 149–198. MR2931228 [Wel87] G. E. Welters, Curves of twice the minimal class on principally polarized abelian varieties, Nederl. Akad. Wetensch. Indag. Math. 49 (1987), no. 1, 87–109. MR883371 (88c:14061)

Department of Mathematics, University of California San Diego, 9500 Gilman Drive # 0112, La Jolla, California 92093-0112 E-mail address: [email protected] Department of Mathematics, University of California San Diego, 9500 Gilman Drive # 0112, La Jolla, California 92093-0112 E-mail address: [email protected]

Contemporary Mathematics Volume 647, 2015 http://dx.doi.org/10.1090/conm/647/12954

Neighborhoods of subvarieties in homogeneous spaces

J´anos Koll´ar Dedicated to C. Herbert Clemens Abstract. We study the holomorphic/meromorphic function theory and the fundamental group of Euclidean open neighborhoods of compact subvarieties in homogeneous spaces; building on results of Hironaka, Hartshorne, Napier and Ramachandran in the ample normal bundle case.

Let X be an algebraic variety over C and D ⊂ X a Euclidean open subset. It is interesting to find connections between the function theory or topology of D and X. There is not much to say if D is affine or Stein. By contrast, strong results are known if D contains a positive dimensional, compact subvariety Z with ample normal bundle: • The field of meromorphic functions Mer(D) is a finite extension of the field of rational functions Rat(X). The proof, by [Hir68, Har68], relies on cohomology vanishing for symmetric powers of the normal bundle of Z. • The image of the natural map π1(D) → π1(X) has finite index in π1(X). More generally, for every Zariski open subset X0 ⊂ X, the image of the 0 0 0 map π1(D ∩ X ) → π1(X ) has finite index in π1(X ). The proof, by [NR98], uses L2 ∂¯-methods. The isomorphism of these function fields and the surjectivity of the maps between the fundamental groups are subtler questions. Mer(D)=Rat(X)was proved for Pn [Hir68, HM68] and for Grassmannians [BH82]. The surjectivity of the maps between the fundamental groups was established for neighborhoods of certain high degree rational curves in [Kol00, Kol03]. It was also observed by [Hir68]thatifMer(D)=Rat(X) for every Z and D then X is simply connected, but the close connection between the two types of theorems was not fully appreciated. I was led to consider these topics while trying to answer some problems about non-classical flag domains raised by Griffiths and Toledo during the conference Hodge Theory and Classical Algebraic Geometry; see Question 7.

2010 Mathematics Subject Classiﬁcation. Primary 14M17, 32M10; Secondary 14D15, 14H30, 32L10. Key words and phrases. Homogeneous space, analytic neighborhood, fundamental group, Hironaka’s G3 condition, non-classical ﬂag domain.

c 2015 American Mathematical Society 91

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It turns out that the answer needs very few properties of non-classical flag domains. The natural setting is to study an arbitrary, simply connected, quasi projective, homogeneous space X, a proper subvariety Z ⊂ X and a Euclidean open neighborhood D ⊃ Z. Theorem 2 gives a complete description of those pairs Z ⊂ X for which the holomorphic/meromorphic function theory of D is determined by the regular/rational function theory of X. The precise connection is established 0 0 through an understanding of the surjectivity of π1(D ∩ X ) → π1(X ). We allow Z to be singular and with non-ample normal sheaf. A slight difference is that, while [Hir68,Har68,HM68] studied the formal completion of X along Z, we work with actual open neighborhoods. In the ample normal bundle case the two versions are equivalent, but I am not sure that this also holds in general; cf. [Gri66]. The main tool is the study of chains made up of translates of Z in X and in D. In the projective case such techniques form the basis of the study of rationally connected varieties; see [Kol96] for a detailed treatment or [AK03] for more in- troductory lectures. For non-proper homogeneous spaces these ideas were used in [BBK96]. Definition 1. Let X = G/H be a simply connected, quasi projective, homogeneous space. The left action of g ∈ G on X is denoted by τg;wecallita translation. An irreducible subvariety Z ⊂ X will be called degenerate if there is a subgroup H ⊂ K ⊂ G such that Z is contained in a fiber of the natural projection pK : G/H → G/K; otherwise we call Z nondegenerate. (If X is not simply connected, these notions should be modified; see Example 14.) For example, if X is a projective homogeneous space of Picard number 1 then every positive dimensional subvariety is nondegenerate. More generally, if the Xi are projective homogeneous spaces of Picard number 1 then Z ⊂ Xi is nondegenerate iff none of the coordinate projections Z → Xi is constant. Our main theorem is the following. Theorem 2. Let G be a connected algebraic group over C, X = G/H aquasi projective, simply connected, homogeneous space and Z ⊂ X acompact,irreducible subvariety. Let D ⊂ X denote a sufficiently small, connected, Euclidean open neighborhood of Z. The following are equivalent. (Finiteness conditions) 0 (1) H (D, OD)=C. (2) H0(D, L) is finite dimensional$ % for every line bundle L on D. (3) dim H0(D, Lm)=O mdim D for every line bundle L on D. (4) H0(D, E) is finite dimensional for every coherent, torsion free sheaf E. (Isomorphism conditions) 0 ∼ 0 (5) H (D, L|D) = H (X, L) for every line bundle L on X. 0 ∼ 0 (6) H (D, F|D) = H (X, F) for every coherent, reflexive sheaf F on X. (7) Mer(D)=Rat(X). (8) The conditions (1–7) hold for every finite, étale cover D˜ → D. (Fundamental group conditions on Zariski open subsets X0 ⊂ X) $ % $ % 0 0 0 (9) π1 D ∩ X → π1 X is surjective for every X .

NEIGHBORHOODS IN HOMOGENEOUS SPACES 93 $ % $ % −1 0 → 0 0 (10) π1 uD (X ) π1 X is surjective for every X and every finite, étale ˜ cover$ uD : D →% D. $ % ∩ 0 → 0 0 ∈ (11) π1$τg(Z) X π%1 X $is surjective% for every X and general g G. −1 0 0 0 (12) π1 (τg ◦ uZ ) (X ) → π1 X is surjective for every X , every finite cover uZ : Z˜ → Z and general g ∈ G. (Geometric characterizations of Z) (13) Z ∩ B = ∅ for every nonzero effective divisor B ⊂ X. (14) For every x1,x2 ∈ X there is a connected subvariety Z(x1,x2) ⊂ X containing them, whose irreducible components are translates of Z. (15) Same as (14) with at most 2dimX irreducible components. (16) Z is nondegenerate in X. 3 (Comments). Although we do not use anything about the structure of G,all examples come from pairs G ⊃ H where G is semi-simple, simply connected and H is connected. We always assume from now on that D is connected. We show that (2.13–16) ⇒ (2.1–12) for every D. The precise conditions for the other implications vary. In all cases of (2.1–7), the space of global sections can only increase as D gets smaller. For (2.9–10) the relevant assumption is that D retracts to Z,orthatitiscontained in a neighborhood that retracts to Z. Many parts of Theorem 2 work even if X is not simply connected, but the deepest statements, (2.5–12) do not. In one of the most interesting cases, when Z is a smooth, rational curve, there are simply connected neighborhoods D ⊃ Z. Thus D→ X lifts to the universal cover D→ X˜, hence the function theory of D is determined by X˜; the embedding D→ X is just an accident. The finite dimensionality statements (2.1–4) fit in the general framework of the papers [Hir68, Har68, HM68]. The isomorphism statements (2.5–7) are more subtle. They were known for Pn [Hir68, HM68] and for Grassmannians [BH82]. In the terminology of [Hir68], property (2.7) is called the G3 condition. It has been investigated in many other cases, see [Spe73, Fal80, B˘ad09, Car12]. Condition (2.8) mixes together some obvious claims with some quite counter intuitive ones. If v : D → D is a finite (possibly ramified) cover and E is a coherent, torsion free sheaf on D then v∗E is also a coherent, torsion free sheaf 0 0 and H (D ,E )=H (D, v∗E ). Thus (2.2–4) hold for D as well. By contrast, one would expect to find more sections and meromorphic functions on D˜.Inparticular, (2.8) implies that a nontrivial finite étale cover D˜ → D is never embeddable into any algebraic variety. A weaker version of the Lefschetz–type properties (2.9–12), asserting finite index image instead of surjectivity, is roughly equivalent to the finiteness assertion (2.2); see [NR98]. The stronger variants are studied in the papers [Kol00,Kol03] when Z is a rational curve. In (2.11–12) the adjective general means that the claim holds for all g in a nonempty Zariski open subset U(X0) ⊂ G that depends on X0. Earlier results gave (2.11) for sufficiently high degree curves only. The stronger forms (2.10) and$ (2.12)% may seem surprising at first since by −1 0 taking étale covers, the groups π1 uD (X ) are getting smaller. However, X itself is simply connected, thus all the fundamental group of X0 comes from loops around X \ X0, and such loops are preserved by étale covers of D.

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As a corollary of (2.12), a Fulton–Hansen-type connectedness theorem is proved in Section 6. Presumably (2.15) also holds with at most dim X irreducible components or maybe with some even smaller linear function of dim X. If X is projective, then the various projections G/H → G/K correspond to the faces of the cone of curves NE(X); see [KM98, Sec.1.3]. This shows that a curve C ⊂ X is nondegenerate iff its homology class [C] ∈ NE(X) is an interior point. Many of the conditions in Theorem 2 are equivalent to each other under much more general conditions. One key assumption for an arbitrary pair Z ⊂ X is that the deformation theory of Z in X should be as rich as for homogeneous spaces. Under such conditions, the properties within any of the 4 groups tend to be equivalent to each other. However, I could not write down neat, general versions in all cases. A rather subtle point is the role of the simple connectedness of X. While this is definitely needed, it seems more important to know that the stabilizer subgroup H is connected. When G is simply connected, these two conditions are equivalent. However, it should be possible to work out a version of these results for Abelian varieties. The equivalence of the 4 groups to each other is more complicated and it depends on further properties. Even when Z is a smooth rational curve with ample normal bundle, the conditions (2.5–12) are much stronger than (2.1–4). The latter case has been studied in the papers [BBK96,Kol00,Kol03,KS03] and most of the arguments of this note have their origins in one of them. See [Kol14] for further results. 4 (Proving that (2.1–15) all imply (2.16)). Assume that Z ⊂ X is degenerate. Thus there is a subgroup H ⊂ K ⊂ G such that Z is contained in one of the fibers of pK : G/H → G/K. Let U ⊂ G/K be a small Stein neighborhood of the point pK (Z). Then −1 ⊂ pK (U) X is an open neighborhood of Z with many holomorphic functions and −1 (2.1–8) all fail for every neighborhood contained in pK (U). Similarly, if U is contractible and U ⊂ Y 0 ⊂ G/K is Zariski open such that 0 −1 0 π1(Y ) is infinite then (2.9–12) fail for every Zariski open subset of pK (Y ). The preimage of a divisor BK ⊂ G/K shows that (2.13) fails and translates of Z can connect only points in the same fiber of pK . Thus (2.14–15) also imply (2.16). Open problems. In connection with Theorem 2 an interesting open problem is to understand which (non-proper) homogeneous spaces X = G/H contain a proper, nondegenerate subvariety. Consider the following conditions. • There is a projective compactification X¯ ⊃ X such that X¯ \ X has codimension ≥ 2. • X contains a proper, nondegenerate subvariety. • There is no subgroup H ⊂ K G such that G/K is quasi affine. It is clear that each one implies the next. Based on [BBK96, Sec.6], one can ask the following. Question 5. Are the above 3 conditions equivalent for a homogeneous space?

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Question 6. If X contains a proper, nondegenerate subvariety, does it contain a proper, nondegenerate, smooth, rational curve? Applications to non-classical flag domains. Our results can be used to study global sections of coherent sheaves over certain homogeneous complex manifolds. While traditionally most attention was devoted to compact homogeneous spaces and to Hermitian symmetric domains, other examples have also been studied [Wol69,FHW06]. The recent paper [GRT14] studies the geometry of non-classical flag domains. Most period domains of Hodge structures are of this type. For our purposes the precise definition is not important, we need only two of their properties. ◦ A flag domain is an open subset of a projective homogeneous space. ◦ A non-classical flag domain contains a compact rational curve with ample normal bundle. The first property is by definition while the second is one of the main results of [GRT14]. They prove that a non-classical flag domain is rationally chain connected; that is, any two points are connected by a chain of compact rational curves contained in it. The existence of an irreducible rational curve with ample normal bundle follows from this by a standard smoothing argument [Kol96, II.7.6.1]. As a simple example, SU(n, 1) ⊂ GL(n +1)actsonPn with two open orbits. One of them is the open unit ball in Cn; a Hermitian symmetric domain. The other is the complement of the closed unit ball; it is a non-classical flag domain for n ≥ 2. We see right away that for n ≥ 2 it contains many lines and in fact it contains a conic through any two of its points. The following questions were raised by Griffiths and Toledo. Question 7. Let X be a projective, homogeneous variety and D ⊂ X anon- classical flag domain. Let LX be an (algebraic) line bundle on X and LD an (analytic) line bundle on D. 0 (1) Is H (D, LD) finite dimensional? 0 0 (2) Is the restriction map H (X, LX ) → H (D, LX |D) an isomorphism? (3) Is Mer(D)=Rat(X)? Theorem 2 answers these questions affirmatively. We note that, by contrast, the two properties marked by ◦ are not sufficient to 1 understand higher cohomology groups, not even H (D, OD). Acknowledgments. This paper grew out of my attempt to answer the above questions of P. Griffiths and D. Toledo. I also thank them for further helpful discussions and L. Lempert for calling my attention to [Ker61]. Partial financial support was provided by the NSF under grant number DMS-07-58275.

1. Chains of subvarieties 8 (Chains of subvarieties in X). Let X = G/H be a quasi projective, homogeneous space, Z an irreducible variety and u : Z → X a morphism. For now we are interested in the case when u : Z→ X is a subvariety, but in Section 5 we use the general setting. A Z-chain of length r in (or over) X consists of

(1) points ai,bi ∈ Z for i =1,...,r and

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(2) translations$ % τi for$ i =1,...,r% such that (3) τi u(bi) = τi+1 u(ai+1) for i =1,...,r− 1.

Thetriple(ai,bi,τi)isalink of the chain. We also write it as $ % τi ◦ u :(Z, ai,bi) → X . $ % $ % We say that the chain starts at τ1 u(a1) ∈ X and ends at τr u(br) ∈ X. The points ai,bi determine a connected, reducible variety Z(a1,b1,...,ar,br) obtained from r disjoint copies Z1,...,Zr of Z by identifying bi ∈ Zi with ai+1 ∈ Zi+1 for i =1,...,r− 1. The morphisms τi ◦ u then deﬁne a morphism

(τ1 ◦ u,...,τr ◦ u):Z(a1,b1,...,ar,br) → X. Its image is a connected subvariety of X that contains the starting and end points of the chain and whose irreducible components are translates of u(Z) ⊂ X.(Formost purposes one can identify a chain with its image in X, but this would be slightly inconvenient when considering deformations of a trivial chain where τ1 = ···= τr. The difference becomes crucial only when we consider properties (2.10–12).) The set of all chains of length r is naturally an algebraic subvariety of Z2r ×Gr. It is denoted by Chain(Z, r). We write Chain(Z, r, x) ⊂ Chain(Z, r)todenotethe subvariety of all chains starting at x ∈ X. Up to isomorphism Chain(Z, r, x)is independent of x. The starting point (resp. the end point) gives a morphism α, β : Chain(Z, r) → X. % Thus β(Chain(Z, r, x) ⊂ X is the set of points that can be connected to x by a Z-chain of length ≤ r. % ⊂ Note that β(Chain($ Z, r,% x) X is constructible; let Wr(x) denote its closure. If there is a translate τ u(Z) that is not contained$ in%Wr(x) but whose intersection with Wr(x) is nonempty, then, by translating τ u(Z) to nearby points we see that dim Wr+1(x) > dim Wr(x); see [Kol96, 4.13]. Thus the sequence W1(x) ⊂ W2(x) ⊂ ··· stabilizes after at most dim X steps with an irreducible subvariety W (x). Furthermore, if x ∈ W (x)thenW (x) ⊂ W (x) hence in fact W (x)=W (x). Since x and x can both be connected by a Z- chain of length ≤ dim W (x) to points in a dense open subset of W (x)=W (x), we see that x and x are connected to each other by a Z-chain of length ≤ 2dimW (x). Note also that if a Z-chain connects x to τ1(x) and another one connects x to τ2(x) then translating$ % the second chain and concatenating gives a Z-chain that connects x to τ1 τ2(x) . We can summarize these considerations as follows. Proposition 9. Let X = G/H be a quasi projective, homogeneous space, Z an irreducible variety and u : Z → X a morphism. Then there is a subgroup H ⊂ K ⊂ G such that two points x1,x2 ∈ X are connected by a Z-chain iff they are contained in the same fiber of the natural projection pK : G/H → G/K. Furthermore, if x1,x2 ∈ X can be connected by a Z-chain then they can also be connected by a Z-chain of length ≤ 2(dim K − dim H). 10 (Equivalence of (2.13–16)). We already saw that (2.13–15) all imply (2.16); we need to see the converse.

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Let X be a homogeneous space under a group G and Z ⊂ X acompact, irreducible, nondegenerate subvariety. Thus the morphism pK : G/H → G/K above is constant and Proposition 9 implies both (2.14–15). In order to see (2.13) let B ⊂ X be a nonzero effective divisor. By (2.14) a ∗O | suitable translate of Z intersects B but is not contained in it, so τg X (B) Z has a ∗O | ◦ O | nonconstant section. In particular, τg X (B) Z is not in Pic (Z). Thus X (B) Z is also not in Pic◦(Z), hence it is not trivial. Therefore Z ∩ B = ∅. 11 (Ampleness of the normal bundle). Let X = G/H be a quasi projective homogeneous space and Z ⊂ X a degenerate subvariety. Let W ⊂ X be the fiber → 2 of pK : G/H G/K as in Proposition 9 that contains Z.ThenIW /IW is a trivial bundle of rank = codim W , hence X $ % ⊕ O ∼ 2 | → 2 i Z = IW /IW Z IZ /IZ is a trivial subsheaf of rank = codimX W . In particular, the normal sheaf of Z ⊂ X is not ample in any sense. Thus if Z ⊂ X is a smooth (or local complete intersection) subvariety with ample normal bundle then Z is nondegenerate. The converse does not hold. For instance, a line in a quadric hypersurface of dimension ≥ 3 is nondegenerate but its normal bundle has a trivial summand. More generally, if X is a projective homogeneous space with Picard number 1 then a line (that is, a minimal degree rational curve) in X has ample normal bundle iff X = Pn.

2. Proof of the finiteness conditions 12 (Chains of subvarieties in D). Using the notation of Paragraph 8, note that forgetting the last component of a chain gives a natural morphism Chain(Z, r + 1,x) → Chain(Z, r, x) whose fibers are isomorphic to Chain(Z, 1,x). Furthermore, Chain(Z, 1,x) ⊂ Z2 × G and the fibers of the projection to Z2 are conjugates of H.ThusifH is connected then Chain(Z, 1,x) is irreducible and so are the other varieties Chain(Z, r, x). 0 Let Chain (Z, n, x) ⊂ Chain(Z, r, x) denote those chains for which ai,bi ∈ Z (as in (8.1)) are smooth points. This is a Zariski open condition. For an open subset D ⊂ X,letChain0(Z, D, r, x) ⊂ Chain0(Z, r, x)denote those chains whose image is contained in D. Assume that D ⊃ Z and x ∈ Z is a smooth point. Then Chain0(Z, D, r, x) ⊂ Chain(Z, r, x) is open and nonempty since it contains the constant chain where ai = bi = x and τi =1foreveryi. If Z ⊂ X is nondegenerate then β :Chain0(Z, r, x) → X is dominant for r ≥ 2dimX and a dominant morphism is generically smooth. Thus β is also smooth at some point of Chain0(Z, D, r, x). We have thus established the following. Lemma 13. Let X = G/H be a quasi projective, homogeneous space with connected stabilizer H and Z ⊂ X a compact, irreducible, nondegenerate subvariety. Let Z ⊂ D $be an open neighborhood% and x ∈ Z a smooth point. 0 Then β Chain (Z, D, r, x) contains a nonempty Euclidean open subset Ur ⊂ D for r ≥ 2dimX. The following example shows that connectedness of H is quite important here. Example 14. Start with X˜ = P2 × P2 \ (diagonal) and let τ be the involution interchanging the 2 factors. Set X = X/τ˜ with the diagonal GL3-action.

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2 Let C1 ⊂ X˜ be a line contained in some P ×{point} and C ⊂ X its image. The preimage of C in X˜ is a disjoint union of a horizontal and of a vertical line. Thus C-chains (of length 4) connect any 2 points X,yetC has an open neighborhood of ∼ 2 2 the form D = D1 × D2 where D1 ⊂ P is a neighborhood of a line and D2 ⊂ C is a ball. Every compact curve in D is contained in D1 ×{p} for some p ∈ D2,thus chains of compact curves in D do not connect two general points of D. In general, let X be a homogeneous space and π : X˜ → X its universal cover. If one (equivalently every) irreducible component of π−1(Z) is nondegenerate, then Z has the good properties one expects based on the simply connected case, but not otherwise. We use Z-chains in D to prove that (2.16) ⇒ (2.1–4). The following lemma, modeledon[Nad91, Thm.2], shows that a section that vanishes to high enough order at one point of a Z-chain will vanish at all points. If Z is smooth, then one needs the semipositivity of the normal bundle NZ,X ; equivalently, the seminegativity of 2 ⊂O IZ /IZ where IZ X is the ideal sheaf of Z.IfZ is singular, the seminegativity of 2 IZ /IZ alone does not seem to be enough, one needs to control successive quotients (m) of the symbolic powers IZ ; see Paragraphs 17–18 for deﬁnitions and details.

Lemma 15. Let D be a normal complex space, Z1,...,Zn ⊂ D compact subvarieties, L alinebundleonD and s ∈ H0(D, L) a section. Assume the following.

(1) For j =1,...n there are smooth points pj ∈ Zj such that pj ∈ Zj−1 for j ≥ 2. (2) For j =1,...n there is a family of irreducible curves {Cj (λ)} passing ≤ through pj and covering a dense subset of Zj such that degCj (λ) L dj for some dj ∈ N. (i) (i+1) (3) I /I are subsheaves of a trivial sheaf ⊕mOZ for i ≥ 1. Zj Zj j (4) s vanishes at p to order c + n d . 1 j=1 j n Then s vanishes along Zr to order c + j=r+1 dj for every r.

Proof. We start with the case i =1andwriteZ := Z1.Chooseq such that $ % $ % ∈ 0 (q) ⊗ \ 0 (q+1) ⊗ s H D, IZ L H D, IZ L . Thus we get a nonzero section $ $ % % ∈ 0 (q) (q+1) ⊗ s¯ H Z, IZ /IZ L , which vanishes at p = p1 to order c + d1 − q. Using assumption (3), we get at least 1 nonzero section $ % 0 s˜ ∈ H Z, OZ ⊗ L that vanishes at p = p1 to order c + d1 − q. Restricting this to the curves C(λ)we see thats ˜ is identically zero on Z, unless q ≥ c. n Returning to the general case, we see that if s vanishes at p1 to order c+ j=1 dj n then it vanishes along Z1 to order c + j=2 dj ,inparticular,s vanishes at p2 to n ··· order c + j=2 dj . Repeating the argument for the shorter chain Z2 + + Zn completes the proof. 16 (Proof of (2.16) ⇒ (2.1–4)). Let X = G/H be a quasi projective, homogeneous space with connected stabilizer H, Z ⊂ X a compact, irreducible, nondegenerate subvariety and D ⊃ Z an open neighborhood. Pick a smooth point x ∈ Z.

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Let L be a line bundle on D and A a very ample line bundle on Z.ThenZ is covered by a family of irreducible curves {C(λ)} passing through x %obtained as dim Z−1 intersections of dim Z − 1membersof|A|.Setd(L):=(L|Z · A . We check in Lemma 18 that Z satisfies the crucial condition (15.3). By Lemma 13, for r ≥ 2dimX there is an open subset Ur ⊂ D whose points can be connected to x by a Z-chain of length r satisfying the assumptions (15.1–3). Thus if a section s ∈ H0(D, L) vanishes at x to order 1+rd(L) then it vanishes at every point of Ur.SinceUr is open, this implies that s is identically zero. This shows that dim X +2dim(X)d(L) dim H0(D, L) ≤ , dim X which proves (2.2). Since d(Lm)=md(L), we also have (2.3). 0 0 Since H (D, OD)isaC-algebra without zero divisors, H (D, OD)=C is equiv- 0 alent to dim H (D, OD) < ∞. Finally consider (2.4). We use induction on rank$ E.Ifrank% E = 1 then its double dual E∗∗ is a line bundle and H0(D, E) ⊂ H0 D, E∗∗ shows that H0(D, E)is finite dimensional. In the higher rank case, we are done if H0(D, E) = 0. Otherwise there is a nontrivial map OD → E and thus a rank 1 subsheaf E1 ⊂ E such that 0 0 0 E/E1 is again torsion free. Thus h (D, E) ≤ h (D, E1)+h (D, E/E1)andweare done by induction. Definition 17 (Symbolic powers). Let X be a variety and Z ⊂ X an irre- ⊂O m ducible, reduced subvariety with ideal sheaf IZ .LetTm X /IZ denote the largest subsheaf whose sections are supported on a smaller dimensional subset of (m) ⊂O Z.LetIZ X denote the preimage of Tm. It is called the m-th symbolic power of IZ ;see[AM69, p.56]. If X is smooth and Z is also smooth (or a local complete intersection) then (m) m IZ = IZ . (m) (m+1) The main advantage of symbolic powers is that the quotients IZ /IZ are m m+1 → (m) (m+1) torsion free sheaves on Z. There are also obvious maps IZ /IZ IZ /IZ that are isomorphisms on a dense open subset. Lemma 18. Let X be a homogeneous space and Z ⊂ X a reduced subscheme. (m) (m+1) ⊕ O Then IZ /IZ can be written as a subsheaf of i Z . Proof. Let us start with the m = 1 case. This is well known but going through it 0 will show the path to the general case. Every tangent vector field v ∈ H (X, TX ) ⊃ Lie(G) gives a differentiation dv : OX →OX which is not OX -linear. However, if φ ∈OX and s ∈ IZ are local sections then dv(φ · s)=dv(φ) · s + φ · dv(s) shows that differentiation composed with restriction to Z gives an OX -linear map →O 0 2 →⊕O dv : IZ Z . Applying this for a basis of H (X, TX )givesIZ /IZ i Z whose (2) kernel is supported at Sing Z. By definition, IZ /IZ has no sections supported on (2) →⊕O a nowhere dense subset. Thus we get an injection IZ /IZ i Z . (m) m If Z is smooth then IZ = IZ , thus we get the required $ % $ % (m) (m+1) m 2 → m ⊕ O IZ /IZ = S IZ /IZ S i Z . However, in general we only have a map $ % m 2 → (m) (m+1) S IZ /IZ IZ /IZ

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(m) (m+1) whichisanisomorphismoverthesmoothlocus.Thus$ % IZ /IZ is more positive m 2 than S IZ /IZ . (m) ··· For IZ we work with m-th order diﬀerential operators D = dv1 dvm .These (m) →O O give well deﬁned maps of sheaves D : IZ Z . X -linearity can be checked over the open set X \ Sing Z. To simplify notation, set M := {1,...,m} and for J = {j1 < ···

3. Meromorphic and holomorphic sections Here we show that property (2.7) and (2.13) imply (2.5–6) in general. Proposition 19. Let X be a normal, quasi projective variety, and D ⊂ X an open subset such that Mer(D)=Rat(X). The following are equivalent. (1) X \ D does not contain any nonzero, effective divisors. (2) Let L be an ample line bundle on X. Then, for every m ∈ N,therestric- 0 m 0 m tion map H (X, L ) → H (D, L |D) is an isomorphism. (3) For every reflexive, coherent sheaf F on X, the restriction map H0(X, F)→ 0 H (D, F|D) is an isomorphism. Proof. Assume (1) and choose m>0 such that Lm has at least one global | m section sm =0.LetsD be a global section of L D.ThensD /sm is a meromorphic function on D, hence, by assumption, it extends to a rational function rm on X. m | m Thus rmsm is a rational section of L such that (rmsm) D = sD . Repeat this with m + 1 to obtain that r := (rm+1sm+1)/(rmsm) is a rational section of L such that r|D = sD.SincesD is holomorphic, the polar set of r must be disjoint from D. However, D meets every divisor, so r has to be a regular section of L. Next we show that (1) and (2) together imply (3). Let L be an ample line bundle on X.ThenF ∗ ⊗ Lm is generated by global sections for m 1. Thus we m m have an injection j : F→⊕iL of F into a direct sum of many copies of L . 0 0 m Since H (D, F|D) ⊂⊕iH (D, L |D), every global section sD of F |D is the m restriction of a global section sX of ⊕iL . We have two subsheaves m F ⊂ F, sX ⊂⊕iL and they agree on D. Thus the support of the quotient F, sX /F is disjoint from D.SinceD meets every divisor, the support of FX ,sX /FX has codimension ≥ 2. 0 Since F is reflexive, this forces FX ,sX = F , hence sX ∈ H (X, F). Theconverse(3)⇒ (2) is clear. Finally we show (2) ⇒ (1). Assuming the contrary, there is an effective divisor B ⊂ X that is disjoint from D.Choosem such that Lm(B) is generated by global

NEIGHBORHOODS IN HOMOGENEOUS SPACES 101 sections. Then $ % $ % $ % $ % 0 m 0 m 0 m 0 m H X, L H X, L (B) → H D, L (B)|D = H D, L |D contradicts (2).

4. Lefschetz property and meromorphic functions Here we show that the Lefschetz–type property (2.9) and (2.3) imply (2.7) in general. Although we do not use it, it is worth noting that, by [NR98], (2.2) implies that the map in (2.9) has ﬁnite index image for every X0. First we show, using (2.3) that Mer(D) is an algebraic extension of Rat(X). Then we establish that having a meromorphic function on D that is algebraic over Rat(X) is equivalent to a failure of (2.9). This idea appears in [Ker61, Hilfsatz 2] and [Ste63, Prop.2.1.1]. Proposition 20. [Hir68,Har68] Let X be a normal, quasi projective variety of$ dimension% n and D ⊂ X a Euclidean open subset. Assume that h0(D, Lm)= O mn for every line bundle L on D.ThenMer(D) is an algebraic extension of Rat(X).

Proof. Let f1,...,fn be algebraically independent rational functions on X and φ a meromorphic function on D.LetB1,...,Bn and B0 be their divisors of poles. Consider the line bundle $ % L := OD B1|D + ···+ Bn|D + B0 . We can view f | ,...,f | and φ as sections of L. Thus the monomials 1 D n!D $ % " ai n a0 · | φ i fi D : i=0ai = m are all sections of Lm. The number of these monomials grows like mn+1 while, by 0 m n assumption, the dimension of H (D, LD ) grows like m .Thus,form 1, the function φ satisfies a nontrivial identity m · i ∈R i=0 hi φ =0 where hi at(X). Proposition 21. Let X be a normal, quasi projective variety and D ⊂ X a ∈ N Euclidean open subset. For every d the following are equivalent. (1) There is a φ ∈Mer(D) such that deg Rat(X)(φ):Rat(X) = d. (2) There is an irreducible (possibly ramified) cover π : X˜ → X of degree d such that the injection j : D→ X lifts to an injection ˜j : D→ X˜. (3) There is a Zariski open subset X0 ⊂ X such that $ % $ % $ % 0 0 0 im π1 D ∩ X → π1 X ⊂ π1 X has index d. Proof. Let φ be a meromorphic function on D that has degree d over Rat(X). Let d · i i=0 hi φ =0, (21.4) be the minimal polynomial of φ where the hi are rational functions on X. Let π : X˜ → X be the normalization of X in the field Rat(X)(φ). The key observation is that we can think of φ in two new ways: either as a rational function φ˜ on X˜ or as a multi-valued algebraic function φX on X whose restriction to D contains a single-valued branch that agrees with φ. Since (21.4) is irreducible over Rat(X), its discriminant is not identically zero, thus there is a dense, Zariski open subset X0 ⊂ X such that π is étale over X0 and

102 JANOS´ KOLLAR´

φ˜ takes up different values at different points of π−1(x) for all x ∈ X0.Thusthe single valued branch φ of φX determines a lifting of the injection j0 : D ∩ X0 → X to ˜j0 : D ∩ X0 → X.˜ (21.5) This shows (3). Next assume (3). The subgroup $ % $ % $ % 0 0 0 im π1 D ∩ X → π1 X ⊂ π1 X determines a degree d étale cover π0 : X˜ 0 → X0 such that (21.5) holds. Further- more, π0 extends to a degree d finite, ramified cover π : X˜ → X.Sinceπ is finite, in suitable local coordinates we can view ˜j0 as a bounded holomorphic function. Thus ˜j0 extends to a lifting ˜j : D→ X˜, hence (3) ⇒ (2). Conversely, it is clear that (2) ⇒ (3). Also, if (2) holds and ψ is a ratio- ˜ ˜ nal function on X that generates Rat(X)/Rat(X)thenψ ◦ ˜j is a meromorphic ˜ ˜ ˜ function on D such that Rat(X)(ψ ◦ j)=Rat(X), hence deg Rat(X)(ψ ◦ j): Rat(X) = d. 22 (Proof of (2.7) and (2.10) ⇒ (2.8)). With the notation and assumptions of Proposition 20, let v : D¯ → D be a finite, possibly ramified, cover. Since Mer(D¯) is an algebraic extension of Mer(D), (2.7) implies that Mer(D¯) is an algebraic extension of Rat(X). $ % $ % 0 0 The assumption (2.10) says that π1 D ∩ X → π1 X is surjective for every X0,thusMer(D¯)=Rat(X) by Proposition 21.

5. Lefschetz–type properties It is clear that (2.10) ⇒ (2.9) and (2.12) ⇒ (2.11). Let uD : D˜ → D be a finite, étale cover. Those g ∈ G for which τg(Z) ⊂ D form a Euclidean open subset of G, thus there are general translations (in the sense of (2.11–12)) such that τg(Z) ⊂ D. There is a finite (étale) cover uZ : Z˜ → Z such that τg ◦ uZ factors through uD. This shows that (2.12) ⇒ (2.10). It remains to show that if Z is nondegenerate then (2.12) holds. Thus let 0 uZ : Z˜ → Z be a finite cover and X ⊂ X a Zariski open subset. For this we use Z˜-chains in general position. 23 (Chains in general position). Let m : G×Z˜ → X be the G-action composed with uZ . Every map between algebraic varieties is a locally topologically trivial fiber bundle over a Zariski open subset, cf. [GM88, p.43]. Thus there is a Zariski 0 open subset G ⊂ G such that the first projection πG : G × Z˜ → G restricts to a topologically trivial fiber bundle 0 −1 0 ∩ −1 0 → 0 πG : m (X ) πG (G ) G . ˜0 ◦ −1 0 We denote its fibers by Zg := (τg uZ ) (X ). Marking$ a pair% of smooth points ˜0 has no significant effect topologically, thus the triples a, b, Zg ,wherea = b are ˜0 smooth points of Zg , are fibers of a topologically trivial fiber bundle over a Zariski open subset of G0 × Z˜2. Wesaythatalink(a, b, τg) ∈ Chain(Z,˜ 1,x)isingeneral position with respect to X0 if g ∈ G0, a = b are smooth points of Z˜ and they are both mapped to X0. The set of all general position links forms a Zariski open subset ∗ Chain (Z,˜ 1,x) ⊂ Chain(Z,˜ 1,x) which is nonempty for general x ∈ X.

NEIGHBORHOODS IN HOMOGENEOUS SPACES 103

For us a key point is that the image of the induced map $ % $ % $ % 0 ˜0 → 0 ⊂ 0 Γ(X ,x):=im π1 Zg ,a π1 X ,x π1 X ,x (23.1) ∗ is independent of (a, b, τg) ∈ Chain (Z,˜ 1,x) whenever the latter is nonempty. We say that a Z˜-chain as in (8.1–3) is in general position with respect to X0 if every link (ai,bi,τgi )isingeneralposition. As before, Z˜-chains in general position with respect to X0 form a Zariski open ∗ subset Chain (Z,r,x˜ ) ⊂ Chain(Z,r,x˜ $ ) which% is nonempty for general x ∈ X. 0 0 Corresponding to Γ(X ,x) ⊂ π1 X ,x as in (23.1) there is an étale cover $ % $ % π0 : X˜ 0, x˜ → X0,x (23.2) X $ % $ % ◦ ˜0 → 0 such that every general position map τg uZ : Zg ,a X ,x lifts to $ % $ % ◦ ˜0 → ˜ 0 τg uZ : Zg ,a X , x˜ . (23.3) (We do not yet know that Γ(X0,x) has finite index, so X˜ 0 → X0 could be an infinite degree cover.) Note further that (23.1) implies the following. Claim 23.4. Letx ˜ ∈ X˜ 0 be any preimage of a point x ∈ X0. Assume that $ % $ % $ % $ % ◦ ˜0 → 0 ◦ ˜0 → ˜ 0 τ uZ : Zg ,a X ,x lifts to τ uZ : Zg ,a X , x˜ ∗ for some (a,b,τ) ∈ Chain (Z,˜ 1,x). Then the lift exists for every (a, b, τ) ∈ ∗ Chain (Z,˜ 1,x).

Proposition 24. Every Z˜-chain in general position with respect to X0 and starting at x lifts to a Z˜-chain on X˜ 0 starting at x˜.

0 Proof. A Z˜-chain is given by the data (ai,bi,τi). By the choice of Γ(X ,x), ◦ ˜0 → ◦ ˜0 → ˜ 0 τ1 uZ :(Z1 ,a1) (X, x1) lifts to τ1 uZ :(Z1 ,a1) (X , x˜1).

If we letx ˜2 denote the image of b1 then we can view the latter map as ◦ ˜0 → ˜ 0 τ1 uZ :(Z1 ,b1) (X , x˜2). Both ◦ ˜0 → ◦ ˜0 → τ1 uZ :(Z1 ,b1) (X, x2)andτ2 uZ :(Z2 ,a2) (X, x2) are in general position with respect to X0, thus by (23.4), if one of them lifts to 0 (X˜ , x˜2) then so does the other. This gives us ◦ ˜0 → ˜ 0 τ2 uZ :(Z2 ,a2) (X , x˜2). We can iterate the argument to lift the whole chain. For r =2dimX,wethusgetaliftoftheendpointmap ∗ ∗ β :Chain (Z,r,x˜ ) → X0 to β˜ :Chain (Z,r,x˜ ) → X˜ 0. r $ r % $ % ∗ 0 Since βr is dominant, the induced map π1 Chain (Z,r,x˜ ) → π1 X has finite index image; cf. [Kol95, 2.10]. Thus X˜ 0 is a finite degree cover of X0 and so it ˜ ˜ uniquely extends to a finite ramified$ cover% πX : X$→ X%where$X is normal.% ˜ 0 0 ˜0 → 0 If X = X then Γ(X ,x)=π1 X ,x ,thusπ1 Zg ,a π1 X ,x is surjective and (2.12) is proved. All that remains is to derive a contradiction from X˜ = X. This is where the simple connectedness of X is finally used. It implies that X˜ → X is not étale, hence

104 JANOS´ KOLLAR´ it has a nonempty branch divisor B ⊂ X. We use the branch divisor to show that some chains do not lift, thereby arriving at a contradiction. Proposition 25. If X˜ = X then there is a Z˜-chain in general position with respect to X0 and starting at x that does not lift to a Z˜-chain on X˜ 0 starting at x˜. Proof. Set d := deg X/X˜ . We apply Kleiman’s Bertini–type theorem as in [Har77, III.10.8] to Z˜sm → X, X˜ sm → X and Bsm → X where the superscript denotes the smooth locus. Let τr+1 ◦ uZ : Z˜ → X be a general translate. Then sm sm sm sm B˜ := B ×X Z˜ ⊂ Z˜ sm sm is a nonempty smooth divisor and the fiber product Z˜ ×X X˜ is also smooth. Thus its first coordinate projection sm sm sm Z˜ ×X X˜ → Z˜ is a degree d coverthatdoesramifyoverB˜sm. The cover could be disconnected, ∼ but, due to the ramification, it can not be a union of d trivial covers Z˜sm = Z˜sm. Let a ∈ Z˜ be a general smooth point anda ˜1,...,a˜d its preimages in Z˜ ×X X˜. ˜ → ˜ Thus, for$ at least one%a ˜i,theidentitymap(Z,a) (Z,a)cannotbeliftedto (Z,a˜ ) → Z˜ ×X X,˜ a˜i .Thusifx ∈ X is the image of a andx ˜1,...,x˜d ∈ X˜ its preimages, then for at least onex ˜ ,themapτ ◦ u cannotbeliftedto i r+1$ Z% τr+1◦ uZ :(Z,a˜ ) → X,˜ a˜i . ˜ ∗ 0 For r ≥ 2dimX consider the dominant map βr :Chain(Z,r,x˜ ) → X˜ and let ∗ ⊂ 0 −1 ∗ ⊂ ˜ X X be a Zariski open subset such that πX (X ) im βr. By choosing the above τr+1 ◦ uZ : Z˜ → X generally, we may assume that there is a smooth point ∗ ∗ ar+1 ∈ Z such that x := (τr+1 ◦ uZ )(ar+1) ∈ X . ∗ ∈ −1 ∗ ˜ ˜ Thus, for everyx ˜i πX (x ) there is$ a Z-chain of% length r whose lift to X ∗ ◦ ˜ → connectsx ˜ andx ˜i . We can add τr+1 uZ : Z,ar+1,br+1 X as the last link of any of these chains. Thus we get d different Z˜-chains of length r +1andatleast one of them can not be lifted to X˜. This completes the proof of the last implication (2.16) ⇒ (2.12).

6. A Fulton–Hansen-type theorem As a far-reaching generalization of the classical Bertini theorem, [FH79]proves that if X1,X2 are irreducible varieties such that dim X1 +dimX2 >nand πi : Xi → Pn × are proper morphisms then the ﬁber product X1 (π1,π2) X2 is connected and × ∈ Pn X1 (π1,τ◦π2) X2 is irreducible for general translations τ Aut( ). Several attempts were made to work out analogs when Pn is replaced by other homogeneous spaces [Fal80,Fal81,Han83,Deb96,Per09] but the optimal forms are not known. It is easy to see that (2.12) is essentially equivalent to the special case when one of the maps π2 : X2 → G/H is dominant, and we get the following. Theorem 26. Let G be a connected algebraic group over C and X = G/H a quasi projective, simply connected, homogeneous space. Let Y be an irreducible variety and π : Y → X a ﬁnite, dominant morphism. Let Z be a proper, irreducible variety and u : Z → X a morphism with nondegenerate image. Then:

NEIGHBORHOODS IN HOMOGENEOUS SPACES 105

(1) the fiber product Y ×(π,u) Z is connected and (2) the fiber product Y ×(π,τ◦u) Z is irreducible for general translations τ ∈ G. Proof. We may harmlessly assume that Y,Z are normal and π, u are finite. Let X0 ⊂ X be a Zariski open subset over which π is étale. Set Y 0 := π−1(X0). Thus 0 0 π corresponds to$ a subgroup π1(%Y )=:ΓY ⊂ π1(X ) whose index equals deg π. −1 0 0 By (2.12), π1 (τ ◦ u) (X ) → π1(X )issurjectiveforgeneralτ ∈ G,thus

0 −1 0 Y ×(π,τ◦u) Z → (τ ◦ u) (X ) is a connected étale cover. Thus, its closure Y ×(π,τ◦u) Z is irreducible, proving (2). In$ order% to prove (1), let W ⊂ Y × Z × G be the set of points (z,y,τ)such that τ u(z) = π(y). The projection of W to Z × Y is surjective and its fibers are homogeneous spaces under the stabilizer subgroup H, hence irreducible. Thus W is irreducible. The 3rd coordinate projection π3 : W → G is a proper morphism whose fiber over τ is isomorphic to Y ×(π,τ◦u) Z. We already proved that the general fiber of π3 is irreducible, hence every fiber is connected, proving (1). Applications of this result to branched covers of homogeneous spaces, generalizing [GL80], are studied in [Tu13].

References [AK03] Carolina Araujo and János Kollár, Rational curves on varieties, Higher dimensional varieties and rational points (Budapest, 2001), Bolyai Soc. Math. Stud., vol. 12, Springer, Berlin, 2003, pp. 13–68, DOI 10.1007/978-3-662-05123-8 3. MR2011743 (2004k:14049) [AM69] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison- Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR0242802 (39 #4129) [B˘ad09] Lucian B˘adescu, On a connectedness theorem of Debarre, Interactions of classical and numerical algebraic geometry, Contemp. Math., vol. 496, Amer. Math. Soc., Providence, RI, 2009, pp. 11–20. MR2555947 (2010j:14087) [BBK96] Frédéric Bien, Armand Borel, and János Kollár, Rationally connected homogeneous spaces, Invent. Math. 124 (1996), no. 1-3, 103–127, DOI 10.1007/s002220050047. MR1369412 (97c:14049) [BH82] Ari Babakhanian and Heisuke Hironaka, Formal functions over Grassmannians, Illinois J. Math. 26 (1982), no. 2, 201–211. MR650388 (84c:14006) [Car12] Jorge Caravantes, Bertini-type theorems for formal functions in Grassmannians,Rev. Mat. Complut. 25 (2012), no. 1, 157–163, DOI 10.1007/s13163-011-0063-x. MR2876923 [Deb96] Olivier Debarre, Théorèmes de connexité pour les produits d’espaces projectifs et les grassmanniennes (French, with English summary), Amer. J. Math. 118 (1996), no. 6, 1347–1367. MR1420927 (98f:14015) [Fal80] Gerd Faltings, A contribution to the theory of formal meromorphic functions, Nagoya Math. J. 77 (1980), 99–106. MR556311 (81c:13025) [Fal81] Gerd Faltings, Formale Geometrie und homogene Räume (German), Invent. Math. 64 (1981), no. 1, 123–165, DOI 10.1007/BF01393937. MR621773 (82m:14006) [FH79] William Fulton and Johan Hansen, A connectedness theorem for projective varieties, with applications to intersections and singularities of mappings, Ann. of Math. (2) 110 (1979), no. 1, 159–166, DOI 10.2307/1971249. MR541334 (82i:14010) [FHW06] Gregor Fels, Alan Huckleberry, and Joseph A. Wolf, Cycle spaces of flag domains, Progress in Mathematics, vol. 245, Birkhäuser Boston, Inc., Boston, MA, 2006. A complex geometric viewpoint. MR2188135 (2006h:32018) [GL80] Terence Gaffney and Robert Lazarsfeld, On the ramification of branched coverings of Pn, Invent. Math. 59 (1980), no. 1, 53–58, DOI 10.1007/BF01390313. MR575080 (81h:14012)

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[GM88] Mark Goresky and Robert MacPherson, Stratified Morse theory, Ergebnisse der Math- ematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 14, Springer-Verlag, Berlin, 1988. MR932724 (90d:57039) [Gri66] Phillip A. Griffiths, The extension problem in complex analysis. II. Embeddings with positive normal bundle,Amer.J.Math.88 (1966), 366–446. MR0206980 (34 #6796) [GRT14] Phillip Griffiths, Colleen Robles, and Domingo Toledo, Quotients of non-classical flag domains are not algebraic,Algebr.Geom.1 (2014), no. 1, 1–13, DOI 10.14231/AG- 2014-001. MR3234111 [Han83] Johan Hansen, A connectedness theorem for flagmanifolds and Grassmannians,Amer. J. Math. 105 (1983), no. 3, 633–639, DOI 10.2307/2374317. MR704218 (85d:14071) [Har68] Robin Hartshorne, Cohomological dimension of algebraic varieties, Ann. of Math. (2) 88 (1968), 403–450. MR0232780 (38 #1103) [Har77] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR0463157 (57 #3116) [Hir68] Heisuke Hironaka, On some formal imbeddings, Illinois J. Math. 12 (1968), 587–602. MR0241433 (39 #2773) [HM68] Heisuke Hironaka and Hideyuki Matsumura, Formal functions and formal embeddings, J.Math.Soc.Japan20 (1968), 52–82. MR0251043 (40 #4274) [Ker61] Hans Kerner, Uberlagerungen¨ und Holomorphiehüllen (German), Math. Ann. 144 (1961), 126–134. MR0137847 (25 #1296) [KM98] János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR1658959 (2000b:14018) [Kol95] János Kollár, Shafarevich maps and automorphic forms, M. B. Porter Lectures, Prince- ton University Press, Princeton, NJ, 1995. MR1341589 (96i:14016) [Kol96] János Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathemat- ics], vol. 32, Springer-Verlag, Berlin, 1996. MR1440180 (98c:14001) [Kol00] János Kollár, Fundamental groups of rationally connected varieties, Michigan Math. J. 48 (2000), 359–368, DOI 10.1307/mmj/1030132724. Dedicated to William Fulton on the occasion of his 60th birthday. MR1786496 (2001k:14045) [Kol03] János Kollár, Rationally connected varieties and fundamental groups, Higher dimensional varieties and rational points (Budapest, 2001), Bolyai Soc. Math. Stud., vol. 12, Springer, Berlin, 2003, pp. 69–92, DOI 10.1007/978-3-662-05123-8 4. MR2011744 (2005g:14042) [Kol14] János Kollár, The Lefschetz property for families of curves, ArXiv e-prints (2014). [KS03] János Kollár and Endre Szabó, Rationally connected varieties over finite fields,Duke Math. J. 120 (2003), no. 2, 251–267, DOI 10.1215/S0012-7094-03-12022-0. MR2019976 (2005h:14090) [Nad91] Alan Michael Nadel, The boundedness of degree of Fano varieties with Picard number one,J.Amer.Math.Soc.4 (1991), no. 4, 681–692, DOI 10.2307/2939285. MR1115788 (93g:14048) [NR98] Terrence Napier and Mohan Ramachandran, The L2 ∂-method, weak Lefschetz theorems, and the topology of Kähler manifolds,J.Amer.Math.Soc.11 (1998), no. 2, 375–396, DOI 10.1090/S0894-0347-98-00257-4. MR1477601 (99a:32008) [Per09] N. Perrin, Small codimension subvarieties in homogeneous spaces, Indag. Math. (N.S.) 20 (2009), no. 4, 557–581, DOI 10.1016/S0019-3577(09)80026-8. MR2776900 (2012b:14095) [Spe73] Robert Speiser, Cohomological dimension of non-complete hypersurfaces, Invent. Math. 21 (1973), 143–150. MR0332788 (48 #11114) [Ste63] Karl Stein, Maximale holomorphe und meromorphe Abbildungen. I (German), Amer. J. Math. 85 (1963), 298–315. MR0152676 (27 #2651) [Tu13] Yu-Chao Tu, Gaffney-Lazarsfeld Theorem for Homogeneous Spaces, ArXiv e-prints (2013).

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[Wol69] Joseph A. Wolf, The action of a real semisimple group on a complex ﬂag manifold. I. Orbit structure and holomorphic arc components, Bull. Amer. Math. Soc. 75 (1969), 1121–1237. MR0251246 (40 #4477)

Princeton University, Princeton New Jersey 08544-1000 E-mail address: [email protected]

Contemporary Mathematics Volume 647, 2015 http://dx.doi.org/10.1090/conm/647/12971

Unconditional noncommutative motivic Galois groups

Matilde Marcolli and Gon¸calo Tabuada Dedicated to Herb Clemens, on the occasion of his non-retirement

Abstract. In this short note we introduce the unconditional noncommutative motivic Galois groups and relate them with those of Andr´e-Kahn.

Motivating questions Motivic Galois groups were introduced by Grothendieck in the sixties as part of his broad theory of (pure) motives. These group schemes are conditional in the sense that their construction makes use of the standard conjectures. Thanks to Kontsevich [6–8], Grothendieck’s theory of motives admits a noncommutative counterpart, with schemes replaced by dg categories. The standard conjectures admit noncommutative analogues and there exist also conditional noncommutative motivic Galois groups; consult [10–12] for details. Recently, via an evolved “⊗-categorification” of the Wedderburn-Malcev’s theorem, André-Kahn [2] introduced unconditional motivic Galois groups. These group schemes GalH (attached to a Weil cohomology theory H) are well-defined up to an interior automorphism and do not require the assumption of any of Grothendieck’s standard conjectures. This lead us naturally to the following motivating questions: NC Q1: Do the Galois groups GalH admit noncommutative analogues GalH ? NC Q2: What is the relation between GalH and GalH ? NC Q3: What is the relation between GalH and the conditional Galois groups? In this short note we provide precise answers to these three questions; see Definition 2.8, Theorem 3.2, and Proposition 4.2, respectively.

1. Preliminaries Let k be a base field and F a field of coefficients. The classical idempotent completion construction will be denoted by (−).

2010 Mathematics Subject Classiﬁcation. 14A22, 14C15, 14F40, 14G32, 18G55, 19D55. Key words and phrases. Motivic Galois groups, motives, noncommutative algebraic geometry. The ﬁrst author was supported by NSF grants DMS-0901221, DMS-1007207, DMS-1201512, and PHY-1205440. The second author was supported by the NSF CAREER Award 1350472 and by the Por- tuguese Foundation for Science and Technology grant PEst-OE/MAT/UI0297/2014. The authors are very grateful to the organizers of the conference “Hodge Theory and Classical Algebraic Geometry” for the opportunity to present this work.

c 2015 American Mathematical Society 109

110 MATILDE MARCOLLI AND GONC¸ ALO TABUADA

Motives. We assume the reader is familiar with the categories of Chow motives Chow(k)F , homological motives Hom(k)F ,andnumericalmotivesNum(k)F ; consult [1, §4]. The Tate motive will be denoted by F (1). At §4 we will assume some familiarity with Grothendieck’s standard conjectures of type C (K¨unneth) and D (homological=numerical) as well as with the sign conjecture C+;see[1, §5].

Dg categories. A diﬀerential graded (=dg) category A is a category enriched over cochain complexes of k-vector spaces; consult [4] for details. Every (dg) k- algebra A gives naturally rise to a dg category A with a single object. Another source of examples is provided by schemes since the category of perfect complexes perf(X) of every quasi-compact quasi-separated k-scheme X admits a canonical dg § enhancement perfdg(X); see [4, 4.6]. When X is quasi-projective this dg enhancement is moreover unique; see [9, Thm. 2.12].

Noncommutative motives. We assume the reader is familiar with the categories of noncommutative Chow motives NChow(k)F , noncommutative homological motives NHom(k)F , and noncommutative numerical motives NNum(k)F ;consult the survey articles [13, §2-3] [14, §4] and the references therein. At §4 we will assume some familiarity with conjectures CNC (the noncommutative analogue of + C )andDNC (the noncommutative analogue of type D); see [13, §4].

2. Construction of the unconditional NC motivic Galois groups Assume that k is of characteristic zero and that k ⊆ F or F ⊆ k.Asprovedin [12, Thm. 9.2], periodic cyclic homology HP gives rise to an F -linear ⊗-functor

(2.1) HP∗ :NChow(k)F −→ sVect(K) with values in the category of ﬁnite dimensional super K-vector spaces (with K = F when k ⊆ F and K = k when F ⊆ k). By deﬁnition of the category of noncommutative homological motives, (2.1) descends to a faithful F -linear ⊗-functor

(2.2) HP∗ : NHom(k)F −→ sVect(K) . Notation . ± 2.3 Let NHom(k)F be the full subcategory of those noncommutative homological motives N whose associated K¨unneth projectors ± ± → πN : HP∗(N) HP∗ (N) HP∗(N) ± ± ± canbewrittenasπN = HP∗(πN ) with πN endomorphisms in NHom(k)F . Example 2.4. Let X be a smooth projective k-scheme. When F ⊆ k,the ± proofof[12, Thm. 1.3] shows us that perfdg(X) belongs to NHom(k)F whenever X satisﬁes the sign conjecture. As proved by Kleiman [5], this holds when X is an ± abelian variety. Using the stability of NHom(k)F under direct factors and tensor products (see [12, Prop. 10.2]) we then obtain a large class of examples. Since by hypothesis k (and hence F ) is of characteristic zero, [2, Prop. 2] applied to the above functor (2.2) gives rise to a new rigid symmetric monoidal category † ± ± NHom (k)F (obtained from NHom(k)F by modifying its symmetric isomorphism constraints) and to a composed faithful F -linear ⊗-functor

† ± ⊂ HP−→∗ forget−→ (2.5) NHom (k)F NHom(k)F sVect(K) Vect(K) .

UNCONDITIONAL NONCOMMUTATIVE MOTIVIC GALOIS GROUPS 111

† ± § ⊗ R± More importantly, NHom (k)F is semi-primary (see [2, 1 page 2]), its -ideals N ± † ± † ± N ± and agree, the quotient category NNum (k)F deﬁned by NHom (k)F / is abelian semi-simple, and the canonical projection ⊗-functor † ± −→ † ± (2.6) NHom (k)F NNum (k)F is conservative; consult [2, §1]. Thanks to [2, Thm. 8 a)], the projection (2.6) admits a ⊗-section and any two such ⊗-sections are conjugated by a ⊗-isomorphism. The choice of a ⊗-section sNC gives then rise to a faithful F -linear ⊗-functor

NC † ± −→s † ± (2.5)−→ (2.7) fHP : NNum (k)F NHom (k)F Vect(K) . † ± The category NNum (k)F , endowed with the fiber functor fHP , becomes a Tan- nakian category; see [12, §7]. Definition 2.8. The unconditional noncommutative motivic Galois group NC ⊗ ⊗ GalHP is the group scheme Aut (fHP )of -automorphisms of the above fiber functor (2.7). A different choice of the ⊗-section sNC gives rise to an isomorphic group scheme † ± (via an interior automorphism). Moreover, since NHom (k)F is abelian semi-simple, NC the group scheme GalHP is pro-reductive, i.e. its unipotent radical is trivial.

3. Relation with André-Kahn’s motivic Galois groups Assume that k is of characteristic zero and that F ⊆ k. As explained by ∗ André-Kahn [2], de Rham cohomology theory HdR gives rise to a well-defined fiber functor † ± −→ fdR :Num (k)F Vect(k) ⊗ and consequently to an unconditional motivic Galois group GaldR := Aut (fdR). † ± Let us denote by F (1) the Tannakian subcategory of Num (k)F generated by the Tate motive F (1) and write GaldR(F (1)) for the group scheme of ⊗-automorphisms of the composed fiber functor

→t † ± −→fdR (3.1) F (1) Num (k)F Vect(k) . As explained in [1, §2.3.3], the inclusion of categories gives rise to an homomorphism NC t :GaldR GaldR(F (1)). The relation between GaldR and GalHP is the following: Theorem 3.2. There exists a comparison group scheme homomorphism NC → (3.3) GalHP Kernel(t :GaldR GaldR(F (1))) .

In the case where k = F , GaldR(F (1)) identiﬁes with the multiplicative group scheme Gm and the above comparison homomorphism (3.3) is faithfully ﬂat NC G (3.4) GalHP Kernel(t :GaldR m) . Intuitively speaking, (3.4) shows us that the ⊗-symmetries of the commutative world which can be lifted to the noncommutative world are precisely those that become trivial when restricted to the Tate motive. It is unclear at the moment if the kernel of the comparison homomorphisms (3.3)–(3.4) is non-trivial.

112 MATILDE MARCOLLI AND GONC¸ ALO TABUADA

Proof. We start by constructing the comparison homomorphism (3.3). Recall from the proof of [12, Thm. 1.7] that we have the following commutative diagram

/ / HP∗/ (3.5) Chow(k)F Chow(k)F/−⊗F (1) NChow(k)F sVect(k)

/ / / Chow(k)F/Ker (Chow(k)F/−⊗F (1))/Ker NHom(k)F sVect(k) HP∗ / / Num(k)F Num(k)F/−⊗F (1) NNum(k)F , where Ker stands for the kernel of the respective horizontal composition towards sVect(k). Since by hypothesis k is of characteristic zero, the proof of [12, Thm. 1.3] shows us that the upper horizontal composition in (3.5) identiﬁes with the functor ∗ −→ → ⊕ n ⊕ n (3.6) sHdR :Chow(k)F sVect(k) X ( HdR(X), HdR(X)) . n even n odd Hence, its kernel Ker agrees with the one of de Rham cohomology theory ∗ −→ →{ n } HdR :Chow(k)F GrVect(k)≥0 X HdR(X) n≥0 .

As a consequence, the idempotent completion of Chow(k)F/Ker agrees with the category Hom(k)F ;see[1, §4]. The two lower commutative squares in (3.5), com- bined with the fact that Num(k)F and NHom(k)F are idempotent complete, give then rise to the following commutative diagram Φ / (3.7) Hom(k)F NHom(k)F

 / Num(k)F NNum(k)F .

By construction of Num(k)F and NNum(k)F , the kernels of the vertical functors in (3.7) are precisely the largest ⊗-ideals of Hom(k)F and NHom(k)F . Hence, the commutativity of diagram (3.7) allows us to conclude that the functor Φ is radical, i.e. that it preserves these largest ⊗-ideals. Now, note that by construction, the following composition

Φ HP∗ Hom(k)F −→ NHom(k)F −→ sVect(k) agrees with the factorization of the above functor (3.6) through the category Hom(k)F . In particular, it is faithful. By applying [2,Prop.2]toHP∗ and HP∗ ◦Φ, ⊗ † ± → † ± we obtain then an induced F -linear -functor Φ : Hom (k)F NHom (k)F .This functor is also radical and therefore gives rise to the following commutative square † ± Φ / † ± Hom? (k)F NHom (k_)F

s sNC † ± / † ± Num (k)F NNum (k)F ; Φ the dotted arrows denote the ⊗-sections provided by [2, Thm. 8 a)]. Note that GaldR can be described as the ⊗-automorphisms of the ﬁber functor † ± −→s † ± −→Φ † ± (2.5)−→ fdR :Num (k)F Hom (k)F NHom (k)F Vect(k) .

UNCONDITIONAL NONCOMMUTATIVE MOTIVIC GALOIS GROUPS 113

Since Φ is radical, [3, Props. 12.2.1 and 13.7.1] imply that the two ⊗-functors ◦ † ± −→ † ± NC ◦ † ± −→ † ± Φ s :Num (k)F NHom (k)F s Φ:Num (k)F NHom (k)F are naturally isomorphic (via a ⊗-isomorphism). As a consequence, the above ﬁber functor fdR becomes naturally ⊗-isomorphic to the following composition

† ± −→Φ † ± −→fHP Num (k)F NNum (k)F Vect(k) . Hence, by deﬁnition of the unconditional motivic Galois groups, the functor Φgives NC → rise to a well-deﬁned comparison group scheme homomorphism GalHP GaldR. It remains then to show that the composition NC −→ t (3.8) GalH GaldR GaldR(F (1)) † ± † ± is trivial. Since the categories Num (k)F and NNum (k)F are abelian semi-simple and ΦisF -linear and additive, we conclude that Φismoreoverexact, i.e. that it preserves kernels and cokernels. Thanks to the commutative diagram (3.5) we ⊗ † ± observe that the image of F (1) under Φ is precisely the -unit k of Num (k)F . Hence, since Φ is also symmetric monoidal, we obtain the commutative diagram

(3.9) F (1) Φ / k _ _

t † ± / † ± Num (k)F NNum (k)F , Φ † ± where k denotes the Tannakian subcategory of NNum (k)F generated by k.The group scheme of ⊗-automorphisms of the ﬁber functor

→ † ± −→fHP k NNum (k)F Vect(k) is clearly trivial. Using the commutativity of diagram (3.9), we conclude finally that the above composition (3.8) is trivial. † ± Letusnowassumethatk = F .Notefirstthat(Num(k)F ,w,F(1)) is a Tate sub-triple of the one described in [12, Example 13.3(i)] (w stands for the weight Z-grading). Since by hypothesis k = F this Tate triple is moreover neutral, with fiber functor given by fdR. As explained in the proof of [12, Prop. 14.1], the weight Z-grading of the Tate triple furnish us the following factorization

(3.1) / F (1) GrVect(k)Z LLL LLL LLL forget (3.1) L& Vect(k) , where GrVect(k)Z stands for the category of ﬁnite dimensional Z-graded k-vector spaces. An argument similar to the one in loc. cit. allows us then to conclude that GaldR(F (1)) % Gm. Let us now prove that the comparison homomorphism (3.4) is faithfully ﬂat. By applying the general [12, Prop. 14.1] to the neutral Tate-triple † ± (Num (k)F ,w,F(1)) we obtain the following group scheme isomorphism † ± −→∼ G (3.10) Gal((Num (k)F/−⊗F (1)) ) Kernel(t :GaldR m) ,

114 MATILDE MARCOLLI AND GONC¸ ALO TABUADA where the left-hand-side is the group scheme of ⊗-automorphisms of the functor

† ± −→Ψ † ± −→fHP (Num (k)F/−⊗F (1)) NNum (k)F Vect(k) . We claim that Ψ is fully-faithful. Consider the composition at the bottom of diagram (3.5), and recall that the functor Num(k)F /−⊗F (1) → NNum(k)F is fully- ± faithful. By ﬁrst restricting ourselves to Num(k)F andthenbymodifyingthe symmetry isomorphism constraints we obtain the following composition

† ± −→ ± † −→ψ † ± Num (k)F (Num(k)F/−⊗F (1)) NNum (k)F . C ± O The general [12, Lem. 4.9] (applied to =Num(k)k and = F (1)) furnish us a ⊗ † ± ± † canonical -equivalence between Num (k)F/−⊗F (1) and (Num(k)F/−⊗F (1)) . Hence, † ± since ψ is fully-faithful and NNum (k)F is idempotent complete we conclude that Ψ is also fully-faithful. As a consequence, we obtain a faithfully flat homomorphism NC † ± GalHP Gal((Num (k)F/−⊗F (1)) ) . By combining it with (3.10) we conclude finally that the comparison homomorphism (3.3) is faithfully flat. This achieves the proof.

4. Relation with the conditional motivic Galois groups. Assuming the standard conjectures of type C and D, we have well-defined † conditional motivic Galois groups Gal(Num (k)F ); see [1, §6]. Similarly, assuming conjectures CNC,DNC, we have well-defined conditional noncommutative motivic † † Galois groups Gal(NNum (k)F ); see [13, §6]. As explained in [2], Gal(Num (k)F ) identifies with GaldR. In the noncommutative world, the following holds: † Lemma 4.1. The conditional Galois group Gal(NNum (k)F ) identifies with NC GalHP . Proof. † † ± Conjecture CNC implies that NNum (k)F = NNum (k)F .Onthe NC other hand, conjecture DNC allows us to choose for the ⊗-section s the identity functor. This achieves the proof.

In what concerns the comparison homomorphism (3.4), the following holds:

Proposition 4.2. The conditional comparison homomorphism

† † (4.3) Gal(NNum (k)F ) Kernel(t :Gal(Num (k)F ) Gm) , constructed originally in [12, Thm. 1.7],identiﬁeswith(3.4).

Proof. It follows from the fact that the comparison homomorphisms (4.3) and (3.4) are induced by the same functor from numerical motives to noncommutative numerical motives, namely by the bottom horizontal composition in diagram (3.5).

Acknowledgments: The authors are grateful to Michael Artin, Eric Friedlander, Steven Kleiman, and Yuri Manin for useful discussions and motivating questions.

UNCONDITIONAL NONCOMMUTATIVE MOTIVIC GALOIS GROUPS 115

References [1] Y. André, Une introduction aux motifs (motifs purs, motifs mixtes, périodes) (French, with English and French summaries), Panoramas et Synthèses [Panoramas and Syntheses], vol. 17, SociétéMathématique de France, Paris, 2004. MR2115000 (2005k:14041) [2] Y. André and B. Kahn, Construction inconditionnelle de groupes de Galois motiviques (French, with English and French summaries), C. R. Math. Acad. Sci. Paris 334 (2002), no. 11, 989–994, DOI 10.1016/S1631-073X(02)02384-1. MR1913723 (2003k:14020) [3] Y. André and B. Kahn, Nilpotence, radicaux et structures mono¨ıdales (French, with English summary), Rend. Sem. Mat. Univ. Padova 108 (2002), 107–291. With an appendix by Peter O’Sullivan. MR1956434 (2003m:18009) [4] B. Keller, On differential graded categories, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 151–190. MR2275593 (2008g:18015) [5] S. L. Kleiman, Algebraic cycles and the Weil conjectures,Dixexposés sur la cohomologie des schémas, North-Holland, Amsterdam; Masson, Paris, 1968, pp. 359–386. MR0292838 (45 #1920) [6] M. Kontsevich, Noncommutative motives. Talk at the Institute for Advanced Study on the occasion of the 61st birthday of Pierre Deligne, October 2005. Video available at http://video.ias.edu/Geometry-and-Arithmetic. [7] M. Kontsevich, Mixed noncommutative motives. Talk at the Workshop on Homological Mirror Symmetry, Miami, 2010. Notes available at www-math.mit.edu/auroux/frg/miami10-notes. [8] M. Kontsevich, Notes on motives in finite characteristic, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, Progr. Math., vol. 270, Birkhäuser Boston, Inc., Boston, MA, 2009, pp. 213–247, DOI 10.1007/978-0-8176-4747-6 7. MR2641191 (2011b:11086) [9]V.A.LuntsandD.O.Orlov,Uniqueness of enhancement for triangulated categories,J.Amer. Math. Soc. 23 (2010), no. 3, 853–908, DOI 10.1090/S0894-0347-10-00664-8. MR2629991 (2012b:14030) [10] M. Marcolli and G. Tabuada, Kontsevich’s noncommutative numerical motives,Compos. Math. 148 (2012), no. 6, 1811–1820, DOI 10.1112/S0010437X12000383. MR2999305 [11] M. Marcolli and G. Tabuada, Noncommutative motives, numerical equivalence, and semi-simplicity,Amer.J.Math.136 (2014), no. 1, 59–75, DOI 10.1353/ajm.2014.0004. MR3163353 [12] M. Marcolli and G. Tabuada, Noncommutative numerical motives, Tannakian structures, and motivic Galois groups. Available at arXiv:1110.2438. To appear in Journal of the EMS. [13] M. Marcolli and G. Tabuada, Noncommutative motives and their applications. Available at arXiv:1311.2867. To appear in MSRI Publications. [14] G. Tabuada, A guided tour through the garden of noncommutative motives, Topics in noncommutative geometry, Clay Math. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 2012, pp. 259–276. MR2986869

Mathematics Department, Mail Code 253-37, Caltech, 1200 E. California Blvd. Pasadena, California 91125 E-mail address: [email protected] URL: http://www.its.caltech.edu/~matilde Department of Mathematics, MIT, Cambridge, MA 02139, USA and Departamento de Matematica´ e CMA, FCT-UNL, Quinta da Torre, 2829-516, Caparica, Portugal E-mail address: [email protected] URL: http://math.mit.edu/~tabuada

Contemporary Mathematics Volume 647, 2015 http://dx.doi.org/10.1090/conm/647/12968

Diﬀerential equations in Hilbert-Mumford Calculus

Ziv Ran To Herb Clemens: mentor, collaborator, friend

Abstract. An evolution-type diﬀerential equation encodes the intersection theory of tautological classes on the Hilbert scheme of a family of nodal curves.

Introduction Let X/B be a family of nodal or smooth curves and L a line bundle on X.Let [m] XB denote the relative Hilbert scheme of length-m subschemes of fibres of X/B, and Λm(L) the tautological bundle associated to L,whichisarank-m bundle on [m] XB . The term ’Hilbert-Mumford Calculus’ refers to the intersection calculus of ’tautological classes’, i.e. polynomials in the Chern classes of Λm(L). This calculus, which is an extension of the classical work of Macdonald [4], was developed in [5], [6] and other papers, where a number of examples and computations were given, with the more involved ones mostly based on the Macnodal computer program developed for this purpose by Gwoho Liu [3]. Our purpose here is to show that this calculus can be encoded in a linear second-order partial differential equation satisfied by a suitable generating function (see (2.38), (2.39) below). While the result is mostly a reformulation of results in [6], it improves on the latter in that it uses the standard language of differential calculus and moreover avoids the complex recursiveness inherent in [6]. In more detail, let W m(X/B) denote the Hilbert scheme of length-m flags in fibres and consider the infinite-flag Hilbert scheme 0 m ⊂ [m] W (X/B) =← lim− W (X/B) XB m which is endowed with discriminant or big diagonal operators Γ(m) pulled back [m] from XB and with classes Li pulled back from the i-th X factor. It was shown in [5] that the Chern numbers of the tautological bundles can be expressed as linear combinations of monomials of the form (working left to right)

a1 a2 (2) k2 ar (r) kr L1 L2 (Γ ) ...Lr (Γ ) .

2010 Mathematics Subject Classiﬁcation. Primary 14N99, 14H99. Key words and phrases. Hilbert scheme, nodal curves, intersection theory. arxiv.org/arXiv:1211.6040.

c 2015 Ziv Ran 117

118 ZIV RAN

Consequently we introduce the ’Hilbert potential’ i G =exp(γΓ) exp( μiL ) in which is external or ’Pontrjagin’ product (whereas the ’implicit’ or ’.’ product is intersection or, in the case of an operator like Γ, composition). Then the intersection calculus of [6] shows how to express G recursively in terms of elements of the so-called tautological module T = T (X/B), and consequently how to read off numerical information. We show in Theorem 2.1 how to encode the latter into an equation in the γ-andμi-derivatives of G and its derivatives with respect to the ’space’ variables corresponding to standard generators of T . This equation can be used to completely determine G. In order to be able to express the appropriate relation in a familiar differential equation form, we introduce a formal model Tˆ for the tautological module T , essentially by replacing suitable generators by independent variables. The use of differential equations to describe intersection theory associated to stable curves is not new. Our evolution equation is somewhat analogous to the ’quantum differential equation’ of Gromov-Witten theory (see [1], Ch. 10 or [2], Ch. 28). Another well-known such equation is Witten’s KdV equation, governing the intersection theory of the moduli space Mg (see [7]). Heuristically, a map of a stable curve X to PN yields a map from the dual projective space to the Hilbert scheme of X, and one may hope in this say to relate the space of stable maps to PN to the space of maps of PN to the Hilbert scheme of the ’universal family’ of stable curves, thus possibly relating Gromov-Witten invariants and their potential to our Hilbert potential. However, the technical details involved are likely to be formidable, and none have been worked out.

1. Big tautologocal module 1.1. Data. We will fix a flat family X/B of nodal, possibly pointed, genus-g curves, which is ’split’ in the sense that its boundary can be covered by finitely many projective families of the form Xθ/B(θ) → X/B,eachendowedwithapair of distinguished sections θx,θy called node preimages, that map to a node θ of X/B. We then have i-th boundary families Xi/Bi where Bi = B(θ1, ..., θi)= B(θ1) ×B ... ×B B(θi)

(θ1,...,θi) (θ1,...,θi) (union over collections of i distinct nodes). This includes the case i =0where B0 = B. To this we associate a coeﬃcient system, in the form of a system of pairs of graded unital Q-algebras # → → (ABi Ai)= (AB(θ1,...,θi) A(θ1,...,θi)) such that → ∗ Q → ∗ Q (i) (ABi Ai)admitsamapto(H (Bi, ) H (Xi, )), whence a ’degree’ or integral map → Q = : ABi .

DIFFERENTIAL EQUATIONS IN HILBERT-MUMFORD CALCULUS 119

(ii) Each A = Ai contains an element ωi that maps to c1(ωXi/Bi ), plus el-

ements that map to the distinguished sections, and each ABi contains elements mapping to Mumford classes and cotangent classes for the distinguished sections (both those coming from X/B and node preimages). These are all mutually compatible, e.g. i | ωi Xθ1,...,θi = ω0 + (θj,x + θj,y),ω0 = ωX/B . j=1 ∗ (iii) For any distinguished section σ over Bi, there is a pullback map σ : → Ai ABi . (iv) There are ’boundary pullback’ maps → → → bi :(ABi Ai) (ABi+1 Ai+1)

compatible with the various data. An element α ∈ Ai may be replaced by its image in Aj,j > i, whenever this makes sense. The maps bi may be assembled together to a ’total boundary pullback’ # → (1.1) b : ABi ABi+1 . 1.2. Generators, product. In [6] we defined the tautotological module # m T = TA(X/B)= TA (X/B). T may be thought of as a poor man’s version (Goldielocks version ?) of the Chow ring of the relative Hilbert scheme of finite subschemes of fibres of X/B:itiscom- putable, unlike the latter, while being large enough to contain many important geometric characters of X/B, such as those redicible to Chern numbers of tautological bundles.We proceed to describe natural generators the module T .NoteT is graded by the weight m which is the ’number of variables’, i.e there is a canonical, not necessarily injective, map to the rational equivalence group m → • [m] T AQ(XB ).

T contains a ’classical’ part T0, which is a commutative algebra under external or Pontrjagin product (as distinct from intersection product), which will be denoted by . Via the correspondence m+m → [m+m ] W (X/B) XB (1.2) &' [m] (m) XB XB

T is a module over T0. A special role will be played by the diagonal classes of T0: the monoblock diagonals Γ(n)[α],α∈ A and their - products, called polyblock diagonals. In fact, if we introduce a formal variable tn,n≥ 1, we have a ring isomorphism

T0 % AB[tnA : n ∈ N].

More concretely, T0 is a direct sum of tensor products of symmetric powers of A over AB, indexed by partitions. The boundary pullback map b in (1.1) may be extended to T0 by setting

b(tnα)=tnb(α)

120 ZIV RAN

In addition to polyblock diagonals, the tautological module also contains (iterated) node scrolls and node sections, of the form n n ∈ θ Fj (θ)[γ],Qj (θ)[γ],γ TAθ (X /B(θ)) (and their iterations). Thus, elements of the tautological module of X/B arise θ n from analogous elements for a boundary family X /B(θ)viaanodescrollFj (θ) n n P1 or a node section Qj (θ). Geometrically, Fj (θ)isa -bundle consisting of schemes given locally at θ by xn−j + tyj ,y ∈ C∗, and their limits as t → 0ott →∞.These n n limits correspond to the node sections Qj ,Qj+1, while t is the ﬁbre coordinate in the P1-bundle (the base coordinates come from the residual length- (m − n) subscheme). To keep track of iterated node/scroll sections systematically, let θF ,θQ be mutually disjoint vectors of distinct nodes of X/B of respective dimensions bF ,bQ,and let jF ,nF ,jQ,nQ be vectors of natural numbers, indexed commonly with θF ,θQ, respectively. Then we get iterated node classes 0 n F nF (θ )Q Q (θ )[ Γ [α ]] jF F jQ Q (mi) i (1.3) 0 n n | | = ...F F,i (θ )...Q Q,i (θ )...[ Γ [α ]] ∈ T m. +bF +bQ jF,i F,i jQ,i Q,i (mi) i n Thus via F nF (θ )Q Q (θ )[∗], we get a map jF F jQ Q (θF θQ) T0(X /B(θF θQ)) → T (X/B).

n F n and Qn are trivial unless 1 ≤ jn+1; − 1 −1 −1 ∗ r(n, j)∗ =0, otherwise.

DIFFERENTIAL EQUATIONS IN HILBERT-MUMFORD CALCULUS 121

Set F (n, j, θ, m, s)=θ∗(s| ) r(n, j)k F n+m(θ),s∈ A. (1.7) x Aθ n+m k k Then we have n n (1.8) Qj (θ) Γ(m)[s]=Qj (θ)[Γ(m)[s]] + F (n, j, θ, m, s). Proof. The case m = 1 is just (1.4). The general case is obtained by applying punctual transfer (cf. [6], §3.3) m−1 times to the result of Γ(1)[s], using [6], Prop. 3.19. Remark . ∗ 1.2 Because θx,θy both map to θ,wehaveθx = θy. Therefore we may denote both by θ∗ and write the map s → F (n, j, θ, m, s)as ∗ k n+m F (n, j, θ, m)=θ r(n, j)n+mFk (θ). k ∗ ∗ Also, θx(ω) = 0 (by residues), θx(1) = 1 (trivially). The same argument shows the following more general statement Proposition 1.3. We have, with the above notations, n F nF (θ )Q Q (θ ) Γ [s ] ... Γ [s ]= jF F jQ Q (m1) 1 (mr ) r nF nQ F (θF )Q (θQ)[Γ(m )[s1] ... Γ(m )[sr]] jF jQ 1 r n \n (1.9) + F nF (θ )F (n ,j ,θ ,m ,s )Q Q Q,i (θ \ θ ) jF F Q,i Q,i Q,i j j jQ\jQ,i Q Q,i i,j · [Γ(m1)[s1] ...Γ(mj )[sj ]...Γ(mr)].

Because F classes are represented by P1-bundles, they automatically have vanishing integrals, so a nice simple consequence of Proporsition 1.3 is Corollary . 1.4 We have n F nF (θ )Q Q (θ ) Γ [s ] ... Γ [s ]= jF F jQ Q (m1) 1 (mr ) r (1.10) n F nF (θ )Q Q (θ )[Γ [s ] ... Γ [s ]]. jF F jQ Q (m1) 1 (mr ) r Note that the tautological module T splits naturally as # T = Tθ. θ. where the sum is over all vectors of distinct nodes and Tθ. consists of the classes that come from the θ. boundary via a node scroll/section construction (though Tθ. is independent of the ordering of θ., it is convenient to specify the ordering). Thus # nF nQ θ. Tθ. = Fj (θF )Qj (θQ)TAθ. (X /B(θ.)) (1.11) F Q θ.=θF θQ, nF ,nQ,jF ,jQ where Xθ./B(θ.) is the desingularized boundary family corresponding to θ.,endowed with the node-preimage sections, and Aθ. is a coeﬃcient ring on Xθ./B(θ.) as above.

122 ZIV RAN

1.3. Standard model. We describe a standard model, actually just a notation change, for the tautological module T . This will be a free module Tˆ over a ˆ ∗ ∗ power series ring T0,inwhichF∗ (∗),Q∗(∗)andΓ(∗)[∗] become variables or formal symbols. This will enable us to express the structure of T in terms of standard operations such as diﬀerential operators. For each n ≥ 1lettn be a formal variable, let t0 =1,andset #∞ #∞ A ∞ = Atn,A ∞ (θ.) = A(θ.)tn n=0 n=0 as AB or AB(θ.)-module, respectively. Then we have an AB-algebra

Tˆ0 = AB[A ∞ ].

We think of generators αtn ∈ A ∞ as corresponding to Γ(n)[α].and assign them weight n. Likewise, ˆ T0,(θ.) = AB(θ.)[A ∞ (θ.)].

We set # # ˆ ˆ T0,∗ = T0,(θ.), T0,i = T0,(θ.). (θ.) |(θ.)|=i n n For each node θ, we designate formal variables φj (θ),χj (θ) corresponding to the n n node classes Fj (θ),Qj (θ). Now let θφ,θχ be disjoint collections of distinct nodes, and let nφ,jφ,nχ,jχ be correspondingly-indexed vectors of natural numbers. Then set 0 0 φnφ (θ )χnχ (θ )= φn(θ) χn(θ), jφ φ jχ χ j j (nφ,jφ,θφ) (nχ,jχ,θχ) # Tˆ = φnφ (θ )χnχ (θ )Tˆ . jφ φ jχ χ 0,θφ θχ Thus, Tˆ is generated by symbols of the form 0 φnφ (θ )χnχ (θ ) (t α ),α ∈ A , jφ φ jχ χ ni i i θφ θχ and is a direct sum of A(θ.) modules for the various collections (θ.) of distinct nodes. Moreover Tˆ is a Tˆ0,∗-module. Note the map

h : Tˆ → T 0 0 (1.12) nφ nχ nφ nχ h(φ (θ )χ (θ ) (t α )) = F (θ )Q (θ ) Γ [α ]. jφ φ jχ χ ni i jφ φ jχ χ (ni) i 

This h is a bijection under which the T0,∗-module structure corresponds to multiplication. We extend the boundary pullback map (1.1) to Tˆ in the obvious way:

b : Tˆ → Tˆ (1.13) 0 0 b(φnφ (θ )χnχ (θ ) (t α )) = φnφ (θ )χnχ (θ ) (t b(α ))). jφ φ jχ χ ni i jφ φ jχ χ ni i

DIFFERENTIAL EQUATIONS IN HILBERT-MUMFORD CALCULUS 123

Remark 1.5. Note that for any AB-linear map ψ : A → A, there is a derivation ψt ∂/∂t of T deﬁned by n n ψ(α),m= n, ψtn ∂/∂tn(tmα)= (1.14) 0,m= n; ∗ ∗ ψtn ∂/∂tn(φ∗(∗)||χ∗(∗)) = 0. → Similarly, if ψ : A AB1 is an AB-linear map, we can deﬁne a derivation ψφn(θ) ∂/∂t : Tˆ → T,ˆ j n n n ψ(α)φj (θ),m= n, (1.15) ψφ (θ) ∂/∂tn(tmα)= j 0,m= n; n ∗ ∗ || ∗ ∗ ψφj (θ) ∂/∂tn(φ∗( ) χ∗( )) = 0. Remark 1.6. Though not critical for our purposes, Tˆ canbemadeintoa commutative associative ring under the proviso that φ|χ monomials must involve only distinct nodes θ: i.e. ∗ ∗ (φ|χ)∗(θ)(φ|χ∗)(θ)=0; otherwise (i.e. where distinct θs are involved) φsandχs multiply formally. 1.4. Γ action. For enumerative purposes, a crucial feature of T is the weight- graded action by the discriminant Γ. The nonclassical (boundary) part of the action is described by the following rules. 0 − 0 j(ni j)ni n (Γ. Γ [α ]) = F i (θ)[ Γ [α ]] (1.16) (ni) i θ j (ni ) i 2 i 0

Here i(θx|y) refers to interior multiplication (see §1.5). The classical or interior part of the action of Γ on T0 is described by (1.19) 0 0 nj − Γ0( Γ [αj ]) = Γ [αj .αj ] Γ [αk] Γn [ωαj ] (nj ) (nj +nj ) (nk) 2 j j

124 ZIV RAN and of course ∂/∂tn =1A ∂/∂tn.Then

Γ0F = h(Γˆ0Fˆ), where 2 (1.20) ∂ n ∂ Γˆ0 := nn tn+n − tnω . ∂tn ∂tn 2 ∂tn n≤n n For example, n n Γˆ ((t α)(t α ))=nn t (α.α )− t (ω.α)(t α )− (t α)t (ω.α ),n= n . 0 n n n+n 2 n n 2 n n This will be amplified below. For later reference, we note the relation between Γ0 on T0(X/B), as given θ by (1.19), and the corresponding operator on T0(X /B(θ)) for a boundary family θ X /B(θ). The only difference is that ω = ωX/B is replaced by ωXθ /B(θ) = ω(−θx − (2) θy), where θx,θy are the node preimage sections. Consequently, if we let i be the derivation with respect to product defined by n (1.21) i(2)(σ)Γ [α]= Γ [σ.α]. (n) 2 (n) then we have

(2) (1.22) ΓXθ /B(θ),0 =Γ0,X/B|Xθ + i (θx + θy)

1.5. Interior multiplication. Given any class α ∈ A, there is an interior multiplication action i(α) on the tautological module T : this is determined by the following conditions (where we recall that a node θ is viewed as a map B(θ) → X ∗ and yields a pullback θ : A → AB(θ)): (i) i(α) is a derivation with respect to product; (ii) i(α)Γ(n)[β]=nΓ(n)[α.β]; n n ∗ n (iii) i(α)Fj (θ)[β]=Fj (θ)[i(α)β]+(θ (α))Fj (θ)[β]; n n ∗ n (iv) i(α)Qj (θ)[β]=Qj (θ)[i(α)β]+(θ (α))Qj (θ)[β]. In applications, α will usually be a section (hence disjoint from the node θ), so the second summand in the last two formulas is trivial. Therefore in such cases i(α) corresponds in the model Tˆ to the operator (1.23) δ(α):= ntnα∂/∂tn. n Similarly, the operator i(2)(α) deﬁned above corresponds to the derivation (2) n (1.24) δ (α)= tnα∂/∂tn. 2 n 1.6. S- transformation. We seek a transformation on the tautological mod- n n k ule taking Qj [α]toQj α. To this end, deﬁne rational numbers r(n, j) as in (1.7) (see Remark 1.2). Then set, as in (1.14) k n+m ∗ (1.25) φ(n, j, θ, m)= r(n, j)n+mφk (θ)θ , k 2 ˆ ∂ (1.26) S = φ(n, j, θ, m) n . ∂χj (θ) ∂tm

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Then Proposition 1.3 shows that Sˆ corresponds to an operator S on T such that n n − n Qj (θ)[Γ(m)[s]] = Qj (θ) [Γ(m)[s]] SQj (θ) Γ(m)[s] hence more generally (1.27) 0 0 n n F nF (θ )Q Q (θ )[ Γ [s.]] = (I − S) F nF (θ )Q Q (θ ) Γ [s.] . jF F jQ Q (m.) jF F jQ Q (m.) Note that Sˆb+1 =0whereb = dim(B). Consequently, (1.28) (I − S)−1 = I + S + ... + Sb.

2. Evolution equation To introduce our evolution equation, we need some notation. First recall the previously introduced derivations corresponding to α ∈ A ( see (1.23), (1.24)): ∂ δ(α)= ntnα ∂tn (2.29) n n ∂ δ(2)(α)= t α . 2 n ∂t n n Then set n − j +1 j (2.30) δn(θ)=−(n − j +1)δ(θ ) − jδ(θ )+ ψ (θ)+ ψ (θ) j x y 2 x 2 y where the ψ terms refer to the appropriate multiplication operators. This is a first-order differential operator. We will need to express the discriminant operator in terms of Tˆ with its Tˆ0- n module structure, which will involve rewriting terms like Qj (θ)[Γ(n)]intermsof n ˆ ˆ ˆ Qj (θ) Γ(n). To this end, let Γ be the operator on T corresponding to Γ. It is Γ whose powers we wish to compute, as this will yield powers of Γ. The idea is to achieve that via a change of variable. Thus set, using the notation of §1.6, (2.31) Γ=(˜ I − S)−1Γ(ˆ I − S). Then via Γˆk =(I − S)Γ˜k(I − S)−1, it suffices to compute powers of Γ.˜ But Γis˜ a relatively ’elementary’ operator: specifically, a second-order differential operator. To describe this, first let Γˆ0 be the extension of Γˆ0 already defined on Tˆ0, determined by the conditions ˆ n | n (2.32) Γ0(φj (θ) χj (θ)) = bθ, where bθ is the boundary operator. In the above notations, we have, by a direct computation, − ˜ ˆ j(n j)n ∗ n ∂ − n ∂ Γ=Γ0 + θxφj (θ) χj (θ) n 2 ∂tn ∂φj (θ) − n n − (2) ∂ (2.33) φj (θ)(δj+1(θ) δ (θx + θy)) n ∂φj (θ) n n − (2) ∂ + χj (θ)(δj (θ) δ (θx + θy)) n ∂χj (θ)

126 ZIV RAN

∗ n ∂ (2) where θ φ (θ) is as in Remark 1.5. Here the δ (θx + θy) term comes from the x j ∂tn difference between ωX/B and ωXθ /B(θ). Notice that because S does not involve the t variables, Γˆ0 coincides with the ’pure- t’ or classical portion of Γ.˜ Now one might simply wish to consider the generating function exp(γΓ)˜ which encodes information about the powers of the discriminant operator Γ (weight un- specified). As discussed in the Introduction, this is not sufficient for enumerative applications, which require monomials involving discriminants of different weights and external multiplications. Fortunately the extension is not difficult to obtain. To this end let α1, ..., αr ∈ A be a set of homogeneous elements. Let X/B be a family of nodal curves with associated flag-Hilbert scheme W m(X/B), and let L be a line bundle over X with associated tautological bundle Λm(L), which is a rank-m vector bundle over W m(X/B). The results of [5]and[6] show that Chern numbers of Λm(L), are given by linear combinations of monomials of the form (read left to right)

k2 kr (2.34) M =(Γ(1)[α1])(Γ(1)[α2])Γ ...(Γ(1)[αr])Γ

ni where αi = L . Accordingly, we define, extending the above, ∈ (2.35) G =exp(γΓ) exp( μiΓ(1)[αi]) T [[γ,μ1, ..., μr]], let (2.36) Gˆ =exp(γΓ)ˆ exp( μiαit1) ∈ Tˆ[[γ,μ1, ..., μr]] be the corresponding element, and (2.37) G˜ =(I − S)−1Gˆ(I − S) (see (1.28)). Note that the first exponential in (2.36) refers to composition of operators while the second refers to product in Tˆ0, which corresponds to product. We will use integral for an element of Tˆ to denote the integral of the corresponding element of T . Theorem 2.1. The following differential equations hold: (2.38) j(n − j)n ∂ G/˜ ∂ γ = Γˆ G˜ + θ∗φn(θ) ∂ G/˜ ∂ t − χn(θ) ∂ G/˜ ∂ φn(θ) 0 2 x j n j j θ,n,j θ,n,j − n n − (2) ˜ n φj (θ)(δj+1(θ) δ (θx + θy)) ∂ G/ ∂ φj (θ) θ,n,j n n − (2) ˜ n + χj (θ)(δj (θ) δ (θx + θy)) ∂ G/ ∂ χj (θ) ˜ ˜ ∗ n+1 ˜ n (2.39) ∂ G/ ∂ μi = t1αiG + θ (αi)φj (θ) ∂ G/ ∂ χj (θ). θ,n,j Moreover, 0 0,n = ∅ (2.40) φnφ (θ )χnχ (θ ) t α = φ jφ φ jχ χ ni i αi,nφ = ∅ X where is the length of nχ, i.e. the number of nodes.

DIFFERENTIAL EQUATIONS IN HILBERT-MUMFORD CALCULUS 127

This Theorem, together with the obvious initial value G˜(0, ..., 0) = 1 enables the computation of G˜, hence of G, hence of monomials M as in (2.34). Proof. To begin with, the first part of relation (2.40) is essentially obvious , as φ variables correspond to P1-bundles of type F . The second part follows from Corollary 1.4, as χ variables correspond to sections of type Q of the F -bundles, and as far as integrals are concerned, Q[α]isequivalenttoQα. Now the relation (2.38) encapsulates the computation of the Γ operator as carried out in [6], §2. Schematically, applying (−Γ) to a class of the form F [y], n F = Fj (θ), yields the sum of (i) the corresponding Q[y] class; n (ii) a class F [dy]whered is analogous to δj+1(θ)above; (iii) the class F [−Γy]. Applying −ΓtoQ[y] yields a sum of only the last two types (with j in place of j +1). The first and second terms on the right of (2.38) correspond to the interior and boundary part of applying Γ to polyblock diagonals and generally to the polyblock factor of an FQ- monomial as in (1.3) (see [6], Thm. 2.23 ). The third term represents item (i) above for the action of Γ on each F factor. In the final summation, the φδ term represents item (ii) above for each F , while the χδ term represents the corresponding term for each Q (see [6], Theorem 2.24 and Remark 2.26 ). The δ(2) term are the result of ’ω adjustment’ as in (1.22), i.e writing

ωXθ /B(θ) = ωX/B ⊗OXθ (−θx − θy). Because diﬀerent nodes θ are disjoint, no products of θsappear. Equation (2.39) is a consequence of the Transfer Theorem of [6](seeTheorem n+1 3.4 and display (3.1.19)). The second term is a reﬂection of the Fj (θ)termin n the transfer of Qj (θ). References [1]D.A.CoxandS.Katz,Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, vol. 68, American Mathematical Society, Providence, RI, 1999. MR1677117 (2000d:14048) [2] K. Hori et al., Mirror symmetry, Clay math. monographs, vol. 1, Amer. Math. Soc., 2003. [3] G. Liu, The macnodal package for intersection theory on hilbert schemes of nodal curves,web interface (small jobs only) at http://gwoho.com/macnodal/index.html.a?format=html;source + executable + instructions at http://math.ucr.edu/~ziv/. [4] I. G. Macdonald, Symmetric products of an algebraic curve, Topology 1 (1962), 319–343. MR0151460 (27 #1445) [5] Z. Ran, Geometry on nodal curves,Compos.Math.141 (2005), no. 5, 1191–1212, DOI 10.1112/S0010437X05001466. MR2157135 (2006i:14024) [6] Z. Ran, Tautological module and intersection theory on Hilbert schemes of nodal curves,Asian J. Math. 17 (2013), no. 2, 193–263, DOI 10.4310/AJM.2013.v17.n2.a1. MR3078931 [7] E. Witten, Two-dimensional gravity and intersection theory on moduli space, Surveys in differential geometry (Cambridge, MA, 1990), Lehigh Univ., Bethlehem, PA, 1991, pp. 243–310. MR1144529 (93e:32028)

Department of Mathematics, University California, Riverside Surge Facility, Big Springs Road, Riverside, California 92521 E-mail address: ziv.ran @ucr.edu

Contemporary Mathematics Volume 647, 2015 http://dx.doi.org/10.1090/conm/647/12957

Weak positivity via mixed Hodge modules

Christian Schnell Dedicated to Herb Clemens

Abstract. We prove that the lowest nonzero piece in the Hodge ﬁltration of a mixed Hodge module is always weakly positive in the sense of Viehweg.

Foreword “Once upon a time, there was a group of seven or eight beginning graduate students who decided they should learn algebraic geometry. They were going to do it by reading Hartshorne’s book and solving some of the exercises there; but one of them, who was a bit more experienced than the others, said to his friends: ‘I heard that a famous professor in algebraic geometry is coming here soon; why don’t we ask him for advice.’ Well, the famous professor turned out to be a very nice person, and oﬀered to help them with their reading course. In the end, four out of the seven became his graduate students . . . and they are very grateful for the time that the famous professor spent with them!”

1. Introduction 1.1. Weak positivity. The purpose of this article is to give a short proof of the Hodge-theoretic part of Viehweg’s weak positivity theorem. Theorem 1.1 (Viehweg). Let f : X → Y be an algebraic fiber space, meaning a surjective morphism with connected fibers between two smooth complex projective ∈ N ν algebraic varieties. Then for any ν ,thesheaff∗ωX/Y is weakly positive. The notion of weak positivity was introduced by Viehweg, as a kind of birational version of being nef. We begin by recalling the – somewhat cumbersome – definition. Let F be a torsion-free coherent sheaf on a smooth algebraic variety X.Weshall denote by Sn(F )then-th symmetric power of F ,andbySˆn(F ) its reflexive hull. A more concrete definition is that

n n ∗ Sˆ (F )=j∗S (j F ), where j is the inclusion of the maximal open subset over which F is locally free. The formula holds because the complement has codimension at least two in X.

2010 Mathematics Subject Classiﬁcation. Primary 14E30; Secondary 14F10. Key words and phrases. Weak positivity; Mixed Hodge module.

c 2015 American Mathematical Society 129

130 CHRISTIAN SCHNELL

Definition 1.2. A torsion-free coherent sheaf F is weakly positive on an open subset U ⊆ X if, for every ample line bundle L on X,andforeverym ∈ N,the restriction morphism $ % 0 ˆmn F ⊗ ⊗n → ˆmn F ⊗ ⊗n H X, S ( ) L S ( ) L U is surjective for all sufficiently large n ∈ N. If we think of Sˆmn(F ) ⊗ L⊗n as being the mn-th symmetric power of the (non-existent) object F ⊗ L⊗1/m, then the definition means the following: after tensoring F by an arbitrarily small fraction of an ample line bundle, all sufficiently large symmetric powers are generated over U by their global sections. Example 1.3. For locally free sheaves, weak positivity on X is equivalent to being nef. For line bundles, being weakly positive is the same thing as being pseudo-effective. The most notable application of the weak positivity theorem is Viehweg’s proof of the Iitaka conjecture over a base of general type. Let f : X → Y be an algebraic fiber space, with general fiber F ; Iitaka’s conjecture predicts that κ(X) ≥ κ(Y )+κ(F ). Viehweg proved the conjecture when Y is of general type. Roughly speaking, he ν uses the weak positivity of f∗ωX/Y , together with the fact that ωY is big, to produce ν sufficiently many global sections of ωX . The survey article [Vie83] contains a nicely written account of these matters.

1.2. Related results. The idea of using methods from Hodge theory to prove the positivity of certain sheaves goes back at least to Fujita and Kawamata. To put Viehweg’s theorem in context, let me briefly mention a few other positivity theorems for the direct image of ωX/Y and its powers: (1) Kawamata [Kaw81] proved that if the fiber space f : X → Y is “nice”, then f∗ωX/Y is locally free and nef. Nice means, roughly speaking, that the singularities of f should occur over a normal crossing divisor, and that the local monodromy should be unipotent. The proof uses Hodge theory. (2) Viehweg deduced from Kawamata’s result that f∗ωX/Y is always weakly positive; by studying certain well-chosen branched coverings of X,he ν ≥ obtained the same result for f∗ωX/Y with ν 2. (3) Kollár gave a simpler proof of Viehweg’s theorem using his vanishing theorem for higher direct images of dualizing sheaves. (4) More recently, Fujino, Fujisawa, and Saito [FF14, FFS14]provedagen- eral nefness theorem for mixed Hodge modules that has Kawamata’s theorem as a special case. There are many other results of this type; the list above contains only those that are most closely related to the topic of this paper.

1.3. Main theorem. Let H be a polarizable variation of Hodge structure on the complement of a divisor D in a smooth projective algebraic variety X, and let p ∈ Z be the largest integer for which the Hodge bundle F pH is nontrivial. The general philosophy that emerges from Kawamata’s work [Kaw81]isthatF pH extends in a canonical way to a nef vector bundle on X, provided that D is a normal

WEAK POSITIVITY VIA MIXED HODGE MODULES 131 crossing divisor and H has unipotent local monodromy. (Without these assumptions, there are examples [FF14, Section 8] where nefness fails.) The analogy with Viehweg’s theorem suggests that, without these assumptions, there should still be a canonical extension of F pH that is weakly positive. This expectation turns out to be correct, if one takes for the extension of F pH the one given by Saito’s theory of mixed Hodge modules. Here is the result, proved in collaboration with Mihnea Popa. Theorem 1.4. Let M be a mixed Hodge module on a smooth complex projective variety X. If the underlying ﬁltered D-module satisﬁes Fk−1M =0, then the coherent sheaf FkM is weakly positive on the open subset where M is a variation of mixed Hodge structure. Viehweg’s theorem (in the case ν = 1) is an immediate consequence. Given a morphism f : X → Y as in Theorem 1.1, we let M be the direct image of the trivial 0 QH Hodge module on X; to be precise, M = H f∗ X [dim X], in Saito’s notation. Setting k =dimY − dim X,itcanbeshownthat

Fk−1M =0 and FkM%f∗ωX/Y , and Theorem 1.4 implies that this sheaf is weakly positive on the smooth locus of f. ν ≥ The weak positivity of f∗ωX/Y for ν 2 follows as in Viehweg’s original argument by considering certain branched coverings of X.

2. Mixed Hodge modules Although it looks more general, Theorem 1.4 is not really a new result, because it could be deduced from Kawamata’s theorem. The point of presenting it is that mixed Hodge modules appear to be the natural setting: as we will see towards the end of the talk, the proof of the theorem is extremely short. This may be only of academic interest in this case, where we are basically reproving an existing result – but in other situations, the use of mixed Hodge modules may allow us to go much further than existing methods.1 Before explaining the proof of Theorem 1.4, I would like to say a few words about mixed Hodge modules and their applications to algebraic geometry; I will also try to motivate their use with a speciﬁc example. To simplify the discussion, let me concentrate on the case of pure Hodge modules.

2.1. Variations of Hodge structure. Hodge modules are a generalization of variations of Hodge structure. For applications to algebraic geometry, the essential features of a variation of Hodge structure H are the following: H ∇ H→ 1 ⊗H (1) A holomorphic vector bundle with a flat connection : ΩX ; (2) a Hodge filtration F •H by holomorphic subbundles, subject to the relation ∇ pH ⊆ 1 ⊗ p−1H (F ) ΩX F , called Griffiths transversality (by everyone except Griffiths, who calls it the infinitesimal period relation).

1At this point in the lecture, Wilfried Schmid interjected, “I think it’s fair to say that mixed Hodge modules have really been greatly underused.”

132 CHRISTIAN SCHNELL

In practice, there are various pieces of additional data: a rational structure; a polarization; a weight filtration (in the mixed case); etc. But the vector bundles H and F •H and their properties are what matters most for algebraic geometers. Variations of Hodge structure are “smooth” objects, arising for example from families of smooth projective varieties. One can include objects with singularities by generalizing vector bundles with flat connection to D-modules. In fact, (H, ∇) is a special case of a regular holonomic D-module. The connection gives rise to an action by tangent vector fields (= linear differential operators of order one), accordingtotheformula

ξ · s = ∇ξ(s)forξ ∈ Γ(U, TX )ands ∈ Γ(U, H). This makes H into a left D-module: on the one hand, ∇ satisﬁes the Leibniz rule

∇ξ(fs)=(ξf)s + f ∇ξ(s), which amounts to the relation [ξ,f]=ξf; on the other hand, ∇ flat means that $ % $ % ∇ξ ∇η(s) −∇η ∇ξ(s) = ∇[ξ,η](s), which amounts to the relation ξη − ηξ =[ξ,η]. This gives H the structure of a left module over the sheaf of linear differential operators DX . The Hodge filtration defines a filtration of H that is compatible with the order of differential operators. −k Indeed, if we set FkH = F H to get an increasing filtration, then we have

TX · FkH⊆Fk+1H because of Griffiths transversality. The resulting D-module is regular holonomic; its so-called characteristic variety, a subset of the cotangent bundle T ∗X, is precisely the zero section. This corresponds to the fact that, locally on X, solutions to the equation ∇s = 0 can be analytically continued in every direction. 2.2. Hodge modules. Now let X be a smooth quasi-projective variety; Hodge modules can be defined much more generally, including on singular varieties, but we shall focus on this case because it simplifies the discussion. The essential features of a Hodge module are then the following:

(1) A regular holonomic DX -module M; (2) a compatible filtration F•M by coherent OX -submodules that is “good”, meaning that TX · FkM⊆Fk+1M, with equality for k 0. As in the case of variations of Hodge structure, there are several additional pieces of data, such as a polarization; a rational structure (in the form of a perverse sheaf with coefficients in Q); a weight filtration (in the mixed case); etc. There is also a long list of axioms that need to be satisfied in order for the pair (M,F•M)tobe called a Hodge module. A basic fact is that every variation of Hodge structure defines a Hodge module; we saw already more or less how this works. In the other direction, every Hodge module is generically a variation of Hodge structure. Example 2.1. In the situation of Theorem 1.4, there is a Zariski-open subset over which M is a variation of mixed Hodge structure. Under the indexing con- −k+1 ventions explained above, the condition Fk−1M = 0 translates into F H =0, and the coherent sheaf FkM in the theorem is an extension of the Hodge bundle F −kH. Note that the theorem allows the possibility that Supp M = X:inthat case, H = 0 and the assertion about weak positivity becomes trivial.

WEAK POSITIVITY VIA MIXED HODGE MODULES 133

2.3. Hyperplane sections and residues. Instead of boring the reader with a list of axioms, let me motivate the use of D-modules by an example. The example is the residue description for the cohomology of hyperplane sections, something that I thought about a lot for my dissertation. This is a good place to acknowledge the great influence that Herb Clemens has had on my mathematical interests: I basically learned about mixed Hodge modules by trying to understand some of his constructions with residues and differential operators. Let X be a smooth projective variety of dimension n, and let Y ⊆ X be a smooth and very ample divisor in X. According to the Lefschetz theorems, the cohomology groups Hk(Y )=Hk(Y,C) are determined by those of X,withthe exception of the so-called variable part $ % n−1 n−1 → n+1 Hvar (Y )=ker H (Y ) H (X)(1) , defined as the kernel of the Gysin morphism. The variable part can be described k ∗ very nicely by residues. We denote by ΩX ( Y ) the sheaf of meromorphic k-forms on X that are holomorphic on X \ Y , but may have poles along Y . By Grothendieck’s theorem, the hypercohomology of the algebraic de Rham complex O ∗ →d 1 ∗ →d ··· →d n ∗ X ( Y )ΩX ( Y ) ΩX ( Y ) computes H∗(X \ Y ); this is a consequence of the fact that X \ Y is affine. From the long exact cohomology sequence ···→Hn−2(Y )(−1) → Hn(X) → Hn(X \ Y ) → Hn−1(Y )(−1) → Hn+1(X) →··· we obtain a short exact sequence

→ n → n \ → n−1 − → 0 H0 (X) H (X Y ) Hvar (Y )( 1) 0

→ → $ % 0 n ∗ ResY H X, ΩX ( Y )

The arrow labeled ResY is the so-called residue mapping; under our assumptions on Y , it is surjective. Carlson and Griffiths [CG80] have shown that, if the line O n−1 bundle X (Y ) is sufficiently ample, then the Hodge filtration on Hvar (Y )isthe filtration by pole order. More precisely, their result is that $ % 0 n → n−k n−1 ResY : H X, ΩX (kY ) F Hvar (Y ) n−1,0 is surjective. For k = 1, this recovers the familiar fact that Hvar (Y ) is generated by residues of logarithmic n-forms on X. The result of Carlson and Griffiths also works in families. Let L be a very ample line bundle on X, and let P = |L| denote the linear system of its sections. Each point p ∈ P corresponds to a hypersurface Yp ⊆ X;wedenotebyP0 ⊆ P the setofpointswhereY is smooth, and by j : P → P the inclusion. Let p ! 0 " Y = (p, x) ∈ P × X x ∈ Yp ⊆ P × X be the incidence variety. Just as above, we can use residues to express the variable n−1 part in the variation of Hodge structure R p1∗QY ; more precisely, what we get is a description of the underlying flat vector bundle (Hvar, ∇) and the Hodge filtration • F Hvar on it.

134 CHRISTIAN SCHNELL

2.4. Extension to singular hyperplane sections. It is an interesting question whether the description from above can also tell us something about singular hypersurfaces. The answer is that we can use residues to construct a natural extension of the bundle H from P0 to all of P ; this idea is due to Clemens. Suppose for a moment that we have a relative n-form $ % ∈ × n ∗Y ω Γ U X, ΩP ×X/P ( ) with poles along Y ,deﬁnedoveranopensubsetoftheformU × X. At each point p ∈ U ∩ P0, the hypersurface Yp is smooth, and so we have a well-deﬁned residue $ % ∈ n−1 ResYp ω {p}×X Hvar (Yp).

In this way, we obtain a holomorphic section of the bundle Hvar over the open set U ⊆ P ; for simplicity, we shall denote it by the symbol Res(ω). We can then deﬁne a subsheaf M⊆j∗H by the following rule: its sections over an open subset U ⊆ P are given by $ M ∈ ∩ H ∈ × n ∗Y Γ(U, )= s Γ(U P0, ) s=Res(ω)forsomeω Γ U X, ΩP ×X/P ( ) .

It has a natural ﬁltration F•M by pole order, deﬁned as follows: $ M ∈ M ∈ × n Y Γ(U, Fk )= s Γ(U, ) s =Res(ω)forsomeω Γ U X, ΩP ×X/P (k ) .

The point is that we consider only those sections of j∗H that are the residue of a meromorphic form with poles along Y ; note that the meromorphic form needs to be defined on all of U, including over points of P \ P0. This makes sense because the incidence variety Y is actually a smooth hypersurface in P × X. Provided that the line bundle L is sufficiently ample, the theorem of Carlson and Griffiths from above shows that we have n−k FkM % Fk−nHvar = F Hvar and M %Hvar . P0 P0

Now M is naturally a left DP -module: differential operators on P act by differentiating the coefficients of ω. More precisely, given a vector field ξ ∈ Γ(U, TP ), let ξ ∈ Γ(U × X, TP ×X ) denote the obvious lifting to the product; then if s =Res(ω), we can take the Lie derivative and define $ % ξ · s =Res Lξ ω . Concretely, this means that we contract dω with ξ, and then take the residue of the resulting relative n-form. It is easy to see that the filtration F•M is compatible with the order of differential operators – after all, differentiating k times will increase the order of the pole by k. One can also compute the characteristic variety Ch(M) and prove in this way that M is holonomic. The characteristic variety turns out to be closely related to the set ! " Y = (p, x) ∈ Y Yp is singular at x of singular points in the fibers of p1 : Y → P ; this is not surprising, because those are exactly the points where one cannot take a residue in the classical sense. Both Y and Y are projective bundles over X, of dimension dim X +dimP − 1and dim P − 1, respectively; they naturally embed into the projectivization of T ∗P .

WEAK POSITIVITY VIA MIXED HODGE MODULES 135

Lemma 2.2. If Hvar =0 , then the characteristic variety of M is given by $ % $ % Ch(M)= the zero section of T ∗P ∪ theconeoverY . In particular, both components of Ch(M) are of dimension dim P , which means that M is holonomic.

One can show that (M,F•+nM) is part of a Hodge module on P that naturally extends the variation of Hodge structure Hvar. You can ﬁnd the details in the paper [Sch12b]; the key point in the proof is that the incidence variety Y is smooth.

2.5. Conclusion. The message to take away from this is that Hodge modules are the correct generalization of variations of Hodge structure. Indeed, here we have one example where we get a natural extension of a variation of Hodge structure (with the help of residues) . . . and it turns out that the extension is precisely the same as the one given by Saito’s theory! I should add that I have found this particular example to be very useful for learning about mixed Hodge modules.

3. Proof of the main theorem The remainder of the paper is devoted to the proof of Theorem 1.4. The proof closely follows Koll´ar’s, but we replace certain geometric arguments by abstract results about mixed Hodge modules. As I have said before, I do not claim that this proof is in any way simpler than the original one; only that it is shorter, and therefore perhaps better suited for generalizations.

3.1. Saito’s vanishing theorem. The main ingredient is the following vanishing theorem for the ﬁrst nonzero sheaf in the Hodge ﬁltration of a mixed Hodge module [Sai91, Theorem 1.2].

Theorem 3.1 (Saito). Let (M,F•) be the ﬁltered D-module underlying a mixed Hodge module on a smooth projective variety X.IfFk−1M =0,then $ % i H X, ωX ⊗ L ⊗ FkM =0 for every i>0 and every ample line bundle L on X.

⊗(n+1) The vanishing theorem implies that ωX ⊗ L ⊗ FkM is 0-regular (in the sense of Castelnuovo-Mumford), and therefore always globally generated (here and below, n =dimX). To prove that FkM is weakly positive, we have to find a way to raise the term FkM in the formula to a large power. In Kollár’s proof, this is accomplished by considering the fiber product X ×Y ···×Y X → Y ;itrequires some analysis of the singularities. We shall replace this geometric argument by the following abstract result about mixed Hodge modules.

3.2. A useful lemma. Next, we recall a useful lemma about restriction to submanifolds; it appears in [Sch12a, Lemma 2.17]. The symbol i!M in the statement refers to a certain pullback operation on mixed Hodge modules (in the derived category); the precise definition does not matter for our purposes. Lemma 3.2. Let M be a mixed Hodge module on a smooth complex algebraic variety X,andlet(M,F) denote the underlying filtered DX -module. Let i: Z→ X be the inclusion of a smooth subvariety of codimension r,andleti!(M,F) denote ! b the complex of filtered DZ -modules underlying the object i M ∈ D MHM(Z).

136 CHRISTIAN SCHNELL

(a) For every k ∈ Z, there is a canonical morphism ! ∗ Fk−r i (M,F) → Li (FkM)[−r] b O in the derived category Dcoh( Z ). (b) The morphism in (a) is an isomorphism over the locus where M is smooth. ! Proof. The functor i is the right adjoint of i∗, and since i is a closed embed- ! b ding, we have the adjunction morphism i∗i M →$ M in D %MHM(X). Passing to ! filtered D-modules, we then get a morphism Fk i∗i (M,F) → FkM.Nowi is a closed embedding of codimension r, and because of how the direct image functor is defined in [Sai88, 2.3], we have a canonical morphism $ % $ % ! ! i∗ ωZ ⊗ Fk−r i (M,F) → ωX ⊗ Fk i∗i (M,F) . ∗ −1 Let ωZ/X = ωZ ⊗ i ω . Composing the two morphisms, we obtain X $ % ! i∗ ωZ/X ⊗ Fk−r i (M,F) → FkM. ! ∗ On the level of coherent sheaves, the functor Li = ωZ/X [−r] ⊗ Li is the right adjoint of i∗; by adjunction, we therefore get the desired morphism ! M → −1 ⊗ ! M ∗ M − Fk−r i ( ,F) ωZ/X Li (Fk )=Li (Fk )[ r] b O in the derived category Dcoh( Z ). Now let us prove (b). After restricting to the open subset where M is smooth, we may assume that M is the mixed Hodge module associated with a variation of mixed Hodge structure; in particular, all the sheaves FkM,aswellasM itself, are locally free. In this situation, the complex i!M is concentrated in degree r,and Hri!M is simply the restriction of M(r)toZ. It follows that ! ∗ Fk−r i (M,F)=i (FkM)[−r]. ∗ ∗ On the other hand, Li (FkM)=i (FkM) because FkM is locally free. It is then easy to see from the construction that the morphism in (a) is an isomorphism. 3.3. The proof. Now consider a mixed Hodge module M as in Theorem 1.4. Let (M,F•M) denote the underlying filtered D-module; without loss of generality, we may assume that Supp M = X. We apply Lemma 3.2 to the diagonal embedding Δ: X→ X ×···×X; if there are m factors, its codimension is r =(m − 1)n.OnXm,wehaveamixed Hodge module M m = M ··· M. The Hodge filtration on the underlying D-module Mm is defined by convolving the filtrations on the individual factors; in particular, m m m Fmk−1M =0 and FmkM =(FkM) . Lemma 3.2 thus gives us a canonical morphism $ % $ % ! m ∗ m Fmk−rΔ M ,F → LΔ FmkM [−r] in the derived category. If we take cohomology in degree r,weobtain ⊗m Fmk−r N→(FkM) , where N is the D-module underlying a certain mixed Hodge module on X.The important thing is that this morphism is an isomorphism over the open set U ⊆ X

WEAK POSITIVITY VIA MIXED HODGE MODULES 137 where M is a variation of mixed Hodge structure. In fact, the proof of Lemma 3.2 shows that, over U, both sides are just isomorphic to the m-th tensor power of the Hodge bundle F −kH. Now we can easily show that FkM is weakly positive on U. Fix a very ample line bundle L on X. Theorem 3.1 implies that the sheaf ⊗(n+1) ωX ⊗ L ⊗ Fmk−r N is 0-regular and therefore globally generated. Because the morphism ⊗(n+1) ⊗(n+1) ⊗m ωX ⊗ L ⊗ Fmk−r N→ωX ⊗ L ⊗ (FkM) is an isomorphism over U, it follows that the sheaf ⊗(n+1) ⊗m ωX ⊗ L ⊗ (FkM) is generated over U by its global sections. Since m was arbitrary, this implies that FkM is weakly positive on U by [Vie83, Lemma 1.4]. Acknowledgement I thank the local organizers (Mirel Caib˘ar, Ana-Maria Castravet, Gary Kennedy, and Emanuele Macr`ı) for making the conference a success. I am grateful to Mihnea Popa for many discussions about Hodge modules and weak positivity, and to Mark Green for pointing out the correct reference for the result by Carlson and Griﬃths. I thank an anonymous referee for several suggestions.

References [CG80] J. A. Carlson and P. A. Griffiths, Infinitesimal variations of Hodge structure and the global Torelli problem,Journées de Géometrie Algébrique d’Angers, Juillet 1979/Alge- braic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980, pp. 51–76. MR605336 (82h:14006) [FF14] O. Fujino and T. Fujisawa, Variations of mixed Hodge structure and semipositivity theorems, Publ. Res. Inst. Math. Sci. 50 (2014), no. 4, 589–661, DOI 10.4171/PRIMS/145. MR3273305 [FFS14] O. Fujino, T. Fujisawa, and M. Saito, Some remarks on the semipositivity theorems, Publ. Res. Inst. Math. Sci. 50 (2014), no. 1, 85–112, DOI 10.4171/PRIMS/125. MR3167580 [Kaw81] Y. Kawamata, Characterization of abelian varieties, Compositio Math. 43 (1981), no. 2, 253–276. MR622451 (83j:14029) [Sai88] M. Saito, Modules de Hodge polarisables,Publ.Res.Inst.Math.Sci.24 (1988), no. 6, 849–995 (1989). MR1000123 (90k:32038) [Sai91] M. Saito, On Kollár’s conjecture, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), 1991, pp. 509–517. MR1128566 (92i:14007) [Sch12a] Ch. Schnell, Complex analytic Néron models for arbitrary families of intermediate Ja- cobians, Invent. Math. 188 (2012), no. 1, 1–81. MR2897692 [Sch12b] Ch. Schnell, Residues and filtered D-modules, Math. Ann. 354 (2012), no. 2, 727–763. MR2965259 [Vie83] E. Viehweg, Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces, Algebraic varieties and analytic varieties (Tokyo, 1981), Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 329–353. MR715656 (85b:14041)

Department of Mathematics, Stony Brook University, Stony Brook, New York 11794 E-mail address: [email protected]

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CONM 647 og hoyadAgbacGeometry Algebraic and Theory Hodge

This volume contains the proceedings of a conference on Hodge Theory and Classical Algebraic Geometry, held May 13–15, 2013, at The Ohio State University, Columbus, OH. Hodge theory is a powerful tool for the study and classiﬁcation of algebraic varieties. This volume surveys recent progress in Hodge theory, its generalizations, and applications. The topics range from more classical aspects of Hodge theory to modern developments in compactiﬁcations of period domains, applications of Saito’s theory of mixed Hodge modules, and connections with derived category theory and non-commutative motives. • end ta. Editors al., et Kennedy

ISBN 978-1-4704-0990-6

9 781470 409906

CONM/647 AMS