Hodge theory and boundaries of moduli spaces
Phillip Griffiths∗
Based on correspondence and discussions with Mark Green, Colleen Robles and Radu Laza
∗Institute for Advanced Study
1/82 I The general theme of this talk is the use of Hodge theory to study questions in algebraic geometry.
I This is a vast and classical subject; here we shall focus on one particular aspect.
I In algebraic geometry a subject of central interest is the study of moduli spaces M and their compactifications M.
I Points in ∂M = M\M correspond to singular varieties X0 that arise from degenerations X → X0 of a smooth variety X .
2/82 I It is well known and classical that X0 may be “simpler” than X and understanding X0 may shed light on X . For an example, any smooth curve X may be degenerated to a stick curve
1 X0 = = configuration of P ’s
and deep questions about the geometry of a general X may frequently be reduced to essentially combinatorial ones on X0.
3/82 I The central thesis of this talk is Hodge theory may be used to guide the study of degenerations X → X0. Underlying this thesis are the points ∗ I the polarized Hodge structure on H (X , Z) (mod torsion) is the basic invariant of a smooth projective variety X ; I the study of degenerations (V , Q, F ) → (V , Q, W , Fo) of polarized Hodge structures is fairly highly developed;
I this understanding may be used to suggest what the possible degenerations X → X0 should be and where to look for them.
4/82 This thesis is well known and has been effectively used in the classical case when the Hodge structures are parametrized by a Hermitian symmetric domain and 1-parameter degenerations are used. It has been utilized in the non-classical case mainly for 1-parameter degenerations of Calabi-Yau threefolds, but to my knowledge not so much in other non-classical cases or for several parameter degenerations other than for the case of algebraic curves.
5/82 I the structure of this talk will be
I discussion of some of the basic aspects of Hodge theory;
I the example of algebraic curves, which indicates how the study might go in the classical case;
I discussion of the first non-classical case, namely that of an algebraic surface X of general type with h2,0(X ) = 2, 1,0 2 h (X ) = 0 and KX = 2; this is based on correspondence with Mark Green, and also with Colleen Robles and Radu Laza, and reflects developments in a very early stage.
6/82 I A final general point: a schematic of how Hodge theory interacts with algebraic geometry for the above surface is canonical 0 2 2,0 ←→ H (ΩX ) = H (X ) series |KX |
bi-canonical variation of Hodge structure ←→ 1 series |2KX | associated to H (C), C ∈ |KX | . . which arises from
KX = KC C. .
7/82 Definition A polarized Hodge structure (PHS) of weight n is given by (V , Q, F •) where
I V is a Q-vector space (in practice we will have V ⊃ VZ); I Q · V ⊗ V → Q is a non-degenerate form with Q(u, v) = (−1)nQ(v, u); • n n−1 0 I F is a decreasing filtration F ⊂ F ⊂ · · · ⊂ F = VC satisfying
p n−p+1 ∼ F ⊕ F −→ VC, 0 5 p 5 n;
p n−p+1 I Q(F , F ) = 0;
I Q(v, Cv¯) > 0 for v 6= 0, where C is the Weil operator.
8/82 q q,p Setting V p,q = F p ∩ F = V , the third condition above is
p,q p,q q,p VC = ⊕V where V = V and Cv = i p−q on V p,q. Example Let X ⊂ PN be a smooth projective variety of dimension d and L ∈ H2(X , Q) the restriction of the generator of a hyperplane class in H2(PN , Z). Then multiplication by L gives a nilpotent operator on H∗(X , Q)
m m+2 L : H (X , Q) → H (X , Q).
9/82 The Hard Lefschetz theorem k m−k ∼ m+k L : H (X , Q) −→ H (X , Q) implies that the action of L may be uniquely extended to the action of an sl2-triple {L, Y , Λ} where m Y = (m − d)Id on H (X , Q) ∗ is the grading. This sl2-action decomposes H (X , Q) into irreducibles whose basic building blocks are the lowest weight spaces
k+1 m−k n m−k L m+k+2 o H (X , Q)prim = ker H (X , Q) −−→ H (X , Q) = V of the irreducible summands.
10/82 Setting Q(u, v) = (Lk u · v)[X ] and using the decomposition n−k H (X , C)prim induced by the Hodge decomposition Hm−k (X , C) = ⊕ Hp,q(X ) gives a PHS of weight p+q=m−k m − k. Instead of the usual Hodge diamond, we will picture the groups in the first quadrant in the (p, q) plane. q
dim X = 1 L r r p ↑r 0 1 H (ΩX ) = holomorphic 1-forms on the compact Rieman surface X .
11/82 dim X = 2 and X is regular (H1,0(X ) = 0)
0 LH (X ) H rr HHj 1,1 H (X )prim r r
0 r 2 H (ΩX )
12/82 dim X = 1 The PHS on H1(X ) for X a compact Riemann surface is classically represented by the matrix of periods
γ 0 1 ω , , ω ∈ H ΩX ˆδ,γ ··· δ Even more classically, if X is given as the Riemann surface associated to an algebraic curve, e.g., the one of genus g given in C2 by
2g+2 2 Y f (x, y) = y − (x − ai ) = 0, ai distinct i=1 g(x) dx 2g(x) dx ω = = , 0 5 deg g(x) 5 g − 1 y fy (x, y)
13/82 the periods are δ ... a1 a2 a3 a4 γ
The closure in P2 of the above affine curve has a singular point at [0, 0, 1] and the condition deg g(x) 5 g − 1 is the one that gives a holomorphic 1-form on X .
14/82 dim X = 2 The surfaces we shall be considering are minimal, of general type and have
2,0 1,0 2 h (X ) = 2, h (X ) = 0, KX = 2.
They are extremal; i.e., they have the maximum h2,0(X ) subject to the other conditions above. They were analyzed classically by Castelnuovo and more recently by Horikawa. A general one is the minimal desingularization of the quartic 4 surface on P given in affine coordinates x1, x2, x3, x4 by ( f (x) = h(x) − k(x)2 = 0 2 g(x) = x1 − x2 = 0 where h(x), k(x) are a general cubic, quadric.
15/82 This surface has as singularity a double curve with eight pinch points. Think of it as an analogue of the smooth curve with affine equation 2g+2 2 Y y − (x − ai ) = 0 i=1 encountered above. It is the first regular, non-classical extremal surface. We will say more about it later.
16/82 The holomorphic 2-forms are
`(x)dx ∧ dx ω = 3 4 , deg `(x) 1 f (x)g (x) 5 x1 x2 and where `(x) vanishes on the double conic so that ω is holomorphic on X . With these conditions we find that h2,0(X ) = 2. Think of ω as the analogue of g(x) dx where fy (x) deg g(x) 5 g − 1 above. One has b2(X ) = 32, so that 1,1 h (X )prim = 31 and the PHS is given by the matrix of periods.
17/82 The surface is more conveniently given in the homogenous coordinate [t0, t1, x3, x4] in the weighted projective space P(1, 1, 2, 2) by
2 2 2 2 2 2 t0 G(t0 , t0t1, t1 , x3, x4) = F (t0 , t0t1, t1 , x3, x4) .
For fixed [t0, t1] we obtain genus 3 plane quartic curves which constitute the pencil |KX |.
18/82 Definition D = period domain given by {(V , Q, F •), dim V p,q = hp,q} Dˇ = compact dual given by {(V , Q, F •), Q(F p, F n−p+1) = 0}. If G = Aut(V , Q), then
D = GR/H, H compact ∩ ˇ D = GC/P, P parabolic and H = P ∩ GR.
19/82 Example
∼ t D = Hg = {Z = g × g matrix, Z = Z and Im Z > 0} = Sp(2g, R)/U(g) ˇ 2g 2g D = GrassL(g, C ) = Lagrangian g-planes in (C , Q) 0 I where Q = −I 0 .
The embedding D ,→ Dˇ is
Z Z → span of the columns of . Ig
1 For g = 1 we have H ,→ P and GR = SL2(R) acting by linear fractional transformations.
20/82 Example
ˇ D = GrassQ (2, VC) = {2-planes E : Q(E, E) = 0} ∪ D = 2-planes E with Q(E, E) < 0.
The GR = SO(4, h) and D = GR/U(2) × SO(h) is not an Hermitian symmetric domain (HSD). Definition A variation of Hodge structure (VHS), or period mapping, is given by a holomorphically varying family of PHS’s parametrized by a complex manifold S and satisfying a differential constraint. In order to have the PHS’s occurring in a fixed vector space (V , Q) we need to use the universal covering Se → S.
21/82 Example S = ∆∗ = {t : 0 < |t| < 1}, Se = H = {z : Im z > 0}, and t = e2πiz . We may take products of this to have S = ∆∗`. Definition A VHS is given by Φ Se e / D
Φ S / Γ\D where we have the monodromy representation ρ : π1(S) → GZ and where Φe is an equivariant holomorphic mapping satisfying
(i) Φ(e γz) = ρ(γ)Φ(e z) for γ ∈ π1(S), z ∈ Se; p dFz p−1 (ii) dz ⊂ Fz (infinitesimal period relation = IPR).
22/82 Example 1 ∗ 2 Q S = ∆ and Xt = {y − x(x − t) (x − ai ) = 0}. The family X → ∆∗ is a differentiable fibre bundle, but we may only 1 2g identify the H (Xt , Z) with Z up to the action of monodromy when t turns around the origin
γ γ ··· −→ ··· δ γ→γ+δ
→·· −→·· ··· 0 t δ γ γ
23/82 At t = 0 we have the singular curve γ δ ··· ···
2 2 Q y = x (x − ai )
The IPR is trivial in this case, as it is whenever the period domain D is a Hermitian symmetric domain. γ γ δ ··· ··· δ
24/82 The pictures of the singularity in Example 1 are
←→ = neighborhood of the node
6 6 real picture topological picture
25/82 Example 2 This is similar to Example 1, but it is harder to draw the pictures. We will be interested in the degeneration of the surface above, e.g., given by
2 f (x, t) = (x1 − t)q(x) − k(x) = 0 where q(x) is a general quadric. It is much harder to draw the pictures.
26/82 The real picture of the singularity is
p
This is a double curve with a pinch point.
27/82 We remark that if we have a family of smooth projective varieties ∗ π ∗ −1 X −→ ∆ , Xt = π (t) the limit lim Xt t→0 is not well defined — it depends on how we complete ∗ ∗ X → ∆ to X → ∆ to have X0. Moreover, even if we have conditions on X that uniquely define X0, the geometry of X0 may be complicated. This situation is even more so when we have a higher dimensional parameter space, e.g., S = ∆∗`. This brings us to the main point of the talk.
28/82 ∗` I Degenerating Hodge structures given by Φ : ∆ → Γ\D are well understood when ` = 1 and a lot is known when ` > 1.
I One may hope that this knowledge can be used to help guide the geometric understanding of X → X0. We will try to (i) explain how this principle works for algebraic curves (classical), (ii) explain the issues when X is a surface of general type (like g = 2 for algebraic curves), and (iii) discuss part of the one surface example where there are very early indications of how the above philosophy might work out quite nicely. Implementing this philosophy in the end of course requires algebro-geometric arguments.
29/82 The analysis of degenerating PHS’s uses Lie theory and the geometry of homogeneous complex manifolds. The starting point is the Monodromy Theorem ∗ For Φ : ∆ → Γ\D the generator T ∈ Aut(VZ, Q) given by ∗ the image of the identity in π1(∆ ) is unipotent — i.e., the eigenvalues of T are roots of unity. Replacing t by tk we may assume that T = exp N where 1 2 N = log T = (T − I ) − 2 (T − I ) + · · · ∈ Aut(VQ, Q) is nilpotent. Moreover, Nn+1 = 0 where n = weight of the PHS’s.
30/82 Definition ∗ Φ k A nilpotent orbit is a VHS ∆r −→{T }\D given by
Φ(e z) = exp(zN) · F0 ˇ where F0 ∈ D and exp(zN) · F0 ∈ D for Im z > log(r). Rescaling t = λt we may assume r = 1 and F0 ∈ D. The IPR is equivalent to p p−1 NF0 ⊂ F0 .
Theorem (Schmid) Any VHS over ∆∗ may be well approximated by a nilpotent orbit.
31/82 Classically it was known (theorem on regular singular points) that any period matrix may be expanded as a polynomial in log t with holomorphic coefficients — the nilpotent orbit is (with a suitable normalization) obtained by evaluating the coefficients at t = 0. The “suitable normalization” is subtle and is given a structure by the further approximating sl2-orbit. Example For y 2 = x(x − t)(x − a)(x − b), complex analysis may be used to show that t dx log t dx = + h(t), = k(t), k(0) 6= 0. ˆ0 y 2πi ˆγ y
32/82 The nilpotent orbit is obtained by using h(0), k(0). Rescaling t and dx/y, we may take h(0) = 0, k(0) = 1. The nilpotent orbit is then " # z 1 (∗) z → ∈ H ⊂ P . 1
0 Note that F0 = 1 ∈ ∂H; this will be a general situation. Definition
An sl2-orbit is given by a representation ρ : SL2(R) → GR with 0 1 ρ∗ ( 0 0 ) = N which induces an equivariant VHS
H ,→ D.
33/82 Theorem (Schmid)
A nilpotent orbit may be approximated by an sl2-orbit.
In this way we may understand limt→0 Φ(t). What is the limiting object? To explain this, we note that any N may be graded by a semi-simple Y with integral eigenvalues and weight spaces Vk where −n 5 k 5 n, and we define the weight filtration
W (N)m = ⊕ Vk+n k5m
W (N)0 ⊂ W (N)1 ⊂ · · · ⊂ W (N)2n = V .
With indices running opposite to the case of the hard Lefschetz theorem we have
N : W (N)k → W (N)k−2.
34/82 Definition A mixed Hodge structure (MHS) is given by (V , W•, F ) where W• is an increasing filtration on V and F is a decreasing
filtration on VC, and where F induces on each W Grk V = Wk /Wk−1 a Hodge structure of weight k. Definition A polarized limiting mixed Hodge structure (we shall just use LMHS) (V , Q, W (N), F ) is given by a MHS (V , W (N), F ) k such that for k = n the bilinear forms Qk (u, v) = Q(N u, v) W (N) induce PHS’s on Gk,primV .
35/82 W (N) Here the primitive spaces are defined on GrV as in the case of a smooth projective variety with ( N ↔ L Y ↔ Y , where as noted above N goes in the opposite direction to L. The above definition depends only on N and not on the Y . The sl2-orbit theorem gives us a choice of Y .
36/82 The punch line of the above is: A VHS
Φ : ∆∗ → Γ\D gives a LMHS
lim Φ(t) = (V , W (N), F0). t→0
The choice of F0 is not unique because rescaling t → λt gives F0 → exp(λN)F0. It may be normalized so that (among other things) F0 ∈ ∂D.
37/82 D
F∞ −1 + exp(iy N )F∞
F0 exp(iyN)F0
F∞ = lim exp(iyN)F0 y→∞ −1 + F0 = lim exp(iy N )F∞ y→0
+ 0 0 where N = ( 1 0 ). The point here is that a lot is known about degeneration of PHS’s.
38/82 Example When the weight n = 1 we have N2 = 0 and the weight filtration is
W0(N) ⊂ W1(N) ⊂ W2(N) = V .
W0(N) = Im N = isotropic subspace (Q = 0 on it) ⊥ W1(N) = Ker N = W0(N) .
39/82 Elementary linear algebra shows that we may choose a ∼ W (N) symplectic basis such that V = ⊕ Grk V and k 0 0 I ! 0 I g0 Q = 0 Q0 0 , Q0 = −Ig 0 −I 0 0 0 0 0 A t N = 0 0 0 , A = A, A > 0. 0 0 0
40/82 Recalling that the period domain D = Hg = {g × g matrices Z, Z = t Z and Im Z > 0}, the nilpotent orbit is
log t ! 2πi A W (N) Z(t) = , Z0 ∈ Hg0 ∈ Gr1 (LMHS). Z0
Taking A = I and having in mind
log t period of the form + h(t) 2πi acquiring a node as X → X plus t 0 ←→ 0 1 , having ωt ∈ H (ΩXt ) where ω0 has a pole at the node
41/82 what is suggested is that we look for a curve X0 where the normalization Xe0 has genus g0 and where X0 has g1 = g − g0 nodes; e.g.,
. . . Xt = . . . - . . . = X0
| {z } | {z } 6 g1 g0
. . . = Xe0 r r r r = curve of genus g0 with g1 sets of pairs of marked points pi , qi
42/82 Monodromy is γi → γi + δi th where δi is the vanishing cycle at the i node. 0 1 rd On Xe0 we have differentials ωi ∈ H Ω (pi + qi ) of the 3 Xe0 kind with Respi ,qi ωi = ±1 and Respi ,qi ωj = 0 for any j 6= i, 0 1 and where ωi is the limit of ωi,t ∈ H ΩXt . The extension data in the LMHS arising from
W (N) 0 → W0(N) → W1(N) → Gr1 → 0 is given by Σ AJ (p − q ) i Xe0 i i In summary, Hodge theory plus heuristic geometric reasoning guides in what to look for when we compactify Mg .
43/82 To complete the story from an algebro-geometric perspective, once we are led to try to add nodal curves to complete Mg , the question arises: Which nodal curves should be added? Hodge theory does not distinguish between
-
-
C
44/82 If we let X0 be the singular curve, then for g = 2 we may impose the condition
ωX0 is ample.
π More precisely, to a family Xe −→e ∆ where I Xe is smooth and has no −1 curves in the fibres,
I Xet is smooth for t 6= 0 and the central fiber Xe0 is nodal we apply the relative pluricanonical mapping given by R0ω⊗k πe Xe/∆ for k = 3 to obtain X −→π ∆,
45/82 which has the effect of contracting all curves C ⊂ Xe0 with
ω · C = 0, Xe/∆ these being the curves such as C above. These turn out to be exactly the stable curves X0 with χ(OX0 ) = 1 − g.
46/82 Another way of describing this from an algebro-geometric perspective is this: The minimal model program (MMP) for surfaces (which is classical, dating to the Italian school) applied to families X → ∆ gives the existence of an abstract compactified moduli space Mg . Hodge theoretic considerations as above then serve as a guide to what the singular curves in the boundary ∂Mg = Mg \Mg should be.
47/82 As we shall explain below, for surfaces of general type the existence of an M has been proved [Kollar, Shepherd-Barron, Alexeev] using the MMP for threefolds, but except in some simple cases what singular surfaces appear in ∂M is not understood. It is here that ones hopes that Hodge theory may provide some guidance.
48/82 Monodromy cones
Above we have taken A = I ; what about other A’s in the cone A = t A, A > 0? For example, for g = 2 and for the degeneration
δ3 δ3=δ1+δ2
δ 1 δ2 2 1 1 0 0 0 1 1 A = ( 1 2 ) = ( 0 0 ) + ( 0 1 ) + ( 1 1 ) = = =
A1 A2 A3
49/82 A is in the interior of the cone spanned by A1, A2, A3. What is suggested is that we consider several parameter families of algebraic varieties giving rise to a VHS
Φ: S → Γ\D where S = S\Z with the sub variety Z corresponding to singular varieties. By localizing around a point of Z we may assume that S = ∆∗` and
Γ = {T1,..., T`}, Ti = exp Ni where Ni ∈ EndQ (V ) are commuting nilpotent elements.
50/82 Definition A several variable nilpotent orbit is given by X exp zi Ni · F0 where the Ni are commuting elements in EndQ (V ) and where the conditions P (i) exp( i zi Ni ) · F0 ∈ D for Im zi 0, p p−1 (ii) Ni F0 ⊂ F0 are satisfied.
51/82 Schmid’s theorem extends to the several variable case, so that from the perspective of the asymptotics of VHS’s it suffices to consider nilpotent orbits. There is also a several variable sl2-orbit theorem [Cattani-Kaplan-Schmid] and [Kashiwara-Kawai], but it is rather involved to state and will not be needed here.
52/82 Note From an algebro-geometric perspective there are well-defined cohomological obstructions to completing the Ni ’s to commuting sl2,i ’s.
possible
not possible
This can be understood for curves but essentially nothing is known in the general case.
53/82 The monodromy cone X σ = {Nλ = λi Ni , λi > 0} i has remarkable properties, including
interior: W (Nλ) is independent of Nλ ∈ σ,
faces: W (N) is a relative weight filtration for W (Ni ). Nilpotent orbits have the property that LMHS associated to (V , W (Nλ), F0) is independent of the Nλ ∈ σ. Here they will also have the property that each Nλ gives a PLMHS, where the primitive pieces will in general depend on Nλ. The above properties are not linear algebra results; they require deep use of Hodge theory.
54/82 Example For the VHS given by independently smoothing nodes in the last example just above, taking ! 0 Ai Ni = 0 0 gives a monodromy cone. This one is in fact maximal. The larger the cone the more singular the varieties corresponding to its interior is.
55/82 Thus
←→ σ = spanR>0 {N1, N2}
?
←→ σmax = spanR>0 {N1, N2, N3}
and σ is a face of σmax
56/82 In general, weight one VHS’s have the special properties that the weight filtrations
⊥ W0 ⊂ W1 ⊂ W2 = V , W1 = W0
∗ ∗ ∗ correspond to maximal parabolic subalgebras 0 ∗ ∗ of 0 0 ∗ EndQ (V ). For each g1 = dim W0
I they are all conjugate,
I their unipotent radicals are abelian. This latter implies that there are maximal monodromy cones; any σ is a face of a maximal one corresponding to all A’s with A = t A, A > 0.
57/82 Returning to limiting mixed Hodge structures we have the Definition The type of a LMHS (V , W (N), F ) is the collection of Hodge W (N) numbers of the PHS’s Grn+k V prim.
58/82 Examples
the type is uniquely determined N n = 1 r r by rank N
r r
N1 r r N r 0 the type is uniquely determined by rank N0 and rank N1 n = 2 r r N r N0 r 1
r r r
59/82 For n = 1 the conjugacy classes of nilpotent cones fall into disjoint subsets indexed by one number, and within each type there is a unique maximal monodromy cone of that type. This is a reflection of the linearity in N of the polarizing condition W (N) Q(Nu, u¯) > 0 for u ∈ Gr2 . We may picture this as
N1 ≺ N2 ≺ · · · ≺ Ng .
A face of a cone in Nk is in some N` for ` 5 k. This partial ordering among the monodromy cones leads to a stratification of ∂Mg .
60/82 For n = 2 with h2,0 = 2, Robles, Pearlstein and Brosnan have classified the nilpotent cones. The story here is quite a bit more subtle. First, as a reflection of the non-linearity in N of the polarizing conditions
2 W (N) Q(N u, u¯) > 0, u ∈ Gr4 V maximal nilpotent cones may not exist. Secondly, the strict partial ordering of the monodromy cones of a given type we saw in the n = 1 case may not hold. This has the algebro-geometric implication that degenerations among certain types of algebraic varieties in the boundary of a moduli space may not be possible. Following is Robles table.
61/82 r r
I N2 = 0, rank N = 2 r rr N
r r
rr
II N3 = 0, rank N = 1 r
r r The notation in II means that N2 6= 0.
62/82 N r
III N2 = 0, rank N = 4 r rr N
r
N1 N r 0 r N3 = 0, N2 = 0 0 1 IV rank N0 = 1, rank N1 = 2 r r r N0 N1
r r
63/82 N r
V N3 = 0, rank N = 2 N r
r Her incidence table is ( ) NII NI ≺ ≺ NIV ≺ NV. NIII
Thus, type II cannot degenerate to type III.
64/82
65/82 For surfaces of general type the KSBA (Koll´ar, Shepherd-Barron, Alexeev) compactification M of the moduli space M is obtained by applying the MMP for threefolds to degenerations X → ∆ to see what type of singular surfaces X0 should be added to the smooth ones.1 Using the general theory from the MMP, this compactified moduli space M may be proved to exist as a complete, separated algebraic space (a somewhat more general notion than an algebraic variety, but it’s OK for present purposes to think variety). 1More precisely, to surfaces X that are smooth except for rational double points (RDP’s), these being the ones that are contracted under all the pluricanonical maps ϕmKX given by |mKX |. The singularities of the surfaces that are added are essentially (i) a double curve with pinch points (no triple points); (ii) a few isolated singularities, including simple elliptic ones and cusps. 66/82 As note above, it seems that there are essentially no non-classical (i.e., whose period domain is not an HSD) examples where the limit surfaces X0 ∈ ∂M needed to compactify the moduli space have been described. If one were to use Hodge theory as a guide to which X0’s to add, one might envision three steps: (A) Use a known stratification of the possible monodromy cones to suggest a stratification of ∂M; (B) Contruct algebro-geometric degenerations of surfaces that realize those suggested by (A); (C) For each stratum in (B), define the “type” of singular surfaces and show that any surface of that type occurs in that stratum in M.
67/82 For curves, the type means the arithmetic genus χ(OX0 ) and the dual graph of X0, where we assume that X0 is nodal and that ω (log D) is ample. Xe0 Example An H-surface is a minimal algebraic surface X that
I is of general type; 2,0 1,0 I has h (X ) = 2 and h (X ) = 0 (regularity); 2 I has KX = 2.
68/82 Then X has only RDP’s, and if it is smooth, the Riemann-Roch theorem gives 1 χ(O ) = K 2 + χ (X ) X 12 X top w 1,1 b2(X ) = 34, h (X ) = 30.
The above H stands for Horikawa, who analyzed in detail the smooth H-surfaces via their bi-canonical map
4 ϕ2K : X ___ / P . The image of this map is given by the equations at the beginning of this talk. We shall use Hodge theory as a guide to construct a degeneration of type I; thus for H-surfaces one has (A) in general and partial results for (B).
69/82 Hodge theory will be used in two ways: (i) to suggest what to look for, and (ii) to suggest where to look. For degenerations of type I, the simplest possibility is that as X → X0
I X0 acquires an elliptic double curve D; 0 2 0 2 I one ω ∈ H (ΩX ) tends to ω0 ∈ H (Ω ). Xe0 Regarding the second point, as X → X0 one expects all of the 0 2 2-dimensional vector spaces H (ΩX ) to go in the limit to 0 2 2,0 H (Ω (log De)). Since we should have h (Xe0) = 1, the ω0 Xe0 above remains holomorphic in the limit and its divisor 0 2 (ω) → D, while a general ψ ∈ H (ΩX ) tends to ψ0 with a log-pole on D and whose residue generates H0(Ω1 ). De
70/82 So where is the double curve D going to come from? The canonical series |KX | is a pencil of curves C. Horikawa shows that for a general X
I |KX | has no fixed component, 2 I the pencil |KX | has KX = 2 base points.
71/82 Blowing up the two base points we get Xˆ with the picture
Ct C∞
Xˆ E1 E 2=E 2=−1 t 1 1
E2 ? 1 P ∞ where a smooth C ∈ |KX | has genus 1 g(C) = (K 2 + K · C) + 1 = 3. 2 X X We note that
KXˆ = (C) + 2(E1 + E2).
72/82 A nice little exercise is to compute (i) # nodal curves δ + 1 = 42 (ii) # hyperelliptic fibres = 1 1 For (i), each node adds +1 to χtop(P ) · χtop(C), which would be the Euler characteristic of Xˆ if there were no singular fibres. For (ii), the argument is more subtle. There is a linear relation among λ = degree (Hodge bundle), δ1, and δ0 = # ˆ 1 hyperelliptic curves in the fires of X → P . Since δ0 is not detectable Hodge-theoretically, this is insufficient to give (ii).
73/82 Lemma 0 ∼ R ω 1 O 1 (1) ⊕ O 1 (1) ⊕ O 1 (3). f Xˆ /P = P P P Proof 3 0 It is known from Hodge theory that Rf ωXˆ / 1 = ⊕ OP1 (ki ) P i=1 where
I all ki = 0 (non-negativity of the Hodge bundles); I in fact, all ki > 0 since q(X ) = 0. 0 0 ∼ 0 2 ∼ 0 2 I H (R ω 1 (−2)) H (Ω ) H (Ω ). f Xˆ /P = Xˆ = X ∼ This map uses OP1 (1) = OP1 (∞) and is given by ϕ(t) −→ ϕ(t) ∧ dt = Φ.
74/82 0 2 Since h (ΩX ) = 2 it follows that k1 + k2 + k3 = 5, which gives the possibilities ( (1, 2, 2) (1, 1, 3). If the first of these holds, then we would have a 0 0 ϕ ∈ H (R ω 1 (−2)) which doesn’t vanish on any fibre f Xˆ /P which implies that KX = OX . 0 From this we find that λ = deg R ω 1 = 5. f Xˆ /P
75/82 To find what D should be we imagine that as X → X0 the nd section ϕ(t) which vanishes to 2 order at t = 0 tends to ϕ0 where lim(ϕ(t) ∧ dt) = C0,∞. ˆ Thus we may hope to have on X0
D = C0,∞ + 2(E1 + E2).
C0,∞ is a curve with χ(OC0,∞ ) = −4 with nodes at the Ei · C0,∞. Thus Ce0,∞ is an elliptic curve. Moreover, the differential ϕ0 ∧ dt on X0 has no zeroes. Thus we may suspect, and it may be proved, that
X0 is a polarized K3 surface.
76/82 Note The above argument gives
(Φ) = (ϕ) + 2(E1 + E2) where (ϕ) is the divisor of 0 ∼ 0 0 ϕ ∈ H (O 1 (1)) H (R ω 2 (−2)). The intersections P = f Xˆ /P (E + E ) · C are bi-tangents to the place quartics ϕ (C ). 1 2 t KCt t
77/82 It remains to actually write down the equations of a family of surfaces that actually realize the above “thought example.” In the weighted projective space P(1, 1, 2, 2) with homogeneous coordinates [t0, t1, x3, x4] we consider the surface given by
2 2 2 2 2 2 (∗) t0 G(t0 , t0t1, t1 , x3, x4) = F (t0 , t0t1, t1 , x3, x4) where F (x), G(x) are general homogeneous polynomials of degrees 2, 3 in x = [x0, x1, x2, x3, x4]. The H-surface is the desingularization X of the surface (∗).
78/82 Geometricaly we are embedding
P(1, 1, 2, 2) / P4 ∈ ∈ 2 2 (t0, t1, x3, x4) / [t0 , t0t1, t1 , x3, x4].
2 4 The image is the singular quadric x0x2 − x1 = 0 in P . 4 Blowing up P along x0 = x1 = x2 = 0 gives a desingularization Pg(1,1, 2, 2) of P(1, 1, 2, 2). In fact,
∼ 0 (1, 1, 2, 2) = (O 1 ⊕ O 1 ⊕ O 1 (2)) (R ω 1 ) Pe P P P P = R f Xˆ /P and the proper transform of the surface (∗) is the surface Xˆ over P1: P(1, 1, 2, 2) ____ / P1 ∈ ∈
[t0, t1, x3, x4] / [t0, t1]. 79/82 The surface (∗) in P4 is the bi-canonical model of X . It is singular along the plane conic
x0 = x1 = F (x) = 0, which is a double curve with eight pinch points. If we now consider the surface
2 2 2 (∗ ∗) t0 t1 Q(x) = F (x) where Q(x) is a general quadric, then we obtain a surface which we may think of as the degeneration of (∗).
80/82 This provides a geometric realization of the thought example above. Thus in conclusion, we have
I Hodge theory ___ / thought example
I thought example ___ / actual example. Although promising, and there is a very nice interplay between Hodge theory and the very beautiful algebraic geometry of the H-surface and its limit, it is only a first step.
81/82 Recent Developments 2 2 0 t0 G = F
? 2 2 2 I t0 t1 Q = F Q Q + QQs 2 2 2 2 t (t0 − t1) t L = F ?IIIII 0 1 Q (t0R + F )(t0R − F ) = 0 (∗) Q QQs + IV in (∗) allow F = R = 0 to acquire a node
? V allow F = R = 0 to acquire a second node
Hodge theory suggests two algebro-geometric components for type III degenerations, and these illustrate two such possibilities. 82/82