A Decomposition of the Moving Cone of a Projective Manifold According to the Harder-Narasimhan ﬁltration of the Tangent Bundle

A decomposition of the Moving cone of a projective manifold according to the Harder-Narasimhan ﬁltration of the tangent bundle

Dissertation zur Erlangung des Doktorgrades der Fakultät für Mathematik und Physik der Albert-Ludwigs-Universität Freiburg im Breisgau

vorgelegt von Sebastian Neumann

November 2009 Dekan: Prof. Dr. Kay K¨onigsmann

1. Gutachter: Prof. Dr. Stefan Kebekus

2. Gutachter: Prof. Dr. Thomas Peternell

Datum der Promotion: 27.01.2010 Contents

Introduction 1

1 Preliminaries 7 1.1 Rationallyconnectedfoliations...... 7 1.2 Intersection numbers, cones and extremal contraction ...... 10 1.3 Minimalrationalcurves ...... 14

2 Rationally connected foliations on surfaces 15 2.1 Rationally Connected Foliations on Surfaces and the MRC-quotient . . . . 15

3 The slope with respect to movable curves 23 3.1 TheHarder-Narasimhanﬁltration ...... 23 3.2 Complete intersection curves versus movable curves ...... 28 3.3 Destabilizing chambers on projective manifolds ...... 31

4 The Harder-Narasimhan filtration of the tangent bundle on Fano threefolds 41 4.1 Preliminary Results for Fano threefolds ...... 42 4.2 PullbackofFoliations...... 49 4.3 Minimal rational curves and Mori fibrations ...... 51 4.4 The Harder-Narasimhan filtration on Fano threefolds ...... 58 4.4.1 Preliminaryremarks ...... 58 4.4.2 Examples of Harder-Narasimhan filtration of special Fano threefolds 59 4.4.3 Picardnumber2 ...... 71 4.4.4 Picardnumber3 ...... 72 4.4.5 Picard number ≥ 4...... 76 4.5 Another method to compute the Harder-Narasimhan filtration of TX .... 84

5 Prospects and open questions 87 5.1 A generalization of Miyaoka’s uniruledness criterion ...... 87 5.2 Maximal rationally connected foliations on higher dimensional manifolds . 88 5.3 The Harder-Narasimhan ﬁltration of TX on higher dimensional Fano manifolds ...... 90

Introduction

Varieties covered by rational curves, i.e. uniruled varieties, play a central role in algebraic geometry. Since Mori’s seminal works [Mor79],[Mor82] it has become clear that not only are many natural varieties uniruled, for example low degree hypersurfaces in projective space, but also they can be eﬀectively studied by looking at the rational curves contained in the variety. Originally Mori developed the theory of rational curves to solve a conjecture of Hartshorne in [Mor79], which states that the only projective variety with ample tangent bundle is the projective space. However, soon it became clear that the theory of rational curves can be applied to a broad spectrum of problems in higher dimensional algebraic geometry, for example to prove the uniqueness of complex contact structures, to prove the deformation rigidity of hermitian symmetric manifolds or to study stability questions of the tangent bundle.

Rationally connected foliations. To better understand the geometry of uniruled varieties one can form the maximal rationally connected fibration or maximal rationally connected quotient based on a construction by Campana [Cam81], [Cam94] and Kollár- Miyaoka-Mori [KMM92], which parametrizes maximal rationally connected subvarieties of a uniruled variety. Roughly speaking, this fibration is a map with the property that the fibres are rationally connected and almost every rational curve in the uniruled variety is contained in fibres of this fibration. Another way to construct rationally connected subvarieties of a given uniruled variety is suggested in [KSCT07]. Let X be a smooth n-dimensional projective variety and suppose C is a complete intersection curve, i.e. there exists ample divisors H1,...,Hn−1 such that the class of C equals the intersections H1 · . . . · Hn−1. Then there exists a unique filtration

0= F0 ⊂ F1 ⊂ . . . ⊂ Fk = TX of the tangent bundle depending on C with some technical defining properties, which are stated precisely in Theorem (1.1.10). This filtration is called the Harder-Narasimhan filtration of TX with respect to C . Kebekus-SoláConde-Toma proved in [KSCT07] that if a sheaf occurring in this filtration fulfills some positivity condition, it is a foliation with algebraic and rationally connected leaves. Evidently, we can ask if the relative tangent sheaf of the maximal rationally connected fibration appears as a term in the Harder-Narasimhan filtration of the tangent bundle with respect to any complete intersection curve. The answer is negative already on surfaces as shown by an example of Thomas Eckl [Eck08]. However, if we ask for the existence of a complete intersection curve with the desired properties, then we are able to give a positive answer in case that X is a surface. This will be done in chapter 2. There our main result is the following.

Theorem. Let X be a uniruled projective surface. Then there exists a polarisation, such that the maximal rationally connected quotient of X is given by the foliation associated to highest positive term in the Harder-Narasimhan ﬁltration with respect to this polarisation.

Destabilizing chambers in the cone of movable curves. If we turn to higher dimensional manifolds the situation gets much more complicated. This is caused mainly by the fact that if we consider the Harder-Narasimhan filtration with respect to complete intersection curves, then we have to understand the numerical classes of these curves on a variety. They are not well understood and a numerical characterization of complete intersection curves seems impossible. To overcome this difficulty we generalize the Harder-Narasimhan filtration in Chapter 3. More precisely, we prove that for any movable class there exists a unique Harder- Narasimhan filtration. Movable classes form a larger class of curves than the classes of complete intersection curves. It is a great advantage, that movable classes or rather the Moving cone, i.e. the cone in the Néron-Severi vector space generated by movable curves, are fairly well understood - the backbone being the duality statement of [BDPP04]. Now having constructed to each movable class a unique Harder-Narasimhan filtration, we ask how this filtration varies with the movable class in the Néron-Severi vector space. To study this question, we will divide the Moving cone into destabilizing chambers. These chambers are defined by the property that two classes lie in the same chamber if and only if the Harder-Narasimhan filtration of these two curves agree. The general properties of the chamber structure are investigated in Chapter 3 where we prove the following Theorem.

Theorem. Let X be a projective manifold. Then the destabilizing chambers are convex subcones of the Moving cone of X. In the interior of the Moving cone, the decomposition is locally ﬁnite and the destabilizing chambers are locally polyhedral.

In addition, we will exhibit a surface having infinitely many destabilizing chambers. This shows that in general the chambers might accumulate at the boundary of the Moving cone. Thus the Theorem above is sharp and we cannot expect the chamber structure to be finite in general. However, we will see that the chamber structure is finite if the Moving cone is polyhedral. This happens for example on Fano manifolds, see [Ara10].

In another direction, we investigate the geometric meaning of the subsheaves occurring in the Harder-Narasimhan filtration on special classes of manifolds. Concerning this aspect, we will study the destabilizing chambers on Fano threefolds. It is known that on a Fano threefold of Picard number one the tangent bundle is stable, i.e. the Harder-Narasimhan filtration is the trivial filtration 0 ⊂ TX with respect to any movable curve, see [Hwa98], [PW95]. If the Picard number of a Fano threefold is greater than or equal to two, there may exist nontrivial terms in the Harder-Narasimhan filtration with respect to some

2 movable curves, see [Ste96]. Our main result is that there is a clear geometric description of the subsheaves occurring in the Harder-Narasimhan ﬁltration of the tangent bundle with respect to any movable class. We will prove that these sheaves are relative tangent sheaves of not necessarily elementary Mori ﬁbrations. More precisely, we will prove the following Theorem in Chapter 4.

Theorem. Let X be a Fano manifold of dimension 3. Then there exists a ﬁnite decomposition of the Moving cone into polyhedral subcones, the destabilizing chambers, such that

(i) in each subcone the Harder-Narasimhan ﬁltration is constant and,

(ii) each term of the Harder-Narasimhan ﬁltration associated to a movable curve is the relative tangent sheaf of a not necessarily elementary Mori ﬁbration.

We ﬁnish the thesis with a chapter that arrange our results in a somewhat broader per- spective. Moreover, we discuss the prospects and constraints to generalize the results of this thesis.

Notation and Conventions

• All varieties and manifolds are deﬁned throughout over the ﬁeld C of complex numbers. We will assume them to be projective and irreducible.

• If two R-divisors D1 and D2 are numerically equivalent, i.e. we have D1 · C = D2 · C for all curves C ⊂ X, then we write D1 ≡ D2. We will use an analogous notation for 1-cycles.

• N1,R(X) denotes the space of 1-cycles on X with coeﬃcients in R modulo numerically equivalence.

1 • NR(X) denotes the space of divisors on X with coeﬃcients in R modulo numerically equivalence.

1 • NR(X), N1,R(X) are R-vector spaces of the same ﬁnite dimension. The dimension of these vector space is the Picard number and is denoted by ρ(X).

1 • We view NR(X) and N1,R(X) with the standard euclidean topology.

⊥ • For a divisor D on X, we deﬁne D := { c ∈ N1,R(X) D · c =0 } .

• For a subset S ⊂ Rd, the linear span of S is denoted by hSi. Moreover, we set

hSiconv := ai · si ai =1,si ∈ S ( i=1 ) X X 3 and n

hSiR+ := ai · si ai ∈ R+,si ∈ S . ( i=1 ) X d • A subset S ⊂ R is a cone, if its closed under multiplication with R+. If it is moreover convex, we call it a convex cone. All cones that we will consider contain no lines.

• The dimension of a cone C ⊂ Rd is the dimension of hCi.

d • If we have a cone C = R+di ⊂ R , then the relative interior of C is the interior of C in the vector space Rdi. For example, if C is ﬁnitely generated, say C := 3 P R+e1 + R+e2 ⊂ R , with e1,e2 linearly independent, then the relative interior is P given by { λ1e1 + λe2 λ1,λ2 > 0 }. Analogously we deﬁne the relative boundary of C.

• The tangent sheaf of a projective manifold X is denoted by TX . When we refer to 1 TX as the tangent bundle, then it will always be mentioned explicitly. We write ΩX for the sheaf of 1-forms on X. If X → Y is a proper morphism then the relative tangent sheaf is denoted by TX/Y .

• The class of a canonical divisor of a manifold X is denoted by KX . If X → Y is a proper morphism then we denote the relative canonical divisor by KX/Y .

4 Acknowledgments

I would like to thank my thesis advisor Prof. Dr. Stefan Kebekus for his supervision and excellent support. The Ph.D. thesis was partially supported by a scholarship of the Graduiertenkolleg Globale Strukturen in Geometrie und Analysis in K¨oln of the DFG. I would like to thank the DFG for the support. I would also like to thank all members of the working group of Prof. Dr. Stefan Kebekus, especially Dr. Sammy Barkowski, Dr. Thomas Eckl, Dr. Daniel Greb and Daniel Lohmann.

Chapter 1 Preliminaries

In this chapter we collect the mathematical facts and terms we will need in the following. The chapter is divided into three parts. In the ﬁrst part we deﬁne rationally connected foliations on a projective manifold. Moreover, we present a way to construct rationally connected foliations, which was suggested in [KSCT07]. The second part contains the background of the minimal model program and in the last part we recall some facts about rational curves.

1.1. Rationally connected foliations

Let X be a projective manifold and let F ⊂ E be coherent sheaves of OX -modules. Consider the quotient Q := E /F . The saturation of F in E is the kernel of the natural map E → Q → Q/TorQ. The sheaf F is called saturated in E if the quotient E /F is torsion free. Definition 1.1.1. The singular locus of a coherent sheaf F is the set Sing(F ) := { p ∈ X F is not locally free in p }. The rank of a coherent sheaf is the rank of the sheaf outside the singular locus. Definition 1.1.2. A foliation F is a coherent saturated subsheaf of the tangent sheaf closed under Lie-bracket. If a coherent subsheaf of the tangent sheaf is closed under Lie-bracket, we call it involutive or integrable. Definition 1.1.3. Let F be a foliation and p ∈ X. The leaf of F through p is the largest subset B ⊂ X containing p with the properties that B is the image of an immersion and is F -invariant.

Remark 1.1.4. Let F ⊂ TX be a foliation. Then for an open set U ⊂ X where F is a subbundle of TX we can apply the classical Frobenius theorem [War83, Theorem 1.60]. It follows, that through any point in U passes a unique leave.

Deﬁnition 1.1.5. Let F ⊂ TX be a foliation. Then a leaf of F is called algebraic, if it is open in its Zariski closure. Next, we collect some basic facts about saturated sheaves and foliations, which are well known, but not explicitly stated in literature. Preliminaries

Proposition 1.1.6. Let F ⊂ TX be a foliation and denote by S the singular locus of TX /F . Then the following conditions are equivalent:

(i) TX /F is torsion free. F (ii) For any open set U ⊂ X and any v ∈ TX (U) such that v U\S ∈ (U\S) it follows F that v ∈ (U).

Proof. Let us prove the direction (ii) ⇒ (i) ﬁrst. Since TX /F is locally free outside S, we only have to prove that TX /F is torsion free in p ∈ S. Let U be an open set around p and v ∈ TX (U). Suppose there exists a nonzero element f ∈ OX (U) such that fv ∈ F (U). We have to show that v ∈ F (U). Because TX /F is locally free outside S, F F we have v U\S ∈ (U\S). By our assumptions we can extend vector ﬁelds in , thus it F follows v ∈ (U).

To prove the other direction, we take a vector ﬁeld v ∈ TX (U) such that v ∈ F (U\S). After possibly diminishing U, we ﬁnd a non zero element f ∈ OX (U) vanishing on S. n Then, since F is coherent, f v ∈ F (U) for some n ≥ 1. As TX /F is torsion free, we have v ∈ F (U).

Proposition 1.1.7. Let F , G ⊂ E be coherent sheaves on a projective manifold X. Suppose that both F and G are saturated in E . We assume that E is locally free. F G F G (i) If there is a Zariski open set U ⊂ X such that U = U then = . F F (ii) Suppose is saturated in TX , then integrability is an open property, i.e. ⊂ TX is involutive on a Zariski open set if and only if it is involutive on X.

(iii) Suppose we have a foliation FU ⊂ TU , which is deﬁned only on a Zariski open set U ⊂ X. Then FU extends uniquely to a foliation on X.

Proof. The proof of (i) is essentially the same as the proof of Proposition (1.1.6) above.

Let us prove (ii). Suppose F is involutive on an open set U ⊂ X. Equivalently, the O’Neill tensor 2

F → TX /F

^ 2 is zero on U. But since TX /F is torsion free, a morphism α ∈ Hom( F , TX /F ) is zero if it is zero on a Zariski open set. This proves (ii). V

To prove (iii), note that we only have to ﬁnd a saturated extension of FU in TX , since then by (ii) the extended sheaf is involutive and by (i) it is unique. This can be done by taking the saturation of the extension of FU as a coherent sheaf [Har77, Ex. 5.15.].

Remark 1.1.8. Let ϕ : X → Y be a ﬁbration. By the foliation associated to the ﬁbration ϕ, we mean simply the relative tangent sheaf TX/Y of ϕ.

8 Chapter 1

Deﬁnition 1.1.9. Let H be an ample line bundle and F a coherent sheaf of positive rank on an n-dimensional projective variety X. We deﬁne the slope of F with respect to H to be c (F ) · Hn−1 µ (F ) := 1 . H rk(F ) Furthermore, we call F semistable with respect to H if for any proper subsheaf G of F with rk(G ) ≥ 1 we have µH (G ) ≤ µH (F ). If there exists a nonzero subsheaf G ⊂ F such that µH (G ) > µH (F ), we will call G a destabilizing subsheaf of F .

Theorem 1.1.10 ([Mar80, Proposition 1.5.]). Let F be a torsion free coherent sheaf on a smooth projective variety and H be an ample line bundle on X. There exists a unique ﬁltration

0= F0 ⊂ F1 ⊂ . . . ⊂ Fk = F of F depending on H, the Harder-Narasimhan ﬁltration, with the following properties:

(i) The quotients Gi := Fi/Fi−1 are torsion free and semistable.

(ii) The slopes of the quotients satisfy µH (G1) >...>µH (Gk).

Deﬁnition 1.1.11. Let F be a coherent torsion free sheaf on a smooth projective variety with Harder-Narasimhan ﬁltration

0= F0 ⊂ . . . ⊂ Fk = F with respect to an ample line bundle H. If the slope of the quotient Fi/Fi−1 is positive with respect to H, then Fi is called positive with respect to H. Note that if µH (F1) > 0, then there exists a maximal number

s := max { i 1 ≤ i ≤ k,µH(Fi/Fi−1) > 0) } .

Then Fs is called the highest positive term in the Harder-Narasimhan ﬁltration of TX with respect to H.

We can now state an important result originally formulated by Miyaoka and shown in [KSCT07]. For a survey on these and related results we refer the reader to [KSC06].

Theorem 1.1.12 ([KSCT07, Theorem 1]). Let X be a smooth projective variety and let

0= F0 ⊂ F1 ⊂ . . . ⊂ Fk = TX be the Harder-Narasimhan ﬁltration of the tangent bundle with respect to a polarisation H. Write µi := µH (Fi/Fi−1) for the slopes of the quotients. Assume µ1 > 0 and set m := max {i ∈ N|µi > 0}. Then each Fi with i ≤ m is a foliation. Furthermore the leaves of these foliations are algebraic and for general x ∈ X the closure of the leaf through x is rationally connected.

9 Preliminaries

Let X be a smooth projective variety and assume the conditions of Theorem (1.1.12) are fulﬁlled. Thus we obtain foliations F1,..., Fm with algebraic and rationally connected leaves. By setting qi : X 99K Im(qi) ⊂ Chow(X) x 7→ Fi-leaf through x we obtain a rational map such that the closure of the general ﬁbre is rationally connected, see [KSCT07, Section 7].

There is another map with the property that the general ﬁbre is rationally connected. This map is called the maximal rationally connected quotient, or MRC-quotient, for short, based on a construction by Campana [Cam81] [Cam94] and Koll´ar-Miyaoka-Mori [KMM92], see also [Kol96, Chapter IV, Theorem 5.2].

Theorem 1.1.13 ([KMM92, Theorem 2.7.]). Let X be a smooth projective variety. There exists a variety Z and a rational map φ : X 99K Z with the following properties:

• The map is almost holomorphic, i.e. there exists an open set U such that φ U is proper.

• The ﬁbres of φ are rationally connected.

• A very general fibre of φ is an equivalence class with respect to rational connectivity, i.e. if x is a general point in a very general fibre, then any rational curve that contains x is automatically in the fibre.

• up to birational equivalence the map φ and the variety Z are unique.

1.2. Intersection numbers, cones and extremal contraction

This section is merely to ﬁx notation and brieﬂy recall some facts about the Minimal Model Program, which will be used in chapter 4.

In this section X and Y are projective Q-factorial varieties with at most terminal singularities.

1 1 1 Notation 1.2.1. We write N (X) for the Néron-Severi group and NQ(X)= N (X) ⊗Z Q 1 1 (resp. NR(X)= N (X) ⊗Z R) for the finite dimensional vector space of Q-divisors (resp. R-divisors) modulo numerical equivalence on X. We write N1(X) for the space of 1- cycles modulo numerical equivalence and we write N1,R(X) for the space of 1-cycles with coefficients in R modulo numerical equivalence. Given a 1-cycle c with real coefficients and a R-divisor D, then we write [c] and [D] for the numerical equivalence class of c and D respectively.

1 Deﬁnition 1.2.2. The dimension of NR(X) is called the Picard number of X. We denote the Picard number of X by ρ(X).

10 Chapter 1

We have a perfect pairing coming from the intersection product.

1 NR(X) × N1,R(X) → R (D,C) 7→ D · C

In the sequel, we will deﬁne the numerical pullback and pushforward for curves and divisor classes which will be used in Chapter 4, see [Ara10, Section 4]. To start, consider a birational map φ : X 99K Y with codimY (Y \im(φ) ≥ 2). Then extending the usual pullback and pushforward on the N´eron-Severi group linearly to numerical classes of R- divisors yield an injective map

∗ 1 1 φ : NR(Y ) → NR(X), and a surjective map 1 1 φ∗ : NR(X) → NR(Y ). ∗ These maps have the property that φ ◦ φ = id 1 . ∗ NR(Y ) Deﬁnition 1.2.3. The numerical pullback of curves

∗ φ1 : N1,R(Y ) → N1,R(X)

1 1 is deﬁned as the dual linear map of the pushforward φ∗ : NR(X) → NR(Y ). Analogously, ∗ we deﬁne the numerical pushforward φ1∗ as the dual linear map of φ . By abuse of ∗ notation, we will write φ and φ∗ for the numerical pullback and pushforward of curves since it will be clear from the context whether we pullback or pushforward curves or divisors.

The following remark follows directly from the deﬁnition of the numerical pullback and pushforward.

1 Remark 1.2.4 (Projection formula). If C ∈ N1,R(Y ) and D ∈ NR(X), then we have ∗ φ C · D = C · φ∗D. 1 ∗ If C ∈ N1,R(X) and D ∈ NR(Y ), then we have C · φ D = φ∗C · D. Deﬁnition 1.2.5 (Cone of curves, Mori cone). The cone of curves

NE(X) ⊂ N1,R(X) is the convex cone generated by all eﬀective 1-cycles on X. Its closure

NE(X) ⊂ N1,R(X) is called the Mori cone of X.

Notation 1.2.6 (Ample cone, pseudoeffective cone). We denote the cone of all ample R-divisors by 1 AmpR(X) ⊂ NR(X). The cone of pseudoeffective divisors is denoted by Eff(X).

11 Preliminaries

For the precise definitions of ample and pseudoeffective R-divisors as well as for a careful discussion of the geometry of the cones introduced above, we refer to [Laz04]. Definition 1.2.7. Let C be a closed convex cone in Rn. An extremal face F of C is a subcone of C having the property that if v + w ∈ F for some vectors v,w in C, then necessarily v,w ∈ F . A one dimensional face is called an extremal ray.

Notation 1.2.8. Let D be a divisor on X and let C ⊂ N1,R(X) be a subset. Then we set

≥0 D := { C ∈ N1,R(X) D · C ≥ 0 }

⊥ D := { C ∈ N1,R(X) D · C =0 }

CD≥0 := { C ∈ C D · C ≥ 0 } . The following Theorem is the backbone of the Minimal Model Program. For a detailed discussion of the Minimal Model Program, we refer to [KM98]. Theorem 1.2.9 ([KM98, Theorem 3.7]). Let X be a Q-factorial projective variety with only terminal singularities. Then:

(i) There are countably many rational curves Ci ⊂ X such that

NE(X)= NE(X) ≥0 + R+[Ci]. KX X (ii) The [C ] are locally discrete classes of rational curves in the half space NE(X) <0 . i KX

(iii) Let F ⊂ NE(X) be a KX -negative extremal face. Then there is a unique morphism ϕF : X → Z to a projective variety such that ϕF ∗OX = OZ and an irreducible curve C ⊂ X is mapped to a point by ϕF iﬀ [C] ∈ F . The key of the Minimal Model Program is the Theorem above. In other words the ⊥ Theorem states that to each extremal ray (or face) lying on the KX -negative side of KX there exists an extremal contraction, contracting exactly those curves on the variety, whose numerical classes lie on the extremal ray (or face).

Deﬁnition 1.2.10. Let F be a KX -negative face of the Mori cone. Then the unique map ϕF as in Theorem (1.2.9) (iii) is called the extremal contraction of F. Fact 1.2.11 ([KM98, Proposition 2.5]). Suppose ϕ : X → Y is the extremal contraction of an KX -negative extremal ray. Then ϕ is one of the following.

• The contraction ϕ is birational and codimX (Exc(ϕ)) = 1. Then ϕ contracts an irreducible divisor and we say that ϕ is a divisorial contraction.

• The contraction ϕ is birational and the codimension of the exceptional locus of ϕ is greater than or equal to 2; the contraction is called a small contraction.

• The last possibility of an extremal contraction is that dim Y < dim X. In this case we call the contraction a Mori ﬁbration.

12 Chapter 1

Furthermore, if X is a smooth threefold, then there exists no small contractions.

Definition 1.2.12. Let ϕ : X → Y be an extremal contraction of an extremal face F of the Mori cone with dim Y < dim X. In this case we say that ϕ is a not necessarily elementary Mori fibration. If we want to emphasize that F is an extremal ray, then we say that ϕ is an elementary Mori fibration.

Deﬁnition 1.2.13. A projective manifold is called a Fano manifold if its anti-canonical class is ample. A Fano surface is called del Pezzo surface.

Note that in case that X is Fano, Theorem (1.2.9) states that NE(X) is a polyhedral cone, or equivalently the Mori cone is generated by ﬁnitely many extremal rays.

Fact 1.2.14. A del Pezzo surface is isomorphic to P2, P1 × P1 or to the projective plane blown up in at most 8 points in general position, see for example [Kol96, III.3.1].

We will now turn to another cone N1,R(X), which will be important for our investiga- tions, the Moving cone of X.

Deﬁnition 1.2.15. A class C ∈ N1,R(X) is called movable if it can represented by an irreducible curve [C] = [Ct0 ] such that Ct0 is a member of an algebraic family (Ct)t∈T , which covers X. We write Mov(X) for the closure of the cone generated by all movable classes in N1,R(X). The cone Mov(X) is called the Moving cone or cone of movable curves of X. We will call an element of the Moving cone diﬀerent from the zero 1-cycle a movable class.

An important characterization of the cone of movable curves is given in [BDPP04]. It states that the cone of movable curves is dual to the cone of pseudoeﬀective divisors.

Theorem 1.2.16 ([BDPP04, Theorem 2.2]). Let X be a projective manifold. Then the cone of movable curves is dual to the pseudoeﬀective cone, i.e.

Mov(X)= c ∈ N1,R(X) c · E ≥ 0 for all E ∈ Eﬀ(X) .

The result of [BDPP04] allows to give an easy description of the cone of movable curves on a Fano threefold, which will be important for us in chapter 4.

Proposition 1.2.17 ([Bar07, Theorem 1.1]). Let X be a Fano 3-fold with exceptional divisors E1,...Ek. Then the Moving cone of X is given by

C ∈ NE(X) C · Ei ≥ 0, i =1,...k .

13 Preliminaries

1.3. Minimal rational curves

In this section, we recall the notion of minimal rational curves in order to ﬁx notation. For a comprehensive discussion see [Kol96]. Let X be a smooth n-dimensional variety and f : P1 → X be a rational curve. By a well known Theorem of Grothendieck, every locally free sheaf on P1 splits into a direct sum of line bundles. Thus we may write

∗ ∼ f TX = O(a1) ⊕···⊕O(an).

∗ Deﬁnition 1.3.1. A rational curve f is said to be free if f TX is generated by its global ∗ sections and it is said to be very free if f TX ⊗ O(−1) is generated by its global sections Deﬁnition 1.3.2. A free rational curve f is called minimal, if

∗ ∼ ⊕d ⊕n−d−1 f TX = O(2) ⊕ O(1) ⊕ O .

If f is a free rational curve, then deformations of f cover the whole manifold X, see for example [Deb01, Proposition 4.8]. This shows the following Remark.

1 Remark 1.3.3. If f is a free rational curve, then the image cycle f∗P is movable. Let f be a free rational curve. Considering the diﬀerential of f and taking into account ∼ that TP1 = O(2) yield that ai ≥ 2 for some i =1,...,n. This shows the following Remark.

1 1 Remark 1.3.4. Let f : P → X be a free rational curve. Then −KX · f∗P ≥ 2. If f is moreover minimal and f an embedding, then we can say even more.

Remark 1.3.5. Let f : P1 → X be a minimal rational curve and f be an isomorphism. The diﬀerential ∗ df : TP1 → f TX . can thus be seen as injective morphism of vector bundles

df : O(2) → O(2) ⊕ O(1)⊕d ⊕ O⊕n−d−1.

Since there exists no nontrivial morphisms O(2) → O(a) with a < 2, df induces an ∗ isomorphism of TP1 and the unique O(2)-term in f TX . An important property of a free rational curve is that a general deformation of this curve avoids subsets of codimension 2.

Proposition 1.3.6 ([Kol96, Proposition 3.7.]). Let X be a smooth projective variety. Let f : P1 → X be a free rational curve and let Z ⊂ X be a subvariety of codimension at least 2. Then g(P1) ∩ Z = ∅ for a general deformation g of f.

14 Chapter 2 Rationally connected foliations on surfaces

Let X be a uniruled manifold and H a polarisation on X. We have seen in Theorem (1.1.12) that the positive terms of the Harder-Narasimhan filtration of TX with respect to H induces foliations with rationally connected leaves. On the other hand the relative tangent sheaf of the MRC-quotient gives a foliation with rationally connected leaves, too. Thus it is natural to ask if there always exists a polarisation such that the highest positive term of the Harder-Narasimhan filtration induces the MRC-quotient. In this chapter we give a positive answer if X is a surface. That is we will show the existence of a polarisation such that the Harder-Narasimhan filtration with respect to the polarisation induces the maximally rationally connected quotient.

2.1. Rationally Connected Foliations on Surfaces and the MRC- quotient

Convention 2.1.1. In this section X denotes a projective surface.

We want to investigate the regions in the ample cone which induce the same Harder- Narasimhan filtration. More precisely we will divide the ample cone into parts, so that in each part we get the same Harder-Narasimhan filtration of the tangent bundle. This will lead to a decomposition of the ample cone into chambers. We will investigate the structure of this decomposition. This will help us to show that the MRC-quotient comes from the Harder-Narasimhan filtration of the tangent bundle with respect to a certain polarisation.

In order to compute the Harder-Narasimhan ﬁltration of the tangent bundle on surfaces, we only have to search for a destabilizing subsheaf whose quotient is torsion free. This is formulated in the next lemma.

Lemma 2.1.2. Let X be a smooth projective surface. If F ⊂ TX is a destabilizing subsheaf with respect to a polarisation such that TX /F is torsion-free, then the Harder- Narasimhan ﬁltration is given by 0 ⊂ F ⊂ TX .

Proof. Let H be a polarisation and F a destabilizing subsheaf of TX with respect to H. Consider the exact sequence

0 → F → TX → TX /F → 0. Rationally connected foliations on surfaces

Using that the rank and the ﬁrst Chern class are additive in short exact sequences, we obtain 1 1 µ (T )= µ (T /F )+ µ (F ). H X 2 H X 2 H

Since µH (F ) > µH (TX ), we therefore have µH (F ) > µH(TX /F ). Since F and TX /F are torsion free and of rank 1, they are semistable. Thus

0 ⊂ F ⊂ TX satisfies the properties of the Harder-Narasimhan filtration and by the uniqueness of the Harder-Narasimhan filtration with respect to H we are done.

Remark 2.1.3. Note that the Harder-Narasimhan filtration depends only on the numerical class of the chosen ample bundle. In particular it makes sense to ask how the filtration of a given sheaf depends on the ample bundle sitting in the finite dimensional vector space of all divisors modulo numerical equivalence. Moreover, the Harder-Narasimhan filtration extends naturally to R-line bundles.

Let us denote the regions in the ample cone which induce the same Harder-Narasimhan 1 filtration. Let H ∈ NR(X) be an ample bundle. To abbreviate notation, let us write HNFH (TX ) for the Harder-Narasimhan filtration of TX with respect to H. Now we consider the ample line bundles which induce the same filtration.

Deﬁnition 2.1.4. Let H be an ample R-divisor. Then we call

′ ∆H := {H ∈ AmpR(X) | HNFH′ (TX ) = HNFH (TX )}. the destabilizing chamber with respect to H. If the tangent bundle is semistable with respect to an ample R-divisor L, then we obtain a destabilizing chamber ∆L such that the tangent bundle is semistable with respect to all ample classes in ∆L. We call ∆L the semistable chamber.

The next Lemma states that a destabilizing chamber is cut out of the ample cone by a half space and thus the Lemma justiﬁes the name “chamber”.

1 Lemma 2.1.5. Let H ∈ NR(X) be an ample bundle such that TX is not semistable with respect to H. Let F be the maximal destabilizing subsheaf of TX with respect to H. Then the destabilizing chamber is given by

1 ∆ = H˜ ∈ Amp (X) c (F ) − c (T ) · H˜ > 0 . H R 1 2 1 X F 1 ˜ Proof. By Lemma (2.1.2) the condition (c1( ) − 2 c1(TX )) · H > 0 ensures that for all polarisations in ∆H we get the same Harder-Narasimhan ﬁltration, namely 0 ⊂ F ⊂ TX . Concerning the structure of these chambers we prove the following Proposition.

Proposition 2.1.6. Let X be a smooth projective surface. We have:

16 Chapter 2

(i) The destabilizing chambers are convex cones in AmpR(X).

(ii) The semistable chamber is closed in AmpR(X).

(iii) A destabilizing chamber diﬀerent from the semistable chamber is open in AmpR(X). (iv) The destabilizing chambers give a decomposition of the ample cone, i.e. the union of all chambers is the ample cone and the chambers are pairwise disjoint. Proof. The convexity property of the destabilizing chambers follows directly from the linearity of the intersection product. We demonstrate the proof for the semistable chamber. ′ ′ Let H,H ∈ AmpR(X) such that TX is semistable with respect to both H and H . Let F ⊂ TX be a coherent subsheaf of rank 1. Then

µH+H′ (F )= µH (F )+ µH′ (F ) ≤ µH(TX )+ µH′ (TX )= µH+H′ (TX ),

′ which shows that TX is semistable with respect to H + H . Statement (iii) is a direct consequence of the continuity of the intersection product, since for a maximal destabilizing subsheaf F ⊂ TX the condition 1 c (F ) − c (T ) · H > 0 1 2 1 X is an open condition. To prove (iv) note that by definition of the chambers, each polarisation appears in at least one chamber. Since for a given polarisation the associated maximal destabilizing subsheaf of TX is unique, the polarisation appears in exactly one chamber. Statement (ii) is a direct consequence of (iii) and (iv). In the proof of our main result, we will use the following corollary. Corollary 2.1.7. Let X be a smooth projective surface. Let ℓ be a line segment in AmpR(X), such that ℓ does not intersect the semistable chamber. Then ℓ is contained in a single destabilizing chamber. Proof. Assume ℓ intersects at least two destabilizing chambers. By Lemma (2.1.6) we get a partition of ℓ into disjoint open sets. This is impossible because ℓ is connected. Notation 2.1.8. Let X be a projective manifold. Then we write Aut(X) for the group of automorphisms on X and Aut0(X) for the connected component of the identity in Aut(X). To prove the semistability of the tangent bundle on certain surfaces having many automorphisms, we will give a useful lemma. Let σ ∈ Aut(X) and F ⊂ TX . By means ∗ ∗ of the differential of σ, we can identify TX and σ TX . Thus we can interpret σ (F ) as a ∗ subsheaf of TX . For instance, if p ∈ X and F := TX ⊗ Ip, then σ (F ) is identified with TX ⊗ Iσ−1(p) ⊂ TX . Proposition 2.1.9. Let X be a smooth projective surface and let σ ∈ Aut0(X). Let F be the maximal destabilizing subsheaf of TX with respect to some polarisation. We then have σ∗F = F . In particular: If F is a foliation then the automorphism σ maps each leaf of F to another leaf of F.

17 Rationally connected foliations on surfaces

1 0 ⊂ T 1 ⊂ T X/P X

2 0 ⊂ T 1 ⊂ T X/P X x

1 2 Figure 2.1: The ample cone of X = Σ0 and the chamber structure. Here TX/P1 and TX/P1 denote the relative tangent bundle of the ﬁrst and second projection.

F Proof. Let H ∈ AmpR(X) and let be the maximal destabilizing subsheaf of TX with ∗ respect to H. We compute the slope of σ (F) ⊂ TX :

∗ ∗ µH σ (F ) = H · c1(σ (F )) = H · σ∗(c (F )) 1 = H · c1(F ) 1 > 2 c1(TX ) · H. We give an explanation of the third equality. Recall that the group of automorphisms acts on the N´eron-Severi group. Since N 1(X) is discrete and Aut0(X) is the connected 0 1 ∗ component of the identity, Aut (X) acts trivially on N (X), i.e. σ (c1(F )) = c1(F ). ∗ We have shown that σ (F) is a destabilizing subsheaf of TX . By Lemma (2.1.2) and the ∗ uniqueness of the maximal destabilizing subsheaf of TX , we conclude that σ F = F .

Example 2.1.10 (Hirzebruch Surfaces). Let Σn be the n-th Hirzebruch surface and let 1 π : Σn −→ P be the projection onto the projective line. We denote the ﬁber under the projection by f and the distinguished section with selﬁntersection −n by C0. Recall that 1 NR(Σn) = hC0, fi and a divisor D ≡ aC0 + bf is ample if and only if a > 0 and b > an, see see [Har77, chapter V.2]. The canonical bundle is given by −KΣn = 2C0 +(2+ n)f. The relative tangent bundle of π is a natural candidate for a destabilizing subbundle. We have the sequence

∗ 1 0 → TΣn/P1 → TΣn → π T P → 0

Let H := xC0 + yf be a polarisation. Then one can compute that TΣn/P1 is destabilizing if and only if −2x − nx +2y > 0. In particular we compute for n ≥ 2: −2x − nx +2y> −2x − nx +2nx = −2x + nx ≥ 0. Therefore, for n ≥ 2 the Harder-Narasimhan ﬁltration is given by

0 ⊂ TX/P1 ⊂ TX for all polarisations. In other words we obtain only one destabilizing chamber. 1 1 For n = 0 we have Σ0 = P × P and we get three chambers. The two destabilizing chambers correspond to the two relative tangent bundles of the projections. They are cut

18 Chapter 2 out by the inequalities x>y and x 2 y the relative tangent bundle is destabilizing. Since Σ1 is the projective plane blown up at a point p, the group of automorphisms is the automorphism group of the projective plane leaving p ﬁxed. The destabilizing foliation corresponds to the radial foliation through p in the plane. So if there were another foliation F coming from the Harder-Narasimhan ﬁltration of

TΣ1 , we could deform the leaves with these automorphisms. Then we would again obtain leaves of this foliation by Lemma (2.1.9). So unless F is the foliation given by the relative tangent bundle of the projection morphism, we could deform each leaf of F while leaving a point on the leaf not lying on C0 ﬁxed. Thus the foliation induced by F would have singularities on a dense open subset of Σ1 which is absurd. So the tangent bundle is 3 semistable for x ≤ 2 y.

0 ⊂ TX/P1 ⊂ TX

Semistable chamber

Figure 2.2: The chamber structure of X := Σ1

Example 2.1.11 (Ruled surfaces). Let π : X → C be a ruled surface over a curve of genus g ≥ 1. We will show that the only destabilizing subsheaf with respect to any polarisation is the relative tangent sheaf of π. We denote the ﬁber under the projection by f and the distinguished section with self- 1 intersection −e by C0. Recall that NR(Σn) = hC0, fiR+ and a divisor D ≡ aC0 + bf is 1 ample if and only if a> 0 and b > ae in case that e ≥ 0 and it is ample if a> 0,b> 2 ae in case e< 0. The canonical bundle is given by −KΣn =2C0 − (2g − 2 − e)f. Using the exact sequence ∗ 0 → TX/C → TX → π TC → 0, we compute c1(TX/C )=2C0 + ef. Let H := aC0 + bf be a polarisation. Then by Lemma (2.1.2) TX/C is maximally destabilizing with respect to aC0 + bf iﬀ 1 c (T ) · H > c (T ) · H. 1 X/C 2 1 X A computation shows that the equality above is equivalent to 1 (2g − 2 − e) · a + b> 0. 2 19 Rationally connected foliations on surfaces

By the ampleness of H the inequality above holds for all polarisations.

In the above examples, the Harder-Narasimhan ﬁltration induces the MRC-quotient with respect to any polarisation. The next example shows that this is false in general as already observed in [Eck08].

Example 2.1.12 ([Eck08, Section 3]). Let X = P1 × C, where C is an elliptic curve. Let 1 p1 : X → C and p2 : X → P be the two projections. Consider the blow up π : X˜ → X of X in three different points such that no two of them lie on the same horizontal or vertical fibre and denote the exceptional divisors by E1, E2, E3. Then the maximal rationally connected quotient is given by the blow up of the ruling of X. Let F be the associated F ∗ foliation. A computation shows that c1( )= p2O(2) − E1 − E2 − E3. ∗ ∗ 1 ′ Let L := p1A + p2B with degC A = 3 and degC P = 4. Then one shows that L := ∗ ′ π L − 2E1 − 2E2 − 2E3 is ample, but L · c1(F )=0. Now we want to answer the question if there always exists a polarisation such that the Harder-Narasimhan filtration gives rise to the MRC-quotient.

Theorem 2.1.13. Let X be a uniruled projective surface. Then there exists a polarisation such that the maximal rationally connected quotient of X is given by the foliation associated to the highest positive term in the Harder-Narasimhan ﬁltration with respect to this polarisation.

Proof. To start, observe that there is always a polarisation A such that c1(TX ) · A > 0. Indeed, there exists a free rational curve f : P1 → X. See [Deb01, Corollary 4.11] for a proof of the existence of such a curve. By the definition of a free rational curve, we can write ∗ f (TX )= O(a1) ⊕ O(a2) with a1 + a2 ≥ 2. Thus 1 −KX · f∗P = a1 + a2 ≥ 2, 1 see also Remark (1.3.2). Write ℓ := f∗P for the image cycle, which is movable by Remark (1.3.3). Since ℓ is movable, it is in particular nef. So for any ample class H and any ε> 0, the numerical class of ℓ + εH will be ample. Thus for sufficiently small ε the numerical class of ℓ + εH will intersect −KX positively. This shows the existence of a polarisation which intersects the anticanonical divisor positively. First let us assume that X is not rationally connected. As we have just seen, we can find a polarisation H with c1(TX ) · H > 0. There exists a destabilizing subsheaf F of TX , since otherwise X would be rationally connected by Theorem (1.1.12). Furthermore the slope of F has to be bigger than c1(TX ) · H and therefore positive. So this sheaf will give a foliation with rationally connected leaves and hence the maximal rationally connected quotient. Next we consider the case where X is rationally connected. We then fix a very free rational curve ℓ on X. For a proof of the existence of a very free rational curve see [Deb01, Corollary 4.17]. This means that TX |ℓ is ample. So we know that each quotient of TX |ℓ has strictly positive degree. Since ℓ is movable, it is in particular nef. Let H be an ample class. Because ℓ is nef, we

20 Chapter 2

1 know that Hε := ℓ + εH is ample in NQ(X) for any ε> 0. Observe that c1(TX ) · Hε > 0 for sufficiently small ε, say for 0 <ε<ε0. If TX is semistable with respect to a certain polarisation Hε with 0 <ε<ε0, the claim follows since TX has positive slope and induces a trivial foliation which gives the rationally connected quotient. If TX is not semistable for all polarisations Hε with 0 <ε<ε0, let Fε be the maximal destabilizing subsheaf of TX with respect to Hε. Because of Corollary (2.1.7) the ray Hε stays in one destabilizing chamber. This ensures that F := Fε remains constant. Now it is clear that for sufficiently small ε both the slope of F and the slope of TX /F will be positive with respect to Hε. Therefore the Harder-Narasimhan filtration of TX with respect to Hε yields the maximal rationally connected quotient. Remark 2.1.14. The proof shows that there not only exists a polarisation such that the highest positive part of the Harder-Narasimhan filtration induces the maximal rationally connected quotient, but that there exists an open subset U in AmpR(X) such that the Harder-Narasimhan filtration of TX with respect to all polarisation in U induces the MRC-quotient.

Chapter 3 The slope with respect to movable curves

In this section we generalize the notion of a slope of a sheaf. So far, the slope of a sheaf depended on a chosen ample line bundle, i.e. we regarded the slope of a sheaf as a function on the space of divisors modulo numerical equivalence. The idea is to measure the slope directly with respect to curves. In other words, given a sheaf F we consider the n−1 intersection c1(F )·C for a curve C instead of the product c1(F )·H . The slope is then a linear function on N1,R(X) whose formal properties are much more well behaved than the original definition. Moreover, we not only consider curves of the form C = Hn−1, that is curves which arises as complete intersections of ample bundles, but more generally we measure the slope with respect to classes of movable curves. In the first section of this chapter we will prove that with this more general notion of a slope, we still have a unique Harder-Narasimhan filtration with the same defining properties as in Theorem (1.1.10). At a first sight one could hope that the interior of the cone of movable curve is the cone of complete intersection curves, but this is false in general. We will give an example in the second section. In the last section of this chapter, we consider the Harder-Narasimhan filtration as a function of the cone of movable curves and investigate its properties.

3.1. The Harder-Narasimhan ﬁltration

To start, we give the generalization of the definition of a slope of a coherent sheaf. Let X be a projective manifold. Definition 3.1.1. Let F be a coherent sheaf of positive rank on X and let C be a movable class. Then we define the slope of F with respect to C as c (F ) · C µ (F ) := 1 . C rkF Let F be a torsion free coherent sheaf and C a movable class on X. Then F is called semistable with respect to C or simply C -semistable, if for all proper coherent subsheaves G ⊂ F of positive rank, we have

µC (G ) ≤ µC (F ). Remark 3.1.2. The linearity of the intersection product directly implies the linearity of the slope, that is for a a sheaf F , classes C,C′ ∈ Mov(X) and λ,λ′ ∈ R we have ′ µλC+λ′C′ (F )= λµC (F )+ λ µC′ (F ). The slope with respect to movable curves

Now that we have defined the slope of a coherent sheaf with respect to movable classes, we explain why we still have a unique Harder-Narasimhan filtration. Theorem 3.1.3. Let F be a coherent torsion free sheaf on a projective manifold. Given a movable class C ∈ Mov(X), there is a increasing filtration,

0= F0 ⊂ F1 ⊂ . . . ⊂ Fr = F such that

(i) µC (Fi/Fi−1) >µC (Fi+1/Fi) for i =1,...r − 1,

(ii) the factors Fi/Fi−1 are torsion free and semistable for all i =1,...r,

(iii) the sheaves Fi are saturated in F . Moreover, this filtration is unique. Actually, we only have to make minor changes in the proof given in [HL97, Section 1.3]. To begin with, we state the properties of the slope which allow us to define the Harder- Narasimhan filtration. We define the degree with respect to a movable class C on a projective variety X to be the map

d : { coherent sheaves on X } −→ Z F 7−→ c1(F ) · C Lemma 3.1.4. The degree is additive in short exact sequences

Proof. This follows since even the ﬁrst Chern class is additive in short exact sequences. Lemma 3.1.5. Let C be a movable class and let F ⊂ F ′ ⊂ G torsion free sheaves, such that F ′ is the saturation of F in G , then

′ µC (F ) ≤ µC(F ).

Proof. By the deﬁnition of the saturation of F in G , the sheaf

F ′/F is a torsion sheaf. By [Kob87, V, Proposition (6.14)], the line bundle associated to ′ c1(F /F ) admits a nontrivial holomorphic section and is therefore eﬀective. Since C is a movable class, it has nonnegative intersection with every eﬀective divisor by Theorem (1.2.16) and thus ′ ′ c1(F /F ) · C = c1(F ) · C − c1(F ) · C ≥ 0.

Now an analysis of the construction of the Harder-Narasimhan ﬁltration explained in [HL97, Section 1.3] yields that Lemma (3.1.4) and (3.1.5) are the only properties of the slope which are needed in the proofs. This has also been observed in [CP07, Proposition 1.3]. Since this ﬁltration is one major object of our studies, we will recall the proof below.

24 Chapter 3

Lemma 3.1.6. Let X be a projective manifold, F a coherent torsion free sheaf on X and C a movable class on X. Then the following are equivalent:

(i) F is semistable.

(ii) For all saturated subsheaves E ⊂ F with 0 < rkE < rkF we have µ(E ) ≤ µ(F ).

(iii) For all quotient sheaves F → G with 0 < rkG < rkF we have µ(F ) ≤ µ(G ).

(iv) For proper torsion free quotient sheaves F → G with 0 < rkG < rkF we have µ(F ) ≤ µ(G ).

Proof. We just write µ for the slope with respect to C. The directions (i) ⇒ (ii) follows directly from the deﬁnition. The proof (ii) ⇒ (i) follows from Lemma (3.1.5), that is the slope of E increases if we replace E by its saturation in F . The implication (iii) ⇒ (iv) is immediately clear. An exact sequence of the form

0 → E → F → G → 0 yields rk(E )+rk(G ) = rk(F ) and c1(E )+ c1(G )= c1(F ). Thus we have

rk(G ) rk(E ) c (F )+ c (F )= c (E )+ c (G ), rk(F ) 1 rk(F ) 1 1 1

This yields rk(E )(µ(E ) − µ(F )) = rk(G )(µ(F ) − µ(G )). This formula yields (i) ⇒ (iii). Moreover, since E is saturated in F iﬀ F /E is torsion free, we have (ii) ⇔ (iv).

Lemma 3.1.7. Let C ∈ Mov(X) and let F and G be coherent torsion free sheaves on the projective manifold X. Assume that F and G are semistable. If µC(F ) > µC (G ), then Hom(F , G )=0.

Proof. Let f : F → G be a nontrivial morphism of sheaves. Then using that F and G are semistable, we have µC (F ) ≤ µC (Im(f)) ≤ µC (G ) and the lemma is proved.

The key of the construction of the Harder-Narasimhan ﬁltration is the following Proposi- tion.

Proposition 3.1.8. Let X be a projective manifold, E a torsion free coherent sheaf on X and µ = µC the slope with respect to a movable class C on X. Then there is a subsheaf F ⊂ E such that for all subsheaves G ⊂ E , we have µ(F ) ≥ µ(G ). Furthermore, if we have µ(F ) = µ(L ) for a sheaf L in E , then we have L ⊂ F . In particular, the sheaf F is uniquely determined and semistable.

25 The slope with respect to movable curves

Proof. Let S be the set of nontrivial coherent subsheaves of E with positive rank. Deﬁne a partial ordering on S as follows:

F1 ≤ F2 iﬀ F1 ⊂ F2 and µ(F1) ≤ µ(F2).

Note that since any ascending chain of subsheaves of a coherent sheaf has to terminate, the set S is inductively ordered. Thus by Zorn’s Lemma there exists a maximal element, that is by deﬁnition an element F ′ such that for all G ∈ S with G ≥ F ′ we automatically have F ′ = G . See for example [Lan65, Appendix 2] for the construction of a maximal element in an inductively ordered set.

Let F be ≤-maximal with minimal rank among all maximal subsheaves. We will prove that F has the asserted properties.

Suppose G is a coherent subsheaf of F with µ(G ) ≥ µ(F ) and F ( F + G .

Step 1: We claim that rk(G ∩ F ) > 0 and µ(G ∩ F ) >µ(F ).

We consider the exact sequence

0 → F ∩ G → F ⊕ G → F + G → 0.

Assume on the contrary G ∩ F = 0. The the exact sequence above would yield F ⊕ G =∼ F + G and thus µ(F + G )= µ(F ⊕ G ). Since F ( F + G , the maximality of F would imply µ(F ) >µ(F + G ). Hence

µ(F ) >µ(F ⊕ G ).

A direct computation shows µ(F ) >µ(G ), a contradiction to µ(G ) ≥ µ(F ).

So let us assume rk(F ∩ G ) > 0. To abbreviate notation, let us write f,g for the rank of F and G respectively. From the equation

fµ(F )+ gµ(G ) = rk(F ∩ G )µ(F ∩ G )+rk(F + G )µ(F + G ) we obtain

µ(F )(rk(F ∩G )+rk(F +G )−g)+gµ(G ) = rk(F ∩G )µ(F ∩G )+rk(F +G )µ(F +G ).

Adding on both sides of the equation rk(G ∩ F )µ(G ) and transposing the equation gives

rk(F ∩ G )(µ(G ) − µ(F ∩ G )) =

rk(F + G )(µ(F + G ) − µ(F )) + (rk(G ) − rk(F ∩ G )) (µ(F ) − µ(G )) .

<0 ≥0 ≤0

| {z } |26 {z } | {z } Chapter 3

This yields µ(F ) ≤ µ(G ) <µ(G ∩ F ). This ﬁnishes the proof of the claim.

Step 2: By Step 1, we can ﬁx H ⊂ F with µ(H ) >µ(F ) which is maximal in F with respect to ≤. Let H ′ be maximal in the set of subsheaves of E containing H . Then we have µ(F ) <µ(H ) ≤ µ(H ′). We claim that the maximality of H ′ and F imply H ′ 6⊂ F . Suppose on the contrary H ′ ⊂ F . If rk(H ′) < rk(F ), we would obtain a contradiction to the minimality of the rank of F and if rk(H ′) = rk(F ), then F /H ′ would be a torsion sheaf thus we would have µ(F ) ≥ µ(H ′) by Lemma (3.1.5), a contradiction. Consequently, F is a proper subsheaf of F +H ′ and the maximality of F yields µ(F ) > µ(F + H ′). Since µ(F ) < µ(H ′), we have µ(H ′) > µ(F + H ′). As in Step 1 this implies µ(H ′) < µ(F ∩ H ′). Hence µ(H ) < µ(F ∩ H ′). Since H ⊂ F ∩ H ′ ⊂ F , we obtain a contradiction to the maximality of H . Thus we have µ(G ) <µ(F ) or G ⊂ F . Since µ(G ) >µ(F ) is impossible, the Proposition is proved.

Deﬁnition 3.1.9. If the unique sheaf F in Proposition (3.1.8) is not equal to E , then we call F the maximal destabilizing subsheaf of E .

The proof of Proposition (3.1.3) is now rather easy.

Proof of Theorem 3.1.3. Let F be torsion free sheaf. Let E1 be the maximal destabilizing subsheaf. Then E1 is semistable and saturated in F , hence F /E1 is torsion free. So we can construct the desired filtration inductively. It is clear that µ(E1) > µ(E2/E1) since otherwise we would have µ(E2) ≥ µ(E1) contra- dicting the maximality of E1. E E ′ Finally, we prove the uniqueness of the filtration. Assume • and • are two Harder- E E ′ E Narasimhan filtrations of . Assume without loss of generality µ( 1) ≥ µ( 1). Let j be E ′ E minimal such that 1 ⊂ j. The composition of morphisms

E ′ E E E 1 → j → j/ j−1

E ′ E E is a nontrivial morphism. Since both 1 and j/ j−1 are semistable and torsion free, we conclude by Lemma (3.1.7)

E E E ′ E E E µ( j/ j−1) ≥ µ( 1) ≥ µ( 1) ≥ µ( j/ j−1).

E E ′ Thus equality holds everywhere, implying j = 1. By the maximality of 1 and 1 this E E ′ forces 1 = 1. By induction on the rank we can assume that uniqueness holds for the E E E ′ E E E Harder-Narasimhan ﬁltration of / 1. This shows i / 1 = i/ 1 and this ﬁnish the uniqueness part of the proof.

27 The slope with respect to movable curves

3.2. Complete intersection curves versus movable curves

Let C be a movable class on a projective manifold of dimension n. If C = [H1 · . . . · Hn−1] for ample divisors H1,...,Hn−1, then obviously the Harder-Narasimhan ﬁltration with respect to C agrees with the original Harder-Narasimhan ﬁltration with respect to H1,...,Hn. In this case Theorem (1.1.12) is true whereas it might be wrong in case that C is a class of a movable curve which does not arise as the intersection of ample divisors.

Deﬁnition 3.2.1. Let X be a projective manifold of dimension n. A class C ∈ N1,R(X) is called complete intersection curve, if there exist ample R-divisors H1,...,Hn−1 such that C = [H1 · . . . · Hn−1]. The cone of complete intersection curves is the closure in N1,R(X) of the convex cone generated by all classes of complete intersection curve. We denote this cone by CIC(X).

Clearly, if C ∈ N1,R(X) such that C = [H1 · . . . · Hn−1], then C · D ≥ 0 for all D ∈ Eff(X). If C ∈ CIC(X), we can approximate C with effective linear combinations of complete intersection curves. The continuity of the intersection product implies C ·D ≥ 0 for all D ∈ Eff(X). Hence C ∈ Mov(X) by Theorem (1.2.16). We have shown the following Remark.

Remark 3.2.2. CIC(X) ⊂ Mov(X).

If X is a surface, then divisors and 1-cycles are the same. Therefore we have that CIC(X) = Amp(X). It is known that Nef(X) = Amp(X), see for example [Laz04, Theorem 1.4.23]. Since on a surface the nef cone is dual to pseudoeﬀective cone, Theorem (1.2.16) implies Nef(X)= Mov(X). Thus we have shown the following Remark.

Remark 3.2.3. If X is a surface, then

CIC(X)= Mov(X).

One might hope that this is still true in higher dimensions. Unfortunately, this is false in general as shown by the next example.

Example 3.2.4. We start with P3. We blow up P3 in a line, say g, to get another smooth ∼ 1 1 threefold, say X˜, with exceptional divisor E˜1 = P × P . We denote the projection map 1 ˜ 3 by φ. The Neron-Severi space NR(X) is spanned by the pullback of a hyperplane H in P 3 and E˜1. Denote by l the class of a line in P and by f˜1 a fibre in E˜1. The space of cycles ∗ ˜ N1,R(X˜) is spanned by φ l and f1. We take a point on a fibre in E˜1 and blow up the point ∼ 2 to get a threefold X with exceptional divisor E2 = P and projection map ψ : X → X˜. ∼ 2 Denote with f2 the class of a line in E2 = P . Let π := φ ◦ ψ be the composition of the ∗ ˜ ∗ two blow ups. Let f1 := ψ (f1) be the pullback of a fibre in E˜1 and E1 := ψ (E˜1) be the pullback of E˜1. Then 1 ∗ NR(X)= hπ H, E1, E2iR+ ∗ N1,R(X)= hπ l, f1, f2iR+

28 Chapter 3

The multiplication of the intersection ring of X is given by

∗ π H E1 E2 ∗ ∗ π H π l f1 0 ∗ E1 f1 −π l +2f1 0 E2 0 0 −f2 and ∗ π H E1 E2 π∗l 1 0 0 f1 0 −1 0 f2 0 0 −1 Claim: The Mori cone and the pseudoeﬀective cone is given by

∗ NE(X)= hπ l − f1 − f2, f1 − f2, f2iR+

∗ Eﬀ(X)= hπ H − E1 − E2, E1 − E2, E2iR+ . Let us ﬁrst compute the Mori cone: Let c be an irreducible curve in X not contained in the exceptional divisors. Consider π∗c ≡ al, where a = deg π∗c. We have ∗ c ≡ aπ l − b1f1 − b2f2. Clearly, b1, b2 ≤ a, since π∗c intersects the blow up line g in less or equal than a = deg π∗c points. So c lies in the claimed cone. Now suppose the irreducible curve c lies in the strict transform of E˜1, which is numer- ∼ 1 1 ically equivalent to E1 − E2. Consider ψ∗c ⊂ E˜1 = P × P . Denote by h1 = f˜1 the class of a vertical line in E˜1 and by h2 the class of a horizontal line in E1. We write ψ∗c ≡ d1h1 + d2h2. The number d1 ∈ N counts the points of intersection of ψ∗c with h2 and d2 ∈ N counts the points of intersection of ψ∗c with h1. A short computations ∗ ∗ ∗ shows that ψ h2 = π l − f1 as cycles in X˜. Therefore c ≡ d2π l − (d2 − d1)f1 − df2 where d ≤ d1,d2. Thus c lies in the claimed cone. Finally, if the curve c lies in E2, then c ≡ af2 with a ≥ 0. ∗ Since all the extremal rays in the cone hπ l − f1 − f2, f1 − f2, f2iR+ can be represented

Figure 3.1: The manifold X by eﬀective curves, we have shown the claimed equality.

29 The slope with respect to movable curves

A similar argumentation shows the claimed equality for the pseudoeﬀective cone.

The duality of the Moving cone and the pseudoeﬀective cone (see [BDPP04]), yields

∗ ∗ ∗ Mov(X)= hπ l, π l − f1, 2π l − f1 − f2iR+ . Since the pseudoeﬀective cone and the nef cone are dual, and the interior of the nef cone ∗ is the ample cone, a class aπ H − b1E1 − b2E2 is ample if and only if a > b1 + b2 and b1 > b2 > 0.

To show that CIC(X) ( Mov(X), we prove the following claim.

∗ Claim: The class 2π l − f1 − f2 cannot be approximated by complete intersection curves.

A complete intersection curve looks like

∗ ∗ (π H − b1E1 − b2E2)(π H − c1E1 − c2E2) ∗ = (1 − b1c1)π l − (c1 + b1 − 2c1b1)f1 − b2c2f2 with 1 > b1 + b2, b1 > b2 > 0 and 1 >c1 + c2,c1 >c2 > 0. (3.1) After multiplying the complete intersection curve with 1 , the class of the curve is 1−b1c1

∗ c1 + b1 − 2c1b1 b2c2 π l − f1 − f2. 1 − b1c1 1 − b1c1

We now have to show, that the coeﬃcient of f1 and f2 cannot get arbitrary close to 1/2.

Let us assume on the contrary that for any ǫ =6 0 with |ǫ|≪ 1, we find coefficients fulfilling the auxiliary conditions (3.1) and c + b − 2c b 1 1 1 1 1 = + ǫ. 1 − b1c1 2 This forces 1 − 2c1 +2ǫ b1 = . 2 − 3c1 +2ǫc1 2 The case if the denominator vanishes, i.e. c1 = 3−2ǫ , can treated separately. Assume that 2 the denominator is positive, i.e. c1 < 3−2ǫ . Since we have to guarantee b1 > 0, we obtain 1 c1 < 2 + ǫ. On the other hand, if the denominator is negative, the condition b1 < 1 is equivalent to 1 − 2c1 +2ǫ> 2 − 3c1 +2ǫc1. This gives c1 > 1, which is impossible.

So all in all, we assume 1 c < + ǫ. (3.2) 1 2 Furthermore a short computation shows, that 1 b < + ǫ. (3.3) 1 2 30 Chapter 3

We now turn to the coeﬃcient of f2 of the complete intersection curve: 2 b2c2 b1c1 1/4+ ǫ + ǫ < ≤ 2 . 1 − b1c2 1 − b1c1 3/4 − ǫ − ǫ The ﬁrst inequality is valid because of condition (3.1) and the second inequality follows from (3.2) and (3.3). So for ǫ small enough, we can bound b2c2 away from 1/2. This 1−b1c2 proves the claim.

3.3. Destabilizing chambers on projective manifolds

Let X be a projective manifold and C ∈ Mov(X) be a movable class. Let

0= F0 ⊂ F1 ⊂ . . . ⊂ Fk = TX be the Harder-Narasimhan filtration with respect to C. To abbreviate notation, let us write HNFC (TX ) for the Harder-Narasimhan filtration of TX with respect to C. Now we consider the set of movable curves which induce the same filtration. Definition 3.3.1. Let C be a movable class. Then the set

′ ∆C := {C ∈ Mov(X) | HNFC′ (TX ) = HNFC (TX )}. is called destabilizing chamber with respect to C. If the tangent bundle is semistable with ˜ respect to a movable class C, we obtain a destabilizing chamber ∆C˜ such that the tangent bundle is semistable with respect to all curves in ∆C˜ . We call ∆C˜ the semistable chamber. In the sequel, we will show that the chamber structure is not too complicated: We will see that the chambers are convex subcones of the Moving cone whose closures are locally polyhedral in the interior of the Moving cone. Moreover, the decomposition of the Moving cone into these chambers is locally ﬁnite in the interior of Mov(X). If in addition Mov(X) is polyhedral, then the chambers are polyhedral and the chamber structure turns out to be ﬁnite. Lemma 3.3.2. Let C ⊂ Rd be a d-dimensional convex cone and let I be an arbitrary set of indices. Suppose for each i ∈ I we have convex subcones Si ⊂ C with the properties that

• C = i∈I Si,

• the conesS Si are pairwise disjoint, • the decomposition of C is locally ﬁnite in the interior of C.

Then the closure of the cones Si are locally polyhedral in the interior of C.

Proof. We consider a subcone of Sj of dimension d and a point p ∈ ∂Sj which lies in the interior of C. Since the decomposition of C around p is ﬁnite, we ﬁnd d-dimensional cones

Si1 ,...,Sin with j =6 i1,...,in such that locally the boundary of Sj is given by Sj ∩ ∂Sik . This shows that the boundary of the chambers is locally convex and thus contained in a hyperplane.

31 The slope with respect to movable curves

Lemma 3.3.3. Let X be a projective manifold. Let C ⊂ Mov(X) be a closed polyhedral subcone. Then the chamber structure is ﬁnite in C. Note that we do not assume that Mov(X) itself is polyhedral. Proof of Lemma 3.3.3. We want to prove that the chamber structure in C is ﬁnite.

Claim: Let E be a torsion free coherent sheaf on X and C ∈ C be a movable class. Then there exists only ﬁnitely many diﬀerent maximal destabilizing subsheaves of E in a neighborhood of C.

Step 1. In this step, we will show that the claim implies the Lemma. Let C ∈ C. The claim shows that there exists only finitely many maximal destabilizing subsheaves of TX around C, say H1,..., Hk for k ∈ N. Since TX /Hi is torsion free for i = 1,...,k by construction of the maximal destabilizing subsheaf, we can apply the claim to TX /Hi. This shows that there exists only finitely many maximal destabilizing subsheaves of TX /Hi for all i. This and the construction of the Harder-Narasimhan filtration, see the proof of Theorem (3.1.3), immediately implies that there exists only finitely many different Harder-Narasimhan filtrations around C. Since the Harder-Narasimhan filtration with respect to a curve C′ is not effected by scaling C′ it is enough to consider the Harder-Narasimhan filtration with respect to curves in a compact subset of C, for example in an affine hyperplane section of C. Thus the chamber structure is finite in C.

In the next steps, we will prove the claim. Step 2. Let E be a torsion free sheaf on X and C ∈ C. We assume on the contrary that there exists a sequence of classes of movable curves ([Cn])n∈N in C converging to [C] such that the Cn-maximal destabilizing subsheaves Gn ⊂ E are pairwise disjoint. We suppose, by passing to a subsequence if necessary, that the rank of Gi stays constant for all i. Since by our assumptions C is polyhedral, we ﬁnd linearly independent classes [c1], [c2],..., [cs] in C ⊂ Mov(X) such that inﬁnitely many [Cn] lie in the relative interior of the convex cone generated by [c1],..., [cs]. Note that in general we cannot guarantee that s = dim C since possibly almost all classes [Cn] lie on the boundary of C.

Consider U := h[c1],..., [cs]iconv. After passing to a subsequence and scaling the classes [Cn], we assume that [Cn] ∈ U for all n. We complete { [c1],..., [cs] } to a basis of N1,R(X) G ρ n and denote the dual basis by H1,...,Hρ. We write c1( n) = j=1 aj [Hj]. Let us show n n n that the coefficients a1 , a2 ,...,as ∈ Z are bounded from above for all n ∈ N. Suppose on the contrary that, after passing to a subsequence, we would findP an index j ∈{ 1,...,s } n such that (aj )n converges to infinity. This would imply G n c1( n) · cj = aj −→ ∞, which is impossible, since the slopes of subsheaves of E with respect to [cj] are bounded from above by Proposition (3.1.8).

n Step 3. Let us prove that the coeﬃcients aj are also bounded from below for 1 ≤ j ≤ s. We assume on the contrary, that there exists a sequence of coeﬃcients converging to

32 Chapter 3

minus inﬁnity. After passing to a subsequence and reordering { [H1],..., [Hs] }, we assume n a1 −→ −∞. n n If the coeﬃcients a2 ,...,as are not bounded from below, then again after passing to n a subsequence and reordering, we can assume a2 −→ −∞. Inductively, we obtain t ∈ { 1,...,s } and a sequence of subsheaves Gn ⊂ E with

t s G n n c1( n)= aj [Hj]+ ai [Hi] + [Dn] j=1 i=t+1 X X n n such that aj → −∞ for j ∈ { 1,...,t } and (aj )n are bounded for j ∈ { t +1,...,s } ρ n and [Dn] = j=s+1 aj [Hj]. Again after passing to a subsequence, we assume that the n n coeﬃcients aj are constant for t +1 ≤ j ≤ s, say aj = aj for all n ∈ N. Now considerP t s G 1 c1( 1)= aj [Hj]+ aj[Hj] + [D1]. j=1 j=t+1 X X n G As (aj )n converges to −∞ for 1 ≤ j ≤ t, we ﬁnd a subsheaf k with k ≥ 2 and

t s G k c1( k)= aj [Hj]+ aj[Hj] + [Dk] j=1 j=t+1 X X and k 1 aj < aj ′ ′ for all j ∈{ 1,...,t }. Since Dn · C = 0 for all C ∈ U and all n ∈ N, it follows G G µCk ( 1) >µCk ( k).

This inequality contradicts that Gk is the maximal destabilizing subsheaf of E with respect to Ck.

n n G Step 4. We have assured ourselves that the coefficients a1 ,...,as ∈ Z of the c1( n) are bounded for all n ∈ N. After passing to a subsequence, we assume that the coefficients are constants. But this shows G G µCn ( n)= µCn ( m) for all m =6 n. Since all Gj are of the same rank, this contradicts the uniqueness of the maximal destabilizing subsheaf. This finishes the proof of the Claim. The structure of the destabilizing chambers is content of the following Theorem.

Theorem 3.3.4. Let X be a projective manifold.

(i) The destabilizing chambers are convex cones whose closures are locally polyhedral in the interior of Mov(X).

(ii) The semistable chamber is closed in Mov(X). If the Harder-Narasimhan ﬁltration of TX with respect to C has the maximal number of terms, then ∆C is open.

33 The slope with respect to movable curves

(iii) The destabilizing chambers are pairwise disjoint and provide a decomposition of the Moving cone. This decomposition is locally ﬁnite in the interior of Mov(X).

Proof. Proof of (i): Note that the slope of a given sheaf F is a linear function on the cone of movable curves. More precisely, for movable classes C,C′ and λ,λ′ ∈ R we have

′ µλC+λ′C′ (F )= λµC (F )+ λ µC′ (F ).

Then it is easy to verify that if F ⊂ E is the maximal destabilizing subsheaf of E with respect to C and C′, then F is also maximal destabilizing with respect to C + C′. The linearity of the slope immediately shows the convexity of the semistable chamber as well. The property that the chambers are polyhedral follows from the previous Lemma (3.3.2) if we grant (iii) for a moment.

Proof of (ii): To see that the semistable chamber is closed, it is enough to recognize that semistability is a closed condition. More precisely, let ∆ be the semistable chamber and suppose we have a sequence ([Cn])n∈N in ∆ converging to [C] ∈ Mov(X). Let F ⊂ TX be a coherent subsheaf of TX . We have F µCn ( ) ≤ µCn (TX ) for all n ∈ N. By the continuity of the intersection product this inequality holds in the limit. Thus C belongs to ∆. To prove the second assertion in (ii), note the following: Suppose the Harder-Narasimhan ﬁltration with respect to C is given by

0= F0 ⊂ F1 ⊂ . . . ⊂ Fn = TX , where n = dim X. Since the quotients Fi/Fi−1 torsion free and of rank 1 for all i = ′ 1,...,n, they are semistable with respect to all curves in Mov(X). Thus C ∈ ∆C iﬀ the condition µC′ (F1/F0) >...>µC′ (Fn/Fn−1) ′ is fulﬁlled. Since this condition holds in neighborhood of a any [C ] ∈ ∆C , the chamber is open in Mov(X).

Let us prove (iii). Since the Harder-Narasimhan filtration is well defined, the chambers are pairwise disjoint. Thus the chambers provide a decomposition of the Moving cone. The decomposition is locally finite by Lemma (3.3.3). Somewhat surprisingly the chamber structure is to some extend determined by the geometry of the cone of movable curves.

Proposition 3.3.5. On a projective manifold whose closed cone of movable curves is polyhedral, the chamber structure is ﬁnite. Proof. This is a direct consequence of Lemma (3.3.3).

Remark 3.3.6. The above Proposition applies in particular to Fano manifolds, since it is known that the closed cone of movable curves is polyhedral, see [Ara10, Corollary 1.2].

34 Chapter 3

On the other hand we can easily have surfaces admitting a decomposition of the cone of movable curves into destabilizing chambers which accumulate at the boundary of the cone of movable curves. This phenomenon is illustrated in the next example.

Example 3.3.7. Consider the projective plane. Write H for the class of the line in P2. Consider two general cubics in |3H| which intersect in exactly 9 points, say p1,...,p9. We may assume that no three are collinear. Then blow up these 9 points to obtain a surface X with projection map π : X → P2. Denote the exceptional divisors of π by ∗ E1,...,E9. Consider the linear system on X given by |3π H −E1 −. . .−E9|. This system has dimension 1 and is basepoint-free. Thus we have a map φ : X → P1. The fibres are elliptic curves and intersect the exceptional divisors E1,...,E9 in exactly one point, that is X is an elliptic surface. Since by our assumptions no three points of p1,...,p9 lie on a line, the fibres of φ are irreducible: Otherwise we would obtain reducible elements in |3H − p1 − . . . − p9|, which is impossible if no three points of p1,...,p9 are collinear. The divisors E1,...,E9 are sections of φ. Choosing the intersection of the fibres of φ with E1 as the zero element in the fibres, we obtain automorphisms of X by translating with the sections given by E2,...,E9, see also [Deb01, Example 6.6]. A priori the translation is only defined on the smooth fibres. To extend the map to the singular fibres, we embed X ∗ 1 2 by the relative very ample line bundle L := π OP2 (1) into P(φ∗(L)) = P × P . Note that singular fibres are then realized as singular cubics in P2. Note also that on the smooth locus of these singular curves we can define a group structure in exactly the same way as on elliptic curves in the projective plane, see [Har77, Example 6.11.4, Ex. 6.7]. Thus we can globally define the translation by the sections. This gives rise to automorphisms on X\{ singular points of fibres of φ }. These maps extend to automorphisms on the whole surface. Thus we obtain infinitely many (−1)-curves as images under the automorphisms of X. The infinitely many (−1)-curves intersect the fibres of φ in one point. Observe that fibres of φ are numerically equivalent to −KX and thus −KX is nef. Moreover, for a curve C ∈ X we have −KX · C = 0 if and only if C is a fibre of φ.

Claim: There are infinitely many extremal rays R+[ℓX ] of the cone of movable curves of X, where the curves ℓX can be obtained as follows. There exists a sequence of maps 1 X → X1 → . . . → X7 → P such that the last map is a Mori fibration, the other maps 1 are contraction of (−1)-curves and ℓX is the pullback of a general fibre of the map X7 → P

Choose a (−1)-curve E obtained by translating one of the E2,...,E9 and consider the blow-down η : X → X1. To start, let us prove that X1 is a del Pezzo surface, that is ∗ 2 −KX1 is ample. Clearly we have −KX = −η KX1 − E and thus (−KX1 ) =1 > 0. So we have to compute

∗ ∗ ∗ −KX1 · C = −KX · π C + E · π C = −KX · π C, where C is a curve in X1. Since −KX is nef, the product is greater than or equal to zero with equality if π∗C is a ﬁbre of φ. But since the exceptional divisor of η intersects all

ﬁbres of φ, this is impossible and thus −KX1 is ample. 2 Thus we can assume that X1 is isomorphic to a P blown up in 8 points, see Fact (1.2.14).

35 The slope with respect to movable curves

Denote by D1,...,D8 the corresponding exceptional divisors. We can contract 7 exceptional divisors, say D1,...,D7, to obtain the first Hirzebruch surface F1. As the last map, 1 we take the fibration F1 → P . We denote the general fibre of this fibration by ℓ. Pulling back ℓ gives us a movable curve ℓX on the boundary of the Moving cone. Since we have infinitely many (−1)-curves, we can repeat the procedure, but starting with another curve (−1)-curve not equal to E,D1,...D8 which will give us another extremal ray. Let lX be a curve which is the pullback of a general fibre in a Mori fibre space. We will see in Proposition (4.2.9) that the Harder-Narasimhan filtration of TF1 with respect to lX and the filtration of TX with respect to the pullback of lX agree over the open sets where the blow down X → X7 is an isomorphism. Furthermore, we have seen in chapter 2 that the

Harder-Narasimhan filtration of TF1 with respect to a fibre of the canonical projection is 1 the relative tangent sheaf TF1/P . Thus we obtain infinitely many destabilizing chambers in Mov(X). One goal in chapter 4 is to determine the subsheaves occurring in the Harder-Narasimhan filtration of the tangent sheaf of Fano threefolds with respect to any movable curve. In the sequel, we will prove that we only have to compute the Harder-Narasimhan filtration on the two dimensional faces of the boundary of the Moving cone when the Picard number of the manifold is greater than or equal to 3, in order to determine all sheaves occurring in the Harder-Narasimhan filtration with respect to any movable curve. In order to simplify the statements in the proofs below, we introduce some definitions. Definition 3.3.8. We say that a destabilizing chamber is a 1-term-chamber, if the Harder- Narasimhan filtration of TX with respect to the chamber has exactly one non trivial term, i.e. the filtration is given by 0 ⊂ F ⊂ TX for a sheaf F ⊂ TX . If the rank of F is one, we say that the 1-term-chamber is of rank 1 and if F is of rank two, we say that the chamber is of rank 2. Analogously, we define a 2-term-chamber as a destabilizing chamber having two non trivial terms, i.e. the filtration is given by 0 ⊂ L ⊂ H ⊂ TX for some sheaves L , H ⊂ TX . Let ∆ be a destabilizing chamber. We will call a chamber Θ a neighboring chamber of ∆ if ∂∆ ∩ Θ or ∆ ∩ ∂Θ is nonempty. Note that if ∆ is a neighboring chamber of Θ, then of course Θ is a neighboring chamber of ∆. We will simply say that Θ and ∆ are neighboring. Lemma 3.3.9. Let X be a projective threefold. (i) Suppose that ∆ is a 1-term-chamber of rank i, with 1 ≤ i ≤ 2 . Then a neighboring chamber Θ of ∆ cannot be a 1-term-chamber of rank i. (ii) Suppose ∆ is a 1-term-chamber of rank 1. Let

0 ⊂ F ⊂ TX be the Harder-Narasimhan ﬁltration with respect to ∆. Suppose Θ is a neighboring chamber of ∆ which is not the semistable chamber. Then there exists a subsheaf G ⊂ TX such that F ⊂ G and the Harder-Narasimhan ﬁltration with respect to Θ is given by 0 ⊂ G ⊂ TX or 0 ⊂ F ⊂ G ⊂ TX .

36 Chapter 3

(iii) Suppose ∆ is a 1-term-chamber of rank 2. Let

0 ⊂ G ⊂ TX

be the Harder-Narasimhan filtration with respect to ∆. Suppose Θ is a neighboring chamber of ∆ which is not the semistable chamber. Then there exists a subsheaf L ⊂ TX such that L ⊂ G and the Harder-Narasimhan filtration with respect to Θ is given by 0 ⊂ L ⊂ TX or 0 ⊂ L ⊂ G ⊂ TX . Proof. Let us prove (i). Suppose on the contrary that Θ is a 1-term-chamber of the same rank as ∆. Suppose the Harder-Narasimhan filtration of TX with respect to ∆ is given by 0 ⊂ F ⊂ TX and the Harder-Narasimhan filtration of TX with respect to Θ is given ′ ′ by 0 ⊂ F ⊂ TX . For a class c in ∂∆ ∩ Θ or ∆ ∩ ∂Θ, we have µc(F ) = µc(F ). Since the maximal destabilizing subsheaf of TX with respect to a movable class is unique, we obtain F = F ′. This shows (i).

Proof of (ii): Suppose ﬁrst that the neighboring chamber Θ of ∆ is a 1-term-chamber. By (i) we know that Θ is of rank 2. Let 0 ⊂ G ⊂ TX be the Harder-Narasimhan ﬁltration with respect to Θ. Then for c ∈ ∂∆ ∩ Θ we have µc(F ) = µc(G ) and by Proposition (3.1.8) we have F ⊂ G . Now suppose Θ is a 2-term chamber. Let F ′ be the maximal destabilizing subsheaf of ′ TX with respect to Θ. Then with respect to c ∈ ∂Θ ∩ ∆ we have µc(F )= µc(F ). Since the maximal destabilizing subsheaf is unique, it follows F = F ′.

Proof of (iii): Suppose first that the neighboring chamber Θ of ∆ is a 1-term-chamber. By (i) we know that Θ is of rank 1. Let 0 ⊂ L ⊂ TX be the Harder-Narasimhan filtration with respect to Θ. Then for c ∈ ∆ ∩ ∂Θ we have µc(L ) = µc(G ) and by Proposition (3.1.8) we have L ⊂ G . Now suppose Θ is a 2-term chamber and the Harder-Narasimhan filtration with respect to Θ is given by 0 ⊂ L ⊂ H ⊂ TX . We have to prove that H = G . With respect to a class in c ∈ ∆ ∩ ∂Θ, the slope of the maximal destabilizing subsheaves of ∆ and Θ agree, i.e. µc(G ) = µc(L ). Proposition (3.1.8) yields L ⊂ G . Moreover, we have

µ(H ) ≤ µ(G ) (*) with respect to classes in ∆ ∩ ∂Θ since G is the maximal destabilizing subsheaf of TX with respect to ∆. Let us prove that we have equality in (*). Assume on the contrary µ(H ) <µ(G ) with respect to C ∈ ∆ ∩ ∂Θ. Then this inequality would hold in a neighborhood of C and thus would contradict that H is the sheaf of maximal slope containing L . Thus we have µ(H )= µ(G ) on ∆ ∩ ∂Θ and again the uniqueness of the maximal destabilizing subsheaf implies H = G . Remark 3.3.10. Consider a projective threefold X. As we know, we have a decomposition of the Moving cone Mov(X) into destabilizing chambers. In the sequel, we will often

37 The slope with respect to movable curves consider the restriction of the chamber structure on subsets S of Mov(X). The chamber structure in S is then given by simply restricting the chambers in Mov(X). If we then consider destabilizing chambers in S, we mean of course just these restricted chambers. Proposition 3.3.11. Let X be a projective threefold with ρ(X) ≥ 3. Let C ⊂ Mov(X) be a convex subcone of dimension greater than or equal to 3 and suppose the chamber structure in C is finite. Suppose F is a term in the Harder-Narasimhan filtration of TX with respect to a movable class in C. Then there exists a class C on the relative boundary of C such that F appears as a term in the Harder-Narasimhan filtration with respect to C. Organization of the proof. The proof is divided into two steps. In the first step we show that each 2-term-chamber contains automatically elements from the boundary of C. This implies the Proposition for all sheaves occurring in a filtration with maximal number of terms. In the second step we will show that each 1-term-chamber with defining Harder- Narasimhan filtration 0 ⊂ F ⊂ TX , which does not contain elements from the boundary of C, has a neighboring 2-term-chamber Θ. Then Lemma (3.3.9) (ii) and (iii) implies that the sheaf F appears in the defining Harder-Narasimhan filtration with respect to Θ. This proves the Proposition.

Proof of Proposition (3.3.11). Let c ∈ rel.int C and let ∆ := ∆c ∩ C be the destabilizing chamber in C with respect to c. Step 1: In this Step, we assume that ∆ is a 2-term-chamber associated to the ﬁltration 0 ⊂ F1 ⊂ F2 ⊂ TX . Since the quotients of the ﬁltration are torsion free and of rank 1, they are semistable and thus ∆ is simply given by the intersection of the two halfspaces

> ˜ F F F H1 := C ∈C| µC˜( 1) − µC˜( 2/ 1) > 0 > n ˜ F F F o H2 := C ∈C| µC˜( 2/ 1) − µC˜(TX / 2) > 0 ,

> > n > > o i.e. ∆ = H1 ∩ H2 . Since dim C ≥ 3 and c ∈ ∆= H1 ∩ H2 , it is clear that ∆ ∩ ∂C= 6 ∅. This shows the Proposition for all sheaves F occurring in a Harder-Narasimhan ﬁltration having the maximal number of terms.

Step 2: Now we assume that ∆ is a 1-term chamber. We can assume that ∆ lies in the relative interior of C since otherwise there is nothing to prove. Recall that Theorem (3.3.4) ensures that the closure of ∆ is polyhedral and since the chamber structure is ﬁnite in C, we know that ∆ is surrounded by neighboring chambers. Claim: There exists a neighboring 2-term-chamber Θ of ∆. We assume on the contrary that each neighboring chambers of ∆ is a 1-term-chamber or the semistable chamber. We intersect C by a suitable aﬃne plane H of dimension 2 not containing the origin with the properties that • H intersects ∆,

• C′ := C ∩ H is a bounded convex set in H ∼= R2

38 Chapter 3

Notice that ∆′ := ∆ ∩ H lies in the interior of C′ ⊂ H and the closure of ∆′ is a convex polytope in H. Let us assume that the dimension of ∆′ is 2, i.e. the dimension of the affine subspace in H generated by ∆′ is two dimensional. Then it is not difficult to see that the convexity of ∆′, the convexity of the neighboring chambers of ∆′ and the fact that ∆′ lies in the interior of C′ implies that ∆′ has at least three neighboring chambers. Recall our assumption that each neighboring chamber of ∆′ is a 1-term chamber or the semistable chamber. By Lemma (3.3.9) (i) all 1-term neighboring chambers of ∆′ have the same rank. Since there exists at least three neighboring chambers of ∆′ and there exists only one semistable chamber, we can find two neighboring 1-term chambers ∆1 and ∆2 of ∆, with the property that ∆1 and ∆2 are neighboring. Since ∆1 and ∆2 are of the same rank, we obtain a contradiction to Lemma (3.3.9) (i). Finally, assume that the dimension of ∆′ is strictly less than 2. If ∆ has at least three neighboring chambers, we can argue exactly as above. At a first sight it might happen ′ that ∆ has only two neighboring chambers, say Θ1, Θ2. In this case Θ1 and Θ2 are obviously not closed in H. The semistable chamber is closed by Theorem (3.3.4) (ii) and ′ hence the semistable chamber is not a neighboring chamber of ∆ in H. Thus both Θ1 and Θ2 are 1-term chambers. Moreover, Lemma (3.3.9) ensures that Θ1 and Θ2 are of the same rank. Since Θ1 and Θ2 are neighboring, this again contradicts Lemma (3.3.9) (i).

Chapter 4 The Harder-Narasimhan ﬁltration of the tangent bundle on Fano threefolds

In this chapter we consider the decomposition of the Moving cone on Fano threefolds into the destabilizing chambers. We already know by Proposition (3.3.5) that the decomposition of the cone of movable curves into the destabilizing chambers is finite on Fano manifolds of any dimension. For Fano threefolds, we are in addition able to give a con- crete geometric characterization of all possible terms occurring in the Harder-Narasimhan filtration of the tangent bundle with respect to any movable curve. It turns out that these terms are relative tangent sheaves of not necessarily elementary Mori fibrations. The main result of this section is the following. Theorem 4.1. Let X be a Fano manifold of dimension 3. Then there exists a finite decomposition of the Moving cone into polyhedral subcones, the destabilizing chambers, such that (i) in each subcone the Harder-Narasimhan filtration is constant and (ii) each term of the Harder-Narasimhan filtration associated to a movable curve is the relative tangent sheaf of a not necessarily elementary Mori fibration. The idea of proving the result is as follows. If we want to determine the subsheaves of the tangent sheaf occurring in the Harder-Narasimhan filtration with respect to a movable curve, then it is enough to consider the Harder-Narasimhan filtration on the boundary of the cone of movable curves by Proposition (3.3.11). Now, somewhat simplified, we can argue as follows. If we have chosen a movable curve which happens to sit on the boundary of the Mori cone, then we use the linearity property of the slope to see that a possible destabilizing subsheaf is also destabilizing with respect to a curve which is a general fibre of a Mori fibration. Then we use the theory of minimal rational curves, to prove that this sheaf is necessarily the relative tangent sheaf of a Mori contraction. If we have chosen a movable curve in the interior of the Mori cone, then by Proposition (1.2.17), there is an exceptional divisor of an extremal contraction which has no intersection with the movable curve. So we can contract the divisor and argue on the contracted variety. The problem here is that the contracted variety is in general neither a Fano manifold nor smooth. So we have to deal with this difficulty. Using the classification of Fano threefolds, we will prove that most Fano threefolds have at most one exceptional divisor, whose contraction leads to a non Fano or non smooth variety. The other cases are ruled out separately. The Harder-Narasimhan filtration of the tangent bundle on Fano threefolds

The organization of this chapter is as follows. In the first section we investigate the Moving and the Mori cone on Fano threefolds. We will focus our investigation on the behavior of extremal rays under extremal contraction. Then, after introducing the notion of a pullback foliation, we can explicitly compute the Harder-Narasimhan filtration with respect to curves lying in fibres of Mori fibrations in Section (4.3). These results, especially Proposition (4.3.8) and (4.3.11), will give the key to prove the main result. In Section (4.4) we will prove Theorem (4.1). Finally, in the last section of this chapter, we present another method to compute the Harder-Narasimhan filtration of the tangent bundle.

4.1. Preliminary Results for Fano threefolds

For smooth threefolds all possible extremal contractions are classiﬁed by Mori in [Mor82, Theorem 3.3 and Theorem 3.5]. We will recall the possible types of contraction in order to ﬁx notation. In the following table, an exceptional divisor of an extremal contraction is denoted by E.

Type contraction φ contracted rational curve

E1 φ(E) is a smooth curve C. The a ﬁbre of the ruled surface E contraction φ is the blow up of a smooth variety in C. 2 ∼ E2 φ(E) is a smooth point. The con- a line in P = E traction φ is the blow up of a point in a smooth variety. 1 1 E3 φ(E) is an ordinary double point s × P and P × t and E ∼= P1 × P1 E4 φ(E) is a double cDV-point and a ruling of the cone E E is a quadric cone E5 φ(E) is a quadrupel non Goren- a line in E stein point; E ∼= P2

Deﬁnition 4.1.1. We say that a contraction φ : X → Y of a Fano threefold X is of type E1a, if it is of type E1 and Y is again a Fano threefold. If not, we say the contraction is of type E1b.

Fact 4.1.2. In case that the contraction of a Fano threefold is of type E1b, the exceptional divisor is contracted to a curve C ∼= P1 with normal bundle O(−1) ⊕ O(−1), see [MM86, Proposition 2.2].

Fact 4.1.3. Let X be a smooth threefold. If φ : X → Y is an extremal contraction of ﬁbering type, then Y is smooth. If dim Y = 2, then the general ﬁbre is a smooth rational curve; if X is moreover Fano, than Y is a del Pezzo surface, see [MM86, Proposition 4.16].

42 Chapter 4

In case dim Y = 1, the general ﬁbre is isomorphic to P2, P1 × P1 or to a del Pezzo surface of degree 1,..., 6, see [Mor82, Theorem 3.5].

Definition 4.1.4. Let X be a smooth threefold and let φ : X → Y be a Mori fibration. In case that dim Y = 2, we call φ a conic bundle and if moreover X is Fano, then we call φ a Fano conic bundle. If dim Y = 1, we call φ a del Pezzo fibration.

The following lemma will be used frequently.

Lemma 4.1.5. Let X be a Q-factorial projective variety with only terminal singularities. Let φ : X → Y be a divisorial contraction. Let E be the exceptional divisor of φ and let ∗ d ∈ N1,R(X) such that d · E =0. Then φ φ∗d = d.

∗ ∗ Proof. We have to show that φ φ∗d · D = d · D for every divisor D ⊂ X. If D = φ H, for some divisor H ⊂ Y the formula follows from the projection formula. Since φ is a divisorial contraction we have that N 1(X) = φ∗N 1(Y ) ⊕ Z[E]. Therefore, it remains to ∗ check that φ φ∗d · E = d · E which also follows from the projection formula.

Lemma 4.1.6. Let X be a smooth threefold and let φ : X → Y be a contraction of type E1. Let D be the exceptional divisor of φ and C ⊂ Y the blow up locus. Let s ⊂ D = P(NX/C ) be a section of the ruled surface D and let f be a contracted ﬁbre of the ruling structure of D. Then φ∗[C] = [s]+ k · [f] for a suitable k ∈ Z.

Proof. To start, it is easy to verify that φ∗[s] = [C]. On the other hand, the projection ∗ ∗ formula implies φ∗φ [C] = [C]. This shows that [φ C] − [s] lies in the kernel of φ∗. Since ker φ∗ is generated by [f], the claim follows.

Remark 4.1.7. Let X be a Fano threefold and let φ : X → Y be a contraction of type ∼ 1 1 E1b. Let D = P × P be the exceptional divisor of φ. Denote the class of a vertical, i.e. the contracted, ﬁbre in D by v and the class of a horizontal ﬁbre in D by h. Let C ⊂ Y be the blow up locus of φ. Then

φ∗C = −v + h.

Proof. We have E · v = −1 and E · h = −1. Thus the claimed equality follows from Lemma (4.1.6).

Lemma 4.1.8. Let X and Y be Q-factorial varieties having at most terminal singularities and let ϕ : X → Y be an extremal divisorial contraction. Let C ⊂ Y be a curve which is not contained in ϕ(Exc(ϕ)). Then the numerical pullback ϕ∗[C] is eﬀective.

Proof. Denote the exceptional divisor of ϕ by E and let f ⊂ E be the class of a contracted curve in E. Let us write C˜ for the proper transform of C. We will show the following equality ϕ∗[C] = [C˜]+ kf (*)

43 The Harder-Narasimhan ﬁltration of the tangent bundle on Fano threefolds

E·C˜ where k = − E·f . Since E · f < 0 and E · C˜ ≥ 0, we have k ≥ 0 and thus the right hand side of the claimed equality (*) is indeed eﬀective. To prove the equality, we have to show that D · ϕ∗[C]= D · ([C˜]+ kf) for all divisors D on X. Let D = ϕ∗H for some divisor class H on Y . Then after possibly replacing H by a numerically equivalent divisor, we may assume that H intersects C away from ϕ(E). Then obviously H · C = ϕ∗H · C˜ and therefore

ϕ∗H · ϕ∗[C]= ϕ∗H · ([C˜]+ kf).

Since N 1(X) ∼= ϕ∗N 1(Y ) ⊕ Z[E] it remains to verify E · ϕ∗[C] = E · ([C˜]+ kf). This E·C˜ equality is clear since k = − E·f . Lemma 4.1.9. Let ϕ : X → Y be a divisorial contraction of a Fano threefold X. Let f be the class of a curve contracted by ϕ. Then for any v ∈ NE(Y ), we have ϕ∗v = w + kf, where k ∈ Z and w ∈ NE(X).

Proof. Suppose v is an irreducible curve in Y . Then the statement is true by Lemma (4.1.6) and Lemma (4.1.8). Now by the linearity of the numerical pullback, the statement is true for all eﬀective 1-cycles with real coeﬃcients. Consider the subset C := hNE(X), −fiR+ in N1,R(X), i.e. the cone generated by NE(X) and −f in N1,R(X). As we have just seen, the numerical pullback ϕ∗ maps NE(X) to C. Simply by the continuity of ϕ∗, the numerical pullback maps NE(X) to C. The Lemma is proved.

Lemma 4.1.10. Let X be threefold and let φ : X → Y be a divisorial contraction. Let f be the class of a contracted curve in the exceptional divisor of φ. Let N be a nef divisor on X which cuts out a face F of NE(X) containing the contracted class f. Then φ∗N is nef.

Proof. Let C be a curve in Y . Then we have to verify

∗ φ∗N · C = N · φ C ≥ 0. (*)

If φ contracts a divisor to a point, then the pullback of any curve in Y is effective by Lemma (4.1.8). In this case the inequality above is clear. Suppose φ is of type E1. If C is not the blow up locus of φ, the numerical pullback of C is effective by Lemma (4.1.10) and thus inequality (*) is true. It remains to verify the above inequality in case that C is the blow up locus. By Lemma (4.1.6) we know that φ∗[C] = [s] + k[f], where s is effective and f is the class of a contracted curve. Since N · f = 0 the claimed inequality (*) holds.

Proposition 4.1.11. Let X be a Fano threefold and let φ : X → Y be a divisorial contraction with exceptional divisor E and let f be the class of a contracted curve. Let d be a movable class such that d · E =0. Then

(i) d is an extremal class of Mov(X) if and only if φ∗d is an extremal class of Mov(Y ).

44 Chapter 4

˚ ˚ (ii) If d ∈ NE(X) then φ∗d ∈ NE(Y ). Equivalently, if φ∗d ∈ ∂NE(Y ) then d ∈ ∂NE(X).

(iii) Suppose φ is of type E1a, E2,...,E5. Then we have φ∗d ∈ ∂NE(Y ) if and only if d ∈ ∂NE(X). If φ is of type E1b, then this is true if we assume that the d and the class of a contracted curve lie on a common proper face of the Mori cone. Proof. Proof of (i): Suppose d is an extremal class of Mov(X). Let D ⊂ Y be an irreducible divisor. Since d ∈ Mov(X) and φ∗[D] is effective, we obtain d · φ∗D ≥ 0. Thus by projection formula φ∗d · D ≥ 0. Thus φ∗d lies in the Moving cone of Y . Let us write φ∗d = r1 + r2 with r1 and r2 movable classes. By Lemma (4.1.5) we obtain ∗ ∗ ∗ ∗ d = φ r1 + φ r2. Since φ r1 and φ r2 are movable classes and R+[d] is an extremal ∗ ∗ ray, there exists λi ∈ R+ such that φ ri = λid. Taking the pushforward of φ ri, yields ri ∈ R+φ∗[d]. This shows that φ∗d is an extremal class. Now we assume that φ∗d is extremal. Let D ⊂ X be an irreducible effective divisor. If D = E, we have d · D = 0. If D =6 E, we can write D = φ∗H − kE for an irreducible effective divisor H and suitable k ≥ 0. Now the projection formula yields d · D ≥ 0. Thus d ∈ Mov(X). Let us write d = v + w with v,w movable. Then φ∗d = φ∗v + φ∗w ∗ and thus φ∗v,φ∗w ∈ R+φ∗d. Since φ φ∗d = d, it follows from the projection formula that both v and w are in R+d.

∗ Proof of (ii) and (iii). Recall that Lemma (4.1.5) guarantees that φ φ∗d = d. If φ∗d ∈ ∂NE(Y ), then there exists a nef divisor N with N · φ∗d = 0. Since the pullback of a nef divisor is nef, the claim follows. This shows (ii). To prove (iii), suppose d ∈ ∂NE(X). Let N be a nef divisor cutting out the face of the Mori cone containing d and the ﬁbre contracted by φ. Then φ∗N is nef by Lemma (4.1.10) and φ∗d · φ∗N = 0 by the projection formula. Proposition 4.1.12. Let X be a Fano threefold with Picard number ρ(X) ≥ 3 and let

Fmori = hr1,r2iR+ be a two dimensional face of the Mori cone of X. Let Fmov be a face of the Moving cone of X contained in Fmori.

(i) Suppose R+r1 corresponds to a divisorial contraction of type E2,...,E5, then Fmov consists of at most one extremal ray.

(ii) Suppose R+r1 corresponds to a divisorial contraction ϕ : X → Y of type E1 and let d ∈ Fmov. Then ϕ∗d spans an extremal ray of NE(Y ). If dim Fmov =2 then R+ϕ∗d corresponds to a del Pezzo ﬁbration.

(iii) Suppose R+r1 and R+r2 correspond to divisorial contractions. Suppose the contraction of R+r1 is of type E2,...,E5. Denote the contraction of R+r2 by φ. Let d ∈ Fmov. Then φ∗d spans an extremal ray of the Mori cone corresponding to a del Pezzo ﬁbration.

Proof. Step 1. Suppose R+r1 corresponds to an extremal contraction ϕ : X → Y .

Let c ∈ hr1,r2iR+ . Then there exist λ1,λ2 ∈ R≥0 such that c = λ1r1 + λ2r2. Since ϕ is the contraction associated to R+r1, we have ϕ∗r1 = 0. Thus we have shown that

ϕ∗c ∈ R+ϕ∗r2 (4.1)

45 The Harder-Narasimhan ﬁltration of the tangent bundle on Fano threefolds

for all c ∈hr1,r2iR+ .

Step 2. We continue to assume that the contraction ϕ : X → Y associated to R+r1 is divisorial. We assume Fmov =6 ∅. Let E be the exceptional divisor of ϕ and let d ∈ Fmori such that R+d =6 R+r1. We show that ϕ∗d spans an KY -negative extremal ray of NE(Y ). Let us prove that R+ϕ∗d is an extremal ray of NE(Y ). Indeed, let N be a nef divisor which cuts out the face spanned by r1 and r2. Then by Lemma (4.1.8) ϕ∗N is nef and by the projection formula ϕ∗N · ϕ∗d = 0. Since ϕ∗d ∈ NE(Y ), we conclude that ϕ∗d lies on an extremal face of the Mori cone of Y . Now consider a class ∗ v ∈ NE(Y ) such that ϕ∗N · v = 0. Then the projection formula yields N · ϕ v = 0. Using Lemma (4.1.9), we can write ϕ∗v = w + kf, where w ∈ NE(X),k ∈ Z and f is the class of a curve contracted by ϕ. Thus N · (w + kf) = 0. By the choice of N, we have N · f = 0 and thus it follows that N · w = 0. Now since w ∈ NE(X) we can conclude that w ∈ hr1,r2iR+ . This and Step 1 gives ϕ∗w ∈ R+ϕ∗d. Since ϕ∗w = v, we have shown that R+ϕ∗d is an extremal ray of the Mori cone. It remains to show that ∗ ϕ∗d is KY -negative. Let d be a movable curve in Fmov. We have KX = ϕ KY + aE, where a ≥ 1 and E denotes the exceptional divisor of ϕ. Then φ∗d·KY = d·(KX −aE) < 0.

Step 3. Let us prove (i): Suppose on the contrary that dim Fmov = 2. Then there exists an extremal class d ∈ Fmov of the Moving cone which intersects the exceptional divisor of the contraction ϕ associated to R+r1 positively. The structure Theorem of [Ara10, Theorem 1.1.] guarantees that there exists a family of generically irreducible curves covering X, such that the numerical classes of these curves are numerically proportional to d. Step 1 and Step 2 show that the pushforward of these curves via ϕ span an extremal ray of NE(Y ). Let us denote the contraction by ψ. If ϕ were of type E2,...,E5, then the images of these curves via ϕ would pass to a common point. This would imply that the contraction ψ contracts Y to a point. This is only possible if the Picard number of Y is 1 and therefore the Picard number of X is 2. This contradicts the assumption ρ(X) ≥ 3.

Step 4. Proof of (ii): Let ϕ be the contraction associated to R+r1. Let E be the exceptional divisor of ϕ and let d ∈ Fmov. By Step 1 and 2 we know that R+ϕ∗d is extremal. Since dim Fmov = 2 there exists an extremal class g of Fmov which intersects E positively. Again by [Ara10, Theorem 1.1] there exists a family of curves, such that the numerical class of these curves is numerically proportional to g. By Step 1, ϕ∗d and ϕ∗g lie on the same extremal ray. Thus the contraction of R+ϕ∗d has at least two dimensional fibres. Since fibres of dimension 3 are impossible, the fibres of the contraction are two dimensional. The classification of extremal contractions on threefolds allows us to say that ϕ∗d corresponds to a del Pezzo fibration.

Step 5. We prove (iii): Denote the contraction associated to R+r1 by σ : X → Z and its exceptional divisor by D. Let E be the exceptional divisor of φ and let f ⊂ E be the class of the contracted curves in E. We already know by Step 1 and Step 2 that R+σ∗c = R+σ∗r2 for all c ∈ Fmori\R+r1 and R+σ∗r2 is an extremal ray of NE(Z). Since d is movable and σ∗d ∈ R+σ∗r2, the contraction associated to R+σ∗r2 corresponds to a Mori ﬁbration, say ν.

46 Chapter 4

On the other hand, consider σ∗f ⊂ σ∗E. Since obviously D · f > 0, each curve f intersect the exceptional divisor D. Hence the pushforward via σ of the exceptional curves in E will pass to a common point. Because σ∗f ∈ R+σ∗r2, ν has at least one fibre of dimension two, namely σ∗(E). Since the relative Picard number of the contraction ν is 1, all fibres of ν are two dimensional. Since the composition ν ◦ σ is the same as first contracting E and then applying the contraction associated to R+φ∗d, we see that φ∗d corresponds to a del Pezzo fibration.

Lemma 4.1.13. Let X be a Fano threefold. Let D1 and D2 be exceptional divisors in X. Assume that D1 is of type E1 or E2 and let φ : X → Y be the corresponding contraction. Suppose furthermore that D1 ∩ D2 = ∅. Then φ∗D2 is an exceptional divisor on Y of the same type as D2.

Proof. Let f be the class of a curve in D2 generating an extremal ray associated to D2. To start, we show that φ∗f is extremal. Indeed, let N be the class of a nef divisor with N · f = 0 and N · c> 0 for all classes c of the Mori cone which do not lie on the extremal ray generated by f. Since φ is an isomorphism outside D1 and D1 ∩ D2 = ∅, we have ∗ φ φ∗f = f. This implies via the projection formula that φ∗N · φ∗f = 0. Since φ∗N is nef by Lemma (4.1.10), the class φ∗f lies on the boundary of the Mori cone of Y . To see that φ∗f is extremal, we take the class of a curve C ⊂ Y with φ∗N · C = 0. The ∗ ∗ projection formula yields N · φ C = 0 and thus φ C = af. This shows that φ∗f generates an extremal ray of the Mori cone of Y . It follows from the adjunction formula and the fact that D1 ∩ D2 = ∅ that φ∗f is KY -negative. It is clear that φ∗D2 is of the same type as D2.

Lemma 4.1.14. Let X be a Fano threefold. Let D1 and D2 be exceptional divisors of extremal contractions. Suppose both divisors are not of type E1a. Then D1 ∩ D2 = ∅.

Proof. By [MM86, Lemma 3.3.] −Di|Di is ample and every eﬀective divisor on Di is movable, i =1, 2. Thus if the intersection of D1 and D2 were not empty, then C := D1∩D2 would be movable on D1 and on D2, and thus for i =1, 2 we would have C · Di ≥ 0. On the other hand the ampleness of −Di|Di implies C · Di < 0, i =1, 2, a contradiction. Lemma 4.1.15. Let X be a smooth projective threefold and Y ⊂ X be a smooth projective subvariety of dimension 2. Suppose we have two numerically equivalent divisors f and g in Y , then f and g are numerically equivalent as 1-cycles in X. Proof. Let D be a divisor in X. We can write D = A − B as a diﬀerence of two very ample divisors, such that A and B are smooth and intersect Y and the components of f and g transversely. We have to show that f · D = g · D. By linearity of the intersection product it is enough to show f · A = g · A. Let C be the intersection cycle A · Y . Since in this situation computing intersection number means counting intersection points we have

f · A = f · C = g · C = g · A.

47 The Harder-Narasimhan ﬁltration of the tangent bundle on Fano threefolds

Proposition 4.1.16. Let X be a Fano threefold with ρ(X) ≥ 3. Suppose X is not isomorphic to the blow up of a cone over the Veronese surface with center in a disjoint union of the vertex and a quartic or the blow up of the cone over a smooth quadric S in P3 with center the disjoint union of an elliptic curve in S and the vertex. Then NE(X) has at most one extremal ray of type E1b, E3, E4, E5.

Proof. We suppose ﬁrst that φ : X → P1 × P1 is a Fano conic bundle with ρ(X)=3. Let D be an exceptional divisor in X. If D =∼ P2 then φ(D) would be a point since there are no nontrivial surjective morphisms P2 → P1 ×P1. This would yield that a class in D is numerically equivalent to a ﬁbre of φ, a contradiction. This shows that D is not of type E5. ∼ 1 1 1 1 In case D is of type E3 we have D = P × P with p × P ⊂ D and P × q ⊂ D numerically equivalent which implies that the morphism φ P1×P1 must be trivial. The same argument yields the claim if D is of type E . From the assumption ρ(X) = 3 it follows that there 4 exists at most one exceptional divisor of type E1b: If there were two E1b-type divisors, they would have empty intersection by Lemma (4.1.14). Thus we could contract them successively to a manifold Y with ρ(Y ) = 1 by a contraction ϕ : X → Y . Let C and C′ be the blow up curves in Y . Then since ρ(X) = 1 a multiple of C is movable, a contradiction.

It follows from the classiﬁcation of Fano threefolds, see [IP99, Appendix] for an overview that in the remaining cases X is the blow up of a Fano threefold Y in a curve such that Y admits no divisorial contraction of type E1b, E2,...,E5. Thus we can assume that there exists an exceptional divisor D of type E1a on X. Recall that D is a smooth ruled surface realized as the projectivation of the normal bundle of the blow up locus. Aiming for a contradiction we assume that there are two exceptional divisors E, E′ of ′ ′ type E1b, E3, E4, E5. Then D has to intersect both E and E since otherwise E or E were exceptional divisors of the same type on Y by Lemma (4.1.13). Let C be an irreducible component of E · D and C′ an irreducible component of E′ · D.

We claim the following: There exists an eﬀective Q−divisor Q on D such that Q ≡ C and no component of Q lies in E or Q ≡ C′ and no component of Q lies in E′

We use the notation of [Har77, Chapter V]: Let C0 be the distinguished section in D. 2 Let −e := C0 and f be a ﬁbre of the ruling structure of D. We can assume that C or ′ C is not numerically proportional to C0. Say C is not numerically proportional to C0. Then C ≡ aC0 + bf with a ≥ 0,b > 0. Since we can move f freely in D we can set Q := aC0 + bf. Thus the claim is proved.

So assume C ≡ Q and no component of Q lies in E. Since C and Q are numerically equivalent on D, they are in particular numerically equivalent as cycles in X by Lemma (4.1.15). Thus E · C ≥ 0. On the other hand C is a curve in an exceptional divisor of type E1b, E2,...,E5 on a Fano threefold which forces C · E to be negative.

Lemma 4.1.17. Let X be a Fano threefold with ρ(X) ≥ 3. Then the each Mori-ﬁbration has one-dimensional ﬁbres.

Proof. Suppose there exists an Mori-ﬁbration onto a curve C. Then we have C ∼= P1. On

48 Chapter 4 the other hand we have an exact sequence 0 → P ic(P1) → P ic(X) → Z → 0 which shows that the Picard number of X is 2.

4.2. Pullback of Foliations

For lack of a reference and sake of completeness, we will introduce the notion of a pullback foliation. Let ϕ : X → Y be a proper morphism of projective manifolds. Then given a foliation on Y , we would like to pull it back to a foliation on X such that the leaves on X are given by the preimages of leaves on Y .

We have defined a foliation as a saturated involutive coherent subsheaf of the tangent sheaf. Given a foliation F ⊂ TX , one can adopt a “dual” point of view, and regard the foliation not generated by vector fields, but generated by the kernel of all 1-forms vanishing on vector fields in F . The advantage is that we can easily pull back differential forms. G 1 Definition 4.2.1. A coherent subsheaf ⊂ ΩX is called involutive if for all p ∈ X and all differential forms ω ∈ Gp, we have

dω ∈ Gp ∧ ΩX,p. Next we want to establish how to switch between the subsheaves of the tangent sheaf and subsheaves of the sheaf of 1-forms: Let F ⊂ TX be a subsheaf of the tangent sheaf. Then we define F ∨ as follows. Definition 4.2.2. For an open set U ⊂ X we set F ∨ 1 F (U) := ω ∈ ΩX (U) ω(v)=0 for all v ∈ (U) . G 1 G ∨ Accordingly, for a sheaf of 1-forms ⊂ ΩX , we define to be the sheaf of vector fields vanishing on all differential forms in G .

Remark 4.2.3. (i) Let F ⊂ TX be a coherent subsheaf of the tangent sheaf. Then we have a natural isomorphism of OX -modules ∨ ∼ ∗ F = (TX /F ) .

F F ∨ 1 (ii) Let ⊂ TX be a coherent subsheaf. Then is saturated in ΩX .

∨ (iii) If F ⊂ TX is saturated in TX , then c1(TX )= c1(F ) − c1(F ).

F G 1 The statements above also hold if we replace by a subsheaf of ⊂ ΩX and TX by 1 G 1 ΩX . For example, the ﬁrst statement encodes in this setting as follows: Let ⊂ ΩX be a coherent sheaf. Then we have a natural isomorphism of OX -modules G ∨ ∼ 1 G ∗ = (ΩX / ) .

49 The Harder-Narasimhan ﬁltration of the tangent bundle on Fano threefolds

F F 1 Proposition 4.2.4. Let X be a projective manifold, and ⊂ TX or ⊂ ΩX . Then we have:

(i) If F is coherent, then so is F ∨.

(ii) If F is involutive, then so is F ∨.

Sketch of proof. Let F be a subsheaf of the tangent sheaf. We have a canonical isomor- phisms of OX -modules ∨ ∼ ∗ F = (TX /F ) .

∗ Part (i) then follows, since (TX /F ) is coherent. G 1 Let us prove (ii): Assume that ⊂ ΩX is involutive. Let p ∈ X and let u, v be local vector ﬁelds around p. Suppose v,w ∈ G ∨. We have to show that the Lie bracket of u G ∨ G 1 and v yields an element in . Since ⊂ ΩX is involutive, it follows that dω(u, v)= 0. Then using the formula [War83, Proposition 2.25]

1 dω(u, v)= u(ω(v)) − v(ω(u)) − ω([u, v]) , (*) 2 we obtain that ω([u, v]) = 0. This implies [u, v] ∈ G ∨. F F ∨ Let ⊂ TX be involutive. Let p in X and ω ∈ p . Formula (*) implies dω(u, v)=0 ∨ for all vector ﬁelds u, v in Fp. Now, using that F is saturated, one can show that F ∨ 1 ω ∈ p ∧ ΩX,p.

Example 4.2.5. Let C2 be the complex plane with coordinates x, y. Consider the foliation ∂ ∂ given by ∂x x + y ∂y . Equivalently, this foliation is given by the kernel of the 1-form ω := ydx − xdy. The leaves of this foliation are given by the lines in C2 through 0. Let us consider the blow up of C2 in 0 with blow up map π. A computation in local coordinates shows that the form π∗ω vanishes along the exceptional divisor. Thus the sheaf given by means of pulling back ω is not saturated in the tangent sheaf of the blow up.

Let us now explain precisely, what we want to understand under the pullback of a foliation.

Definition and Construction 4.2.6. Let ϕ : X → Y be a proper morphism of projective manifolds. Let FY ⊂ TY be a foliation. We define the pullback foliation FX of FY F ∨ 1 N 1 as follows. First we consider Y ⊂ ΩY . Then we obtain a subsheaf X ⊂ ΩX by means F ∨ of pulling back 1-forms of X . Finally we define

F N ∨ X := X .

Remark 4.2.7. The pullback of a foliation is indeed involutive and saturated in the tangent sheaf, thus it deﬁnes a foliation.

In our studies, we have to compute the Chern classes of pullback foliation in the case that ϕ is a conic bundle.

50 Chapter 4

Lemma 4.2.8. Let X be a projective threefold and Y a projective surface. Let ϕ : X → Y be a conic bundle and let FY ⊂ TY be a foliation. Then we have

∗ c1(FX )= −KX/Y + φ c1(FY ), where FX is the pullback of the foliation FY .

1 Proof. Let us denote the subsheaf of ΩX which is obtained by pulling back 1-forms from ∨ F , by NX . The structure of conic bundles are well known. In particular the local structure of ϕ : X → Y is explicitly computable, see [MM86, Proposition 4.2]. It follows from [MM86, Proposition 4.2] that ϕ is smooth except in a set of codimension greater than or equal to two. This shows that the zero locus of the pulled back differential forms, which give the foliation in Y , has codimension greater than or equal to two. Since the first Chern class of a torsion free coherent sheaf is not affected by tensoring with an ideal sheaf of an algebraic set of codimension at least two, the first Chern class of NX equals the first N 1 F N Chern class of the saturation of X in ΩX . This yields that −KX = c1( X ) − c1( X ). ∨ ∗ ∨ On the other hand we have −KY = c1(FY ) − c1(F ) and c1(NX )= ϕ c1(F ). Now the claimed formula follows from these equations.

Let F |U ⊂ TX |U be a distribution which is deﬁned on an open subset of X. In the F ⋆ F following Proposition we denote by |U the saturation of a coherent extension of |U in TX .

Proposition 4.2.9. Let φ : X˜ → X be a blow up of a projective manifold and denote the blow up locus in X by B. Let C˜ be a movable class on X˜ which does not intersect the exceptional locus E of φ. Set U := X˜\E. Suppose the Harder-Narasimhan ﬁltration of TX with respect to φ∗C˜ is given by

0 ⊂ F1 ⊂ . . . ⊂ Fk ⊂ TX . ˜ Then the Harder Narasimhan ﬁltration of TX˜ with respect to C is given by ∗ F ⋆ ∗ F ⋆ 0 ⊂ (φ|U ( 1|X\B)) ⊂ . . . ⊂ (φ|U ( k|X\B)) ⊂ TX˜ .

Proof. We have ∗ F ⋆ ∗ F det(φ|X\B( i|X\B)) = φ (det i)+ kE, where E denotes the exceptional divisor. Since the class C˜ does not intersect the excep- F ∗F ⋆ tional divisor the slopes of i and φ i|X\B are the same. From this one can deduce that the filtration on X˜ fulfills the defining properties of the Harder-Narasimhan filtration.

4.3. Minimal rational curves and Mori ﬁbrations

In this section, we will compute the Harder-Narasimhan filtration of the tangent bundle on a Fano threefold with respect to classes of curves which lie in fibres of Mori fibrations.

51 The Harder-Narasimhan ﬁltration of the tangent bundle on Fano threefolds

Lemma 4.3.1. Let X be a projective threefold. Let 0 ⊂ F ⊂ G ⊂ TX be the Harder- Narasimhan ﬁltration with respect some movable curve c. Then

µc(G ) >µc(TX ).

Proof. By construction of the Harder-Narasimhan ﬁltration, we have

µc(G /F ) >µc(TX /F ) and thus 1 1 c (G ) · c> c (T ) · c + c (F ) · c. 1 2 1 X 2 1 F 1 Since we have c1( ) · c> 3 c1(TX ) · c, the assertion in the lemma follows. Deﬁnition 4.3.2. Let F be a coherent torsion free sheaf on a smooth projective variety with Harder-Narasimhan ﬁltration

0= F0 ⊂ . . . ⊂ Fk = F with respect to a movable class C. If the slope of the quotient Fi/Fi−1 is positive with respect to C with i ∈{ 1,...,k }, then Fi is called positive with respect to H.

Proposition 4.3.3. Let X be a threefold. If F ⊂ TX is a positive term in the Harder- Narasimhan ﬁltration with respect to a movable class, then F is a foliation.

Proof. Since F is already saturated in TX , we only have to prove that F is closed under Lie bracket. This is clear if rk(F ) = 1. So we assume that F is of rank 2.

Step 1 Let ϕ : E → G be a map between coherent sheaves and assume that G is torsion free. Then Hom(E , G ) = 0 if and only if Hom(E /Tor(E ), G ) = 0, where Tor(E ) ⊂ E denotes the torsion subsheaf of E . Indeed, a morphism E → G factorizes over E /Tor(E ) → G as soon as Tor(E ) ⊂ ker(ϕ). Since G is torsion free, we necessarily have Tor(E ) ⊂ ker(ϕ).

Step 2. We have to prove that the O’Neill tensor

F → TX /F . ^ is generically zero. To simplify the notation, let us write T for the torsion subsheaf of 2 F . By the construction of the Harder-Narasimhan ﬁltration TX /F is torsion free. Thus by Step 1, it is enough to show that V 2

( F ) T → TX /F ^ 2 is zero. We prove slightly more, namely we will show that Hom ( F )/T,TX /F = 0. Let β :( 2 F )/T → T /F be a morphism of O -modules. Let G := imβ. If rk(G )=0 X X V we are done. V 52 Chapter 4

So assume on the contrary that rk(G ) = 1. Let c be the movable class, such the F is a positive term in the Harder-Narasimhan ﬁltration with respect to c. We have a surjection ( 2 F )/T → G . Since both ( 2 F )/T and G are of rank 1 the kernel is a torsion sheaf in ( 2 F )/T . Since ( 2 F )/T is already torsion free, the kernel is zero and consequently V V 2µc(F ) = µc(G ). Since F is a positive term in the Harder-Narasimhan ﬁltration, we V V have µc(F ) > 0 and thus µc(G )=2µc(F ) >µc(F ). (+)

Since TX /F is torsion free and of rank 1, we obtain µc(G ) ≤ µc(TX /F ). Furthermore, we have µc(F ) >µc(TX ) which is clear in case that F is the maximally destabilizing subsheaf of TX and which follows from Lemma (4.3.1) otherwise. Now an easy computation shows that µc(TX /F ) <µc(F ) and thus

µc(G ) ≤ µc(TX /F ) <µc(F ). (++)

The equations (+) and (++) give a contradiction, and thus rk(G )=0.

Remark 4.3.4. Recently, it has been shown in [CP07, Theorem 5.1 and Appendix] that the tensor product modulo torsion of two semistable sheaves with respect to a movable class is again semistable. Then, similar as above, one can prove that the positive terms of the Harder-Narasimhan ﬁltration of a projective manifold of any dimension are foliations.

Corollary 4.3.5. Let φ : X → S be a Fano conic bundle. Suppose that

0 ⊂ TX/S ⊂ G ⊂ TX (∗) is the Harder-Narasimhan ﬁltration of TX with respect to a movable curve. Then G is a foliation.

Proof. Let ∆ be the destabilizing chamber in the Moving cone determined by the ﬁltration (∗). By Proposition (4.3.3) we have to show that

µC (G /TX/S) > 0. for some C ∈ ∆. It follows from the construction of the Harder-Narasimhan ﬁltration that

µC′ (G /TX/S) >µC′ (TX /TX/S ) for all C′ ∈ ∆. Thus we obtain

1 ′ 1 ∗ ′ µ ′ (G /T ) >µ ′ (T /T )= c (T ) − c (T ) · C = − φ K · C (4.2) C XS C X X/S 2 1 X 1 X/S 2 S for all C′ ∈ ∆. Since the chamber ∆ is open by Theorem (3.3.4) (ii), we ﬁnd a class ∗ C˜ ∈ ∆ which is not contracted by φ. Moreover −KS is ample. Thus −φ KS · C˜ > 0.

Proposition 4.3.6. Let φ : X → S be a conic bundle. Suppose there is a foliation G with 0 ⊂ TX/S ⊂ G ⊂ TX . Then G is the pullback of a foliation on S.

53 The Harder-Narasimhan ﬁltration of the tangent bundle on Fano threefolds

Proof. Let B be a two dimensional leaf of G and p ∈ B. Since G is a foliation and −1 TX/S ⊂ G , it follows that the fibre φ (φ(p)) of φ through p lies in B. On the other hand B has no intersection with fibres of φ which do not lie in B. This implies that C := φ(B) is one dimensional and B = φ−1(C). Next we determine the maximal destabilizing subsheaves of the tangent sheaf on del Pezzo surfaces. We will show that these sheaves are relative tangent sheaves of Mori 1 1 fibrations. This has been already verified for P × P and F1 in chapter 2 with the help of the automorphism group of these surfaces. This approach is hopeless for del Pezzo surfaces of higher Picard number since the automorphism groups are very small. However, using minimal rational curves, we easily have the following result.

Proposition 4.3.7. Let S be a del Pezzo surface. Then any destabilizing saturated subsheaf of the tangent sheaf with respect to a movable curve is the relative tangent sheaf of a not necessarily elementary Mori ﬁbration.

Proof. Since S is a del Pezzo surface, the Moving cone of S is polyhedral. The extremal rays of the Moving cone correspond to pullbacks of general fibres in Mori fibre spaces obtained by running the Minimal Model Program, [Ara10, Corollary 1.2]. Let G ⊂ TS be a destabilizing sheaf with respect to a movable curve. By the linearity of the slope, G destabilizes TS with respect to an extremal ray of the Moving cone. Proposition (4.2.9) implies that we can assume that G destabilizes TS with respect to fibre of a Mori fibration. Let φ : S → Y be a Mori fibre space and ℓ a general fibre of φ. Since ℓ is a general free rational curve, it avoids a subset of codimension greater or equal 2, see Proposition (1.3.6). Since the singular locus of G has codimension greater than or equal to two, we can assume that the restriction of G to ℓ is locally free. ∼ 2 If dim Y = 0, then X = P and thus TX is semistable with respect to ℓ. If dim Y = 1, then restricting G and TS to ℓ yields G ∼ ∼ ℓ = O(a) ⊂ O(2) ⊕ O = TS ℓ. G Since destabilizes TX with respect to ℓ, it follows a > 1. On the other hand we have a ≤ 2 since otherwise there are no nontrivial maps from O(a) to O(2) ⊕ O. Thus a =2 and consequently G is the relative tangent sheaf of the Mori fibration. The following two Propositions will be the key to prove Theorem (4.1).

Proposition 4.3.8. Let φ : X → S be a conic bundle with dim X =3. Let ℓ be a general ﬁbre of φ. Suppose there is a foliation F ⊂ TX . We have

(i) if µℓ(F ) >µℓ(TX ) and rk(F )=1, then F is the relative tangent sheaf of the conic bundle φ.

(ii) if µℓ(F ) ≥ µℓ(TX ) and rk(F )=2 then F is the pullback of a foliation on S. If F is moreover maximal destabilizing with respect to some movable class and S is a del Pezzo surface, then it is the pullback of the relative tangent sheaf of a Mori ﬁbration on S.

54 Chapter 4

(iii) If the Harder-Narasimhan ﬁltration with respect to a movable class is given by

0 ⊂ TX/S ⊂ G ⊂ TX , (+) and S is a del Pezzo surface, then G is the pullback of the relative tangent sheaf of a Mori ﬁbration on S. Proof. To start, note that since ℓ is a general free rational curve, it avoids a subset of codimension greater or equal 2, see Proposition (1.3.6). Since the singular locus of both F and TX /F has codimension greater than or equal to two, we assume that the restriction of F and TX /F to ℓ is locally free.

Proof of (i): Note that ℓ is a minimal rational curve. Thus by restricting F and TX to ℓ we obtain

F ∼ ∼ ℓ = O(a) ⊂ O(2) ⊕O⊕O = TX ℓ. F The inequality µℓ( ) > µℓ( TX ) yields a ≥ 1. On the other hand a ≤ 2 since otherwise F we would not even have a map form ℓ to TX ℓ. Note that O(a) has to map to the O(2)-term in T . Since the quotient of T by F restricted to ℓ is torsion free, a = 1 is X ℓ X impossible. This proves (i).

Proof of (ii): Again by restriction to ℓ we obtain F ∼ ℓ = O(a) ⊕ O(b) ⊂ O(2) ⊕O⊕O.

The assumption on the slopes yields a + b ≥ 2. On the other hand we have an injection O(a) ⊕{ 0 } → O(2) ⊕O⊕O and { 0 } ⊕ O(b) → O(2) ⊕O⊕O. Thus a ≤ 2 and b ≤ 2. This implies a = 2 and b = 0 modulo order. Consequently, the general ﬁbre of φ is contained in a leaf of F . This in turn implies that TX/S ⊂ F and hence F is the pullback of a foliation on S by Proposition (4.3.6). Suppose F is maximal destabilizing with respect to a class C ∈ Mov(X). Then we have certainly the inequalities

µC(F ) ≥ µC (TX/S) (*) and µC (F ) >µC (TX ). (**) Using that F is the pullback of a foliation on S and using Lemma (4.2.8), we obtain

∗ c1(F )= −KX/S + φ c1(FS) for a suitable foliation FS on S. Then the inequality (**) reads as 1 1 − K + φ∗K + φ∗c (F ) · C > (−K ) · C, 2 X S 1 S 3 X 55 The Harder-Narasimhan ﬁltration of the tangent bundle on Fano threefolds equivalently written as 1 1 1 φ∗c (F ) · C > K · C + φ∗(−K ) · C. (♯) 2 1 S 6 X 2 S The inequality (*) is 1 − K + φ∗K + φ∗c (F ) · C > −K · C + φ∗K · C, 2 X S 1 S X S equivalently written as

∗ ∗ KX · C > − φ c1(FS)+ φ KS · C . (♯♯)

Now we use the inequality (♯♯), to estimate the right hand side of (♯). This yields 1 φ∗c (F ) · C > φ∗(−K ) · C, 1 S 2 S hence FS destabilizes TS with respect to φ∗C. As we have seen in Proposition (4.3.7), this forces FS to be the relative tangent sheaf of a Mori ﬁbration on S.

Proof of (iii): Observe that by Proposition (4.3.6) there exists a foliation GS on S such that G is the pullback of GS. As in the proof of (ii), we will show that GS destabilizes TS with respect to a movable curve on S. Let C be a movable class in the destabilizing chamber associated to (+). Now by the properties of the Harder-Narasimhan ﬁltration, we have

µC (G /TX/S) >µC (TX /TX/S ).

This is equivalent to 1 1 c (G ) · C > c (T )+ c (T ). (*) 1 2 1 X 2 1 X/S Since by Lemma (4.2.8), we have

∗ c1(G )= −KX/S + φ c1(GS), and since the ﬁrst Chern class of the relative tangent sheaf is given by

∗ c1(TX/S)= c1(TX ) − φ c1(TS), the above inequality (*) is equivalent to 1 φ∗c (G ) · C > − φ∗K · C. 1 S 2 S

Hence GS destabilizes TS with respect to φ∗C. Finally, Proposition (4.3.7) guarantees that G is a the Pullback of a Mori ﬁbration. Remark 4.3.9. Let X be a Fano threefold. If X → S is a Fano conic bundle, then S is a del Pezzo surface, see Fact (4.1.3). Thus we can apply Proposition (4.3.8) to X → S.

56 Chapter 4

Lemma 4.3.10. Let X be a smooth threefold and φ : X → P1 be a del Pezzo fibration. Then X is covered by minimal rational curves lying in fibres of φ. Moreover through a general point in X passes at least two minimal rational curves in fibres having distinct tangent directions.

Proof. This Lemma follows since the general fibre of φ is either P1 × P1, P2 or a del Pezzo surface of degree d with 1 ≤ d ≤ 6: If the general fibre is P2 we take the family of lines in the fibre. If the general fibre is P1 × P1 the horizontal and vertical fibres in P1 × P1 give us minimal rational curves. Let Xp be a general fibre of a del Pezzo fibration. Then Xp is the blow up of P2 in at least 3 points. The strict transforms of lines through a blow up point are minimal rational curves on X. Then, since Xp is the blow up in at least 3 points, there are two distinct minimal rational curves with distinct tangent directions through a general point on X.

Proposition 4.3.11. Let φ : X → P1 be a del Pezzo fibration, i.e. the relative Picard number of ϕ is 1, and let ℓ be a general curve in a fibre of this fibration. Let F be a foliation. If µℓ(F ) >µℓ(TX ) then F is the relative tangent sheaf of φ.

Proof. Since all curves in fibres of φ are numerically proportional, we can take any curve in the fibre of φ to measure the slope of F and TX . Thus we can assume that ℓ is a minimal rational curve in the fibre of φ, avoiding the singular set of both F and TX /F . Now we proceed similar to the proof of Proposition (4.3.8). Let us assume that F is of rank one. Then by restricting F and TX to ℓ, we obtain

F ∼ ℓ = O(a) ⊂ O(2) ⊕ O(1) ⊕ O, or F ∼ ℓ = O(a) ⊂ O(2) ⊕O⊕O. The condition µ (F ) > µ (T ) yields a = 2. Thus the curve ℓ is contained in a leaf of ℓ ℓ X F . But there are two minimal rational curves with distinct tangent direction through a general point on X by Proposition (4.3.10). This shows that the rank of F is two.

So assume rk(F ) = 2. Again, we restrict F and TX to minimal rational curves in the ﬁbres of φ. This yields

F ∼ ℓ = O(a) ⊕ O(b) ⊂ O(2) ⊕ O(1) ⊕ O, or F ∼ ℓ = O(a) ⊕ O(b) ⊂ O(2) ⊕O⊕O. (+) Let us consider the equation (+). Since F destabilizes T with respect to ℓ, this forces X a + b > 2 and as in the proof of Proposition (4.3.8), we see that the only possibility is a = 2, b = 0. This implies that ℓ is contained in a leaf of F . Now Proposition (4.3.11) guarantees that through the general point on X we find at least two minimal rational curve in a fibre of φ. This shows that the general fibre of φ is a leaf of F and thus the Proposition is proved.

57 The Harder-Narasimhan ﬁltration of the tangent bundle on Fano threefolds

4.4. The Harder-Narasimhan ﬁltration on Fano threefolds

4.4.1. Preliminary remarks

In order to prove Theorem (4.1) and to discuss the examples in subsection (4.4.2), we introduce some notation and collect some preliminary remarks. Notation 4.4.1. Let us introduce some notation in order to have compact statements. Let X be a Fano threefold and let R+d be an extremal ray of the Moving cone of X. It might happen that R+d is also an extremal ray of the Mori cone of X. Obviously the extremal contraction associated to R+d is then a Mori fibration. In this case, we just say that R+d corresponds to a Mori fibration. By the Harder-Narasimhan filtration with respect to an extremal ray we mean the filtration with respect to a movable class on the ray. Since the destabilizing chambers are cones, this notion does not depend on the choice of the class on the ray. The following observations are easy consequences of the description of the Moving cone stated in Proposition (1.2.17). Since we will use it several times, let us state it explicitly. Remark 4.4.2. Let X be a Fano threefold with ρ(X) ≥ 3. Suppose we have a two dimensional face F = hd1,d2iR+ of the Moving cone. (i) If F is not contained in a two dimensional face of NE(X), then there exists a divisorial contraction with exceptional divisor D, such that F ⊂ D⊥.

(ii) If moreover d1 does not lie on a two dimensional face of the Mori cone, then there exists another divisorial contraction with exceptional divisor E with E =6 D such that E · d1 = 0.

(iii) Let Fmori be a face of the Mori cone. Suppose that R+r1,..., R+rs are the extremal rays of Fmori which correspond to extremal divisorial contractions. Denote the exceptional divisors of the contraction associated to R+ri by Ei for i = 1,...,s. Then Mov(X) ∩ Fmori = { c ∈ Fmori c · Ei ≥ 0, i =1,...,s } .

Remark 4.4.3. One key to prove our main result is the following easy observation. Let C ⊂ Mov(X) be a polyhedral subcone of the Moving cone. Suppose we have a subsheaf F ⊂ TX such that µC (F ) >µC (TX ) for some class C ∈ C. We claim that this inequality holds on at least one extremal ray of C. Indeed, the inequality µc(F ) >µc(TX ) determines a half space H := { c ∈ N1,R(X) µc(F ) − µc(TX ) > 0 } in N1,R(X). Since C ∈ H, the half space intersects the cone C. Now since C is polyhedral, it is clear that one extremal ray of C lies in this half space. Notation 4.4.4. Let ϕ : X → Y be the blow up of a smooth threefold Y . Let U ⊂ X and V ⊂ Y be the largest open sets such that ϕ U : U → V is an isomorphism. Denote the inverse map of ϕ by ψ. Let F ⊂ T be a coherent saturated subsheaf of T . Then U X X we obtain a coherent saturated subsheaf of TY by taking the saturation of the coherent ∗ F F extension of ψ ( U ). In the sequel, we will denote this sheaf simply by ϕ⋆ .

58 Chapter 4

Lemma 4.4.5. Let ϕ : X → Y be the blow up of a smooth threefold Y with exceptional divisor E ⊂ X. Let d be a movable class on X with d · E =0. Let F ⊂ TX be a coherent saturated subsheaf of TX . Then F F µd( )= µϕ∗d(ϕ⋆ ) ∗ Proof. This follows since c1(F ) = ϕ c1(ϕ⋆F ) + kE for a suitable k ∈ Z and since d · E = 0.

4.4.2. Examples of Harder-Narasimhan ﬁltration of special Fano threefolds

To begin with, we will give some examples of the chamber structure on selected Fano threefolds. These are exactly the Fano threefolds which have to be treated separately in the proof of Theorem (4.1). We start with a remark, which will often be used in the following examples. Remark 4.4.6. Let X be Fano threefold. Suppose F is a covering family of minimal rational curves on X and suppose that ∼ TX ℓ = O(2) ⊕O⊕O or ∼ TX ℓ = O(2) ⊕ O(1) ⊕ O F F for general ℓ ∈ F . Assume there is a foliation ⊂ TX such that µℓ( ) > µℓ(TX ) for general ℓ ∈ F . Then F F (i) ℓ is contained in a leaf of and Tℓ ⊂ ℓ. (ii) if T ∼ O(2) ⊕ O(1) ⊕ O, then rk(F )=2. X ℓ = Proof of Remark (4.4.6). This was already shown implicitly in the proof of Proposition

(4.3.8) and (4.3.11). Let us nevertheless recall the arguments: If we restrict TX and F to ∼ F general ℓ, then both sheaves are locally free. Suppose TX ℓ = O(2) ⊕O⊕O and has rank one. Then F ∼ ℓ = O(a) ⊂ O(2) ⊕O⊕O for a suitable a ∈ Z. Since there is no morphism from O(a) to O(b) if a > b we have a ≤ 2. On the other hand µℓ(F ) > µℓ(TX ) implies that a ≥ 1. Since the quotient of TX /F is torsion free, a = 1 is impossible. Hence a = 2 which shows that F is regular along ℓ and ℓ is contained in a leaf of F . If F has rank 2, then restricting F and TX to ℓ yields F ∼ ℓ = O(a) ⊕ O(b) ⊂ O(2) ⊕O⊕O for suitable a, b ∈ Z. Again, as shown in the proof of Proposition (4.3.8) (ii), we can conclude that a =2, b = 0 modulo order and thus ℓ is contained in the leaf of F and F ∼ is regular along ℓ. The same argumentation shows (i), if TX ℓ = O(2) ⊕ O(1) ⊕ O. Thus (i) is proved.

To prove (ii) note that by [Kol96, Chapter II. Proposition 3.10], deformations of ℓ through a ﬁxed point cover a subset of dimension two. Since by (i), these curves lie in a leaf, F has rank 2.

59 The Harder-Narasimhan ﬁltration of the tangent bundle on Fano threefolds

Example 4.4.7. Let X be the blow up of a cone in P6 over the Veronese surface S =∼ P2 with center in the disjoint union of the vertex and a quartic on the Veronese surface. Another description of X is given by the diagram

φ W o P(O ⊕ O(2)) o X

π P2 where W is the cone over the Veronese surface, P(O ⊕ O(2)) is the blow up of W in the vertex and φ is the blow up of a quartic in S. Let us concentrate on Z := P(O ⊕ O(2)) ﬁrst. There are two distinguished classes of sections of π, namely

′ • the section E1 ⊂ Z corresponding to the surjection O ⊕ O(2) → O. It has normal ′ bundle O(−2) and E1 is the exceptional divisor of a contraction Z → W .

• A movable section H := [OP(O⊕O(2))(1)] corresponding to the surjection O⊕O(2) → O(2). It has normal bundle O(2).

Denote the pullback of a class of a line in P2 via π by L. Moreover we write g for the ∼ 2 ′ ′ class of a line in H = P , l for a ﬁbre of π and f1 for a line in E1. The intersection product is given by

L H l 0 1 g 1 2

′ ′ The equalities [g] = [2l + f1] and [2L] = [H − E1] are easily verified and the canonical bundle formula yields [−KZ ]=[2H + L]. Using the automorphism group of Z, one can directly show that the only possible term in the Harder-Narasimhan filtration of TZ is the relative tangent sheaf of π. Another possibility to show that the only possible term in the Harder-Narasimhan filtration is the relative tangent sheaf TZ/P2 of π is the following. A computation shows that the Moving cone is spanned by the classes l and g. The first Chern class of the relative tangent bundle of π is given by [2H − 2L]. A computation immediately yields that the relative tangent bundle of π destabilizes TZ on both extremal rays of the Moving cone. Consider the destabilizing chamber ∆l. That is the Harder-Narasimhan filtration of all classes c ∈ ∆l is given by 0 ⊂ TZ/P2 ⊂ TZ . Then consider a hypothetical neighboring chamber Θ of ∆l. Since TZ/P2 destabilizes TZ on both rays of the Moving cone, the linearity of the slope immediately implies that TZ/P2 destabilizes TZ with respect to all movable curves. Thus the destabilizing chamber Θ is not the semistable chamber. By Lemma (3.3.9), the only possibilities for the defining Harder-Narasimhan filtration with respect to Θ are the following: There exists a sheaf G of rank 2 with TZ/P2 ⊂ G such that the Harder-Narasimhan filtration with respect to Θ is given by 0 ⊂ TZ/P2 ⊂ G ⊂ TZ or by 0 ⊂ G ⊂ TZ . This shows that G is the pullback of a foliation on P2, see Proposition (4.3.6). Now Proposition (4.3.8) (i) and (ii) implies

60 Chapter 4 that G is the pullback of a Mori ﬁbration on P2 which is impossible. Thus there is no neighboring chamber of ∆l and consequently TZ/P2 is the maximal destabilizing subsheaf of TZ with respect to any movable class.

Remark 4.4.8. We remark for later application that in this example we have c1(TX/P2 ) · g = 2 and c1(TX ) · g =5.

φ∗l

H˜

Figure 4.1: The blow up of P(O ⊕ O(2)) in a quartic

Now we blow up a quartic C ⊂ H ∼= P2, where H ∈|H|, to obtain the variety X. Denote the exceptional divisor of the blow up by E2, the proper transform of the distinguished ˜ ′ section H by H and a ﬁbre in E2 by f2. We write E1 for the pullback of E1 and f1 for ′ the pullback of f1. The intersection product is then given by

∗ E1 E2 φ H f1 −2 0 0 f2 0 −1 0 φ∗l 1 0 1

We claim that the Mori cone is given by

∗ ∗ NE(X)= hf1, f2,φ l − f2, 2φ l + f1 − 4f2iR+ =: C

To see this, we start with an irreducible curve Q ⊂ X not contained in H,˜ E1, E2. Then ∗ φ∗Q ≡ al + bf and Q ≡ φ φ∗Q − kf2 where k counts the points of intersection of φ∗Q with C. Thus k = #(φ∗Q ∩ C) ≤ φ∗Q · H =(al + bf) · H = a, and thus Q lies in the claimed cone. If the curve lies in one of H,˜ E2, E2, then we also see that the curve is numerically equivalent to a class in C. Moreover, all classes in C can be represented by curves. This shows the claimed equality.

61 The Harder-Narasimhan ﬁltration of the tangent bundle on Fano threefolds

∗ ∗ Denote the proper transform of π C by D. Note that [D]=[2φ H − 2E1 − E2]. We have the following exceptional divisors on X. The divisors E2 and D are exceptional divisors whose contractions result in smooth varieties. The exceptional divisors E1 and H˜ correspond to divisorial contraction with singular target varieties. The exceptional divisors are sketched in Figure (4.1).

Denote the contraction of D by α. Then the contraction of D results again in a variety isomorphic to P(O ⊕ O(2)), where the roles of D and E2 are reversed. So we get the following diagram:

φ W o P(O ⊕ O(2)) o X

α P2 o P(O ⊕ O(2)) Using Proposition (1.2.17), we can compute the cone of movable curves. The cone of movable curve is given by

∗ ∗ ∗ ∗ Mov(X)= hφ l, 4φ l + f1 − 4f2, 2φ l + f1 − 2f2, 2φ l + f1iR+ .

We want to visualize the corresponding curves: Clearly, φ∗l is just the pullback of the 1 ∗ fibres of the P -bundle π. The classes of 2φ l + f1 − 2f2 can be represented by the proper transform of the family of lines in the movable sections H ∼= P2 which intersect the blow ∗ up locus twice. Let us denote this family by F . The class 2φ l + f1 is just the pullback ∗ of a class of a line in H. Note that by the symmetry 4φ l + f1 − 4f2 is also spanned by the class of such a line – viewed under the blow up α. By Proposition (3.3.11), we only have to compute the Harder-Narasimhan filtration with respect to classes from two dimensional faces of the Moving cone in order to determine all sheaves occurring in the Harder-Narasimhan filtration with respect to any movable class. Note that for movable classes lying on a two dimensional face which is cut out by E2 or D, the filtration comes from the corresponding filtration on P(O ⊕ O(2)) by Proposition (4.2.9).

∗ Claim: TX is semistable with respect to d := 2φ l + f1 − 2f2.

Recall that we denote the families of proper transforms of lines in a movable section H ⊂ P(O ⊕ O(2)) which intersect the blow up locus twice by F . A computation of the ∗ splitting type of TX restricted to ℓ ∈ F with [ℓ]=2φ l + f1 − 2f2 yields that TX ℓ = O(2)⊕O(1)⊕O and thus a hypothetical destabilizing subsheaf F has rank 2 and Remark

(4.4.6) ensures that each line in F is contained in a leaf of F . Consider a general movable section H′ ⊂ P(O ⊕ O(2)) with H′ ∈ |H|. Then H′ intersects the distinguished section H with C ∈ H in a quadric Q. Thus C ∩ Q consists of 8 points. Strict transforms of lines in H′ through every two of these points give elements in F . These lines intersect away from the blow up locus. Thus they lie in a single leaf, say B˜. Note that d can be approximated by complete intersection curves. Since the sheaf F remains maximal destabilizing in a neighborhood of d, we can use Theorem (1.1.12) and conclude that the

62 Chapter 4

D⊥ f2

⊥ E2 ∗ 2φ l + f1

⊥ f1 H˜ ∗ φ l − f2 d

∗ 2φ l + f1 − 4f2

⊥ E1

Figure 4.2: The Mori cone of P(O⊕O(2)) in a quartic. The tinted area is the Moving cone of P(O⊕O(2))

leaves of F are algebraic. Let us write B for φ∗B˜. Then the class of B can be written as [B]= a[H]+ b[L]. Now intersecting B with H′ and using that B contains the lines in H′ mentioned above, yields that 2a + b ≥ 28. Thus a leaf B and any line ℓ′ in F not contained in B have nonempty intersection outside the blow up locus which contradicts that F which is absurd. This shows that TX is semistable with respect to d. ⊥ ∗ Now consider the face Fmov of the Moving cone contained in E1 , i.e. Fmov = h2φ l + f1,diR+ . The extremal rays of this faces are shown in Figure (4.2). Suppose that we have a maximal destabilizing subsheaf G of TX with respect to c ∈ Fmov. Since TX is semistable with respect to d the linearity of the slope immediately implies that G destabilizes TX ∗ with respect to 2φ l + f1. Let TX/P2 be the relative tangent sheaf of π ◦ φ. We know that ∗ c1(TX/P2 )·(2φ l +f1) = 2 and that TX/P2 is the maximal destabilizing subsheaf of TX with ∗ ∗ 5 respect to 2φ l + f1. Moreover, we have c1(TX ) · (2φ l + f1) = 3 . Now since we assume G ∗ G 2 that destabilizes TX with respect to 2φ l + f1 and since µ2φ∗l+f1 ( ) ≤ µ2φ∗l+f1 (TX/P ), G it follows that µ2φ∗l+f1 ( ) = 2. Because the maximal destabilizing sheaf is unique, we obtain G = TX/P2 . This shows that the sheaf TX/P2 is the only possible term in the Harder-Narasimhan ﬁltration with respect to classes from the face Fmov. By the symmetry of X, the same argumentation is valid if we consider the face h4φ∗l + f1 − 4f2,diR+ . Example 4.4.9. Let X be the blow up of the cone over a smooth quadric in P3 along the disjoint union of the vertex and an elliptic curve on the smooth quadric. Similar to

63 The Harder-Narasimhan ﬁltration of the tangent bundle on Fano threefolds the previous example, we get the following picture:

β V o P(O ⊕ O(1, 1)) o X

P1 × P1 Let us first take a look at π : P(O ⊕ O(1, 1)) → P1 × P1. To abbreviate notation, let us write Z for P(O ⊕ O(1, 1)). The Picard number of Z is 3. Let us denote the exceptional ′ ∼ 1 1 ′ divisor of the blow up β by E1 = P × P and the two classes of the ruling of E1 with f1 ∼ 1 1 and f2. Furthermore, let H = P × P be a movable section of π. ∗ The bundle structure of P(O ⊕ O(1, 1)) gives us two divisor classes, namely L1 := π l1 ∗ and L2 = π l2 where l1 and l2 denotes the fibres of the first and second projection of P1 × P1 onto the projective line. Finally we denote a fibre of π by l.

A computation of the Mori cone shows

NE(Z)= hl, f1, f2iR+ .

′ Note that we can contract the exceptional divisor E1 in two directions. These contractions are examples of contractions of type E1b. Proposition (1.2.17) yields that the cone of movable curves is given by

Mov(Z)= hl, l + f1, l + f2iR+

The classes of l1 and l2 satisfy the equalities [l1] = [l + f1] and [l2] = [l + f2]. We claim that the only possible terms in the Harder-Narasimhan filtration are relative tangent sheaves of not necessarily elementary Mori fibrations. We prove this in exactly the same manner as we will proceed in the proof of Proposition (4.4.19). We only have to compute the Harder-Narasimhan filtration of TZ with respect to the two dimensional faces of Mov(Z) in order to determine the the subsheaves of the Harder-Narasimhan filtration with respect to any movable class, see Proposition (3.3.11). Let us consider exemplarily F the face F := hl, l + f1iR+ . Suppose we have a maximal destabilizing subsheaf of TZ with respect to a class c in F . Then by the linearity of the slope, F destabilizes with respect to l or l + f1, see Remark (4.4.3). If F destabilizes TZ with respect to l, then F is the relative tangent sheaf of the projection π or the pullback of a Mori fibration 1 1 on P × P by Proposition (4.3.8) (i) and (ii). If F destabilizes with respect to l + f1, then we consider the contraction ϕ : Z → Y associated to R+f1. Then ϕ∗(l + f1) is an extremal ray of the Mori cone of Y corresponding to a del Pezzo fibration ξ, see Proposition (4.1.12) (ii). Thus ϕ⋆(F ), see Notation (4.4.4), is the relative tangent sheaf of the del Pezzo fibration ξ by Proposition (4.3.11). If the rank of F is one, there might exists a second term G in the Harder-Narasimhan filtration with respect to the movable class c. Then µc(G ) > µc(TX ) by Lemma (4.3.1) and by the linearity of the slope, G destabilizes with respect to l or l + f1. Now, as above, we conclude that G is the relative tangent sheaf of a non elementary Mori fibration. The remaining two faces of the Moving cone can be treated similarly.

64 Chapter 4

Now we blow up P(O ⊕ O(1, 1)) along an elliptic curve C in a movable section H ∈|H|. Note that C ∈ |O(2, 2)|. What happens is very similar to Example (4.4.7). Denote the exceptional divisor of the blow up map by E2, a ﬁbre in E2 by f and the proper transform ∗ ′ of π C by D. Furthermore, we denote the pullback of E1 by E1.

E1 D

H˜

Figure 4.3: The blow up of P(O ⊕ O(1, 1)) along an elliptic curve

The Mori cone is given by ∗ ∗ ∗ ∗ ∗ NE(X)= hf,φ l1 − 2f,φ l2 − 2f,φ f1,φ f2,φ l − fiR+ . Denote the proper transform of the distinguished section H ∼= P1 × P1 by H˜ . The ∗ ∗ extremal rays R+(φ l1 − 2f1) and R+(φ l2 − 2f2) correspond to divisorial contractions ∗ with the exceptional divisor H˜ of type E1b. The extremal ray R+(φ l − f) corresponds to the exceptional divisor D. The contraction of D results again in variety isomorphic to P(O ⊕ O(1, 1)), where the roles of H and E are reversed or where the variety is turned upside down. The variety together with all exceptional divisors of the extremal contractions are shown in Figure (4.3).

Claim: Each term of the Harder-Narasimhan ﬁltration with respect to any movable class is the relative tangent sheaf of not necessarily elementary Mori ﬁbration.

Our argumentation is very similar to the proof of Proposition (4.4.27). We only have to compute the Harder-Narasimhan ﬁltration of TX with respect to two dimensional faces of the Moving cone in order to determine all possible sheaves occurring in the Harder-Narasimhan ﬁltration with respect to any movable curve, see Proposition (3.3.11).

Let Fmov be a two dimensional face of the Moving cone. We suppose ﬁrst that Fmov is contained in the boundary of the Mori cone, i.e. there exists a three dimensional face Fmori of the Mori cone with Fmov ⊂ Fmori. A computation shows that in case that Fmori ∗ ∗ ∗ ∗ contains φ f1,φ f2 or φ l1 − 2f,φ l2 − 2f, then there exists no movable class on Fmori. ∗ More precisely, the only three dimensional faces of NE(X) that contain both φ f1 and ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ φ f2 are hφ f1,φ f2, fiR+ and hφ f1,φ f2,φ l + φ f1 − 2f,φ l + φ f2 − 2fiR+ . Then computing Mov(X) ∩ Fmori via Remark (4.4.2) (iii) shows that there exists no movable class on these faces.

65 The Harder-Narasimhan ﬁltration of the tangent bundle on Fano threefolds

Let us consider for example the three dimensional face

∗ ∗ ∗ Fmori := hf1,φ l + φ f1 − f,f,φ l − fiR+

˜ ≥0 ≥0 ≥0 ≥0 of the Mori cone. Then the Mov(X) ∩ Fmori is given by Fmori ∩ H ∩ E1 ∩ E2 ∩ D . This is illustrated in Figure (4.4). Let us determine the sheaves occurring in the

∗ ∗ ˜ ⊥ φ l + φ f1 − 2f H

φ∗l − f

∗ φ f1

⊥ E1

Figure 4.4: The Figure shows a slice through the face Fmori of the Moving cone. The hatched area is the Moving cone in Fmori.

Harder-Narasimhan filtration with respect to the two dimensional faces of the Moving cone in Fmori. In this case there are four two dimensional faces of the Moving cone as shown in Figure (4.4). ⊥ Suppose Fmov is contained in D . The contraction of D results again in a variety isomorphic to P(O⊕O(1, 1)). Let us denote this contraction by α. Then the Harder-Narasimhan filtration with respect to a class c in Fmov is the pullback of the Harder-Narasimhan filtration with respect to α∗c, see Proposition (4.2.9). Since we have already convinced ourselves that the only possible terms in the Harder-Narasimhan filtration of the tangent sheaf with respect to a movable class of P(O ⊕ O(1, 1)) are relative tangent sheaves of Mori fibration, we can conclude that the filtration with respect to c contains only relative tangent sheaves of Mori fibrations, too. ⊥ If we suppose that Fmov is contained in E1 , then we can argue similarly. ˜ ⊥ Now suppose Fmov is contained in H , i.e. Fmov = hd1,d2iR+ as shown in Figure (4.4). ∗ ∗ Denote the contraction of the face hf1,φ l + φ f1 − 2fiR+ by ξ : X → V . Since the exceptional divisors E1 and H˜ are disjoint, this map contracts exactly these divisors. Now one can verify that R+ξ∗d1 is an extremal ray of NE(V ) corresponding to a del Pezzo fibration. ′ ∗ ∗ Let ψ : X → Z be the contraction of the face hϕ l + f1 − sf,ϕ l − fiR+ . Then one ′ verifies that R+ψ∗d2 is an extremal ray of NE(Z ) corresponding to a del Pezzo fibration. Now let us determine the Harder-Narasimhan filtration of TX with respect to a class c ∈ Fmov. Let F be the maximal destabilizing subsheaf of TX with respect to c. Since the slope is a linear function on the Moving cone, we have µ(F ) >µ(TX ) with respect to d1 F F F or d2, say for instance µd1 ( ) >µd1 (TX ). Now by Lemma (4.4.5), µξ∗d1 (ξ⋆ )= µd1 ( ). F Furthermore we have µd1 (TX )= µξ∗d1 (TV ). Hence we have µξ∗d1 (ξ⋆ ) >µξ∗d1 (TV ). Now Proposition (4.3.11) guarantees that F is the relative tangent sheaf of the Mori fibration

66 Chapter 4

associated to R+ξ∗d1 that is F is the relative tangent sheaf of a del Pezzo fibration. If F destabilizes TX with respect to d2, we can argue similarly. A similar argumentation ⊥ also holds, if we consider the face of the Moving cone which is contained in E1 . Hence we have shown that with respect all classes on the two dimensional faces of the Moving cone in Fmori the Harder-Narasimhan filtration is given by relative tangent sheaves of not necessarily elementary Mori fibrations. By now we have considered all three dimensional faces of the Mori cone except the face ′ ∗ ∗ ∗ Fmori := hf2,φ l+φ f2 −2f,f,φ l−fiR+ . Since we can determine the Harder-Narasimhan ′ filtration with respect to all two dimensional faces of the Moving cone contained in Fmori in exactly the same way as above, we omit this case.

So far, we have computed the Harder-Narasimhan ﬁltration with respect to all movable classes on two dimensional faces of the Moving cone inside a three dimensional face of the Mori cone. It remains to prove the Claim in case that we have a two dimensional face of the Moving cone which is not contained in the boundary of the Mori cone.

Let Fmov = hh1, h2iR+ be a two dimensional face of the Moving cone not contained in boundary of the Mori cone. There exists two exceptional divisors D1,D2, with Fmov ⊂ ⊥ ⊥ D1 ∩ D2 . If one of these divisors is E2 or D, then the Harder-Narasimhan filtration with respect to a class in Fmov is the pullback of the corresponding Harder-Narasimhan filtration on P(O⊕O(1, 1)) by Proposition (4.2.9). Thus we only have to consider the case ⊥ ˜ ⊥ ∗ ∗ that Fmov is contained in E1 ∩H . In this case we have R+h1 = R+d1 = R+(φ l+φ f1−f), ∗ ∗ see Figure (4.4), and R+h2 = R+(φ l + φ f2 − f). Now a maximal destabilizing subsheaf F with respect to a class on Fmori, will destabilizes on h1 or h2, say for instance with respect to h1. Then we have already seen above that F is the pullback of a del Pezzo fibration. We have proved the Claim.

Example 4.4.10. Let π : X → P3 be the blow up of P3 in a plane cubic C. Denote the distinguished plane which contains the cubic C by H. Let H be the class of a plane in P3 and l be the class of a line in P3; denote the exceptional divisor by E and a line in E by f. A computation shows that the Mori cone is spanned by f and π∗l − 3f. The proper transform H has the numerical class π∗H − E an is the exceptional divisor of the ∗ contraction associated to R+(π l − 3f). The proper transform of H can be contracted to a singular point. A computation using Proposition (1.2.17) shows that the cone of movable curves is spanned by π∗l − f and π∗l. We claim that the tangent bundle is semistable with re- ∗ ∗ ∗ spect to both π l and π l − f. The semistability of TX with respect to π l follows from Proposition (4.2.9) and the fact that TP3 is stable. Observe that the class π∗l − f can be represented by the proper transforms of lines in P3 which intersect the blow up locus once. These curves give a family F of minimal rational curves with splitting type ∼ TX ℓ = O(2) ⊕ O(1) ⊕ O, where ℓ denotes a rational curve in the family F with class π∗l − f.

Suppose there exists a destabilizing subsheaf F of TX with respect to ℓ ∈ F . By Remark (4.4.6) F has rank 2 and the curves ℓ ∈ F are contained in the leaves of F . One easily sees that the class π∗l − f can be approximated by complete intersection curves. Since F

67 The Harder-Narasimhan ﬁltration of the tangent bundle on Fano threefolds stays maximal destabilizing in a neighborhood of π∗l − f, we can apply Theorem (1.1.12) to see that F is a foliation with algebraic leaves. For any point x∈ / E consider the curves in X through x which are obtained as the proper transform of lines through π(x) which intersect the blow up locus. These curves have to 3 lie in a leaf B of F . The pushforward π∗B in P has numerical class a · [H]. Since π∗B · H contains the blow up locus of π, it follows that a ≥ 3. This shows that a general element ℓ ∈ F intersects a leaf not containing ℓ, which is absurd. Thus TX is semistable with respect to π∗l − f. ∗ ∗ Since TX is semistable with respect to π l and π l −f and since the semistable chamber is convex by Theorem (3.3.4) (i), it follows that TX is semistable with respect to all movable classes.

Example 4.4.11. Let π : X → P3 be the blow up of P3 in curve C which is the intersection of a quadric Q and a cubic. We adopt the notation of the previous example. A computation shows that the Mori cone is spanned by π∗l − 3f and f Note that the class 2π∗H − E can be represented by the proper transform of the quadric Q. This proper transform of Q is the exceptional divisor of an extremal contraction to a singular point. More precisely, it can be contracted to a ordinary double point in case the quadric is smooth and to a double cDV point in case the quadric is a singular cone. Using Proposition (1.2.17) one shows that the cone of movable curves is spanned by π∗l ∗ ∗ ∗ and π l − 2f. We will prove that TX is semistable with respect to both π l and π l − 2f. The semistability of the tangent bundle with respect to π∗l follows from Proposition ∗ (4.2.9) and the fact that TP3 is semistable. Thus we consider the class π l − 2f. Note that this class can be represented by the proper transforms of secants of the blow up locus. Since the secant variety of the blow up locus is three dimensional, these curves cover X. We denote the family of these curves by F . The splitting type of TX on a curve ℓ ∈ F is given by

TX ℓ = O(2) ⊕O⊕O. Remark (4.4.6) implies that the curves in F are contained in the leaves of a hypothetical destabilizing subsheaf F . Note that for any curve ℓ ∈ F we can ﬁnd another curve ℓ′ ∈ F such that ℓ ∩ ℓ′ =6 ∅. Since the curves in F lie in leaves of F , it follows that F is of rank two. Since the class of π∗l − 2f can be approximated by complete intersection curves, we can assume that the leaves of F are algebraic, see Theorem (1.1.12). Consider a general hyperplane H′ in P3. Then the intersection of H′ with the blow up locus consists of six points. Note that there are 15 lines in H′ through each two points. Since the foliation F is regular along the proper transforms of these lines and since these lines intersect away from the blow up locus, the proper transforms of these lines have to lie in the same leaf of F . Let B be a general leaf of F . We can write π∗B ≡ aH. As we have seen, a general hyperplane in P3 contains 15 secant lines of C which lie in a single leaf. This implies that a ≥ 15. This in turn implies that a general element ℓ ∈ F not contained in the leaf B intersects B - a contradiction. Since the semistable chamber is convex by Proposition (3.3.4) and TX is semistable with respect to both extremal rays of the Moving cone, we can conclude that TX is semistable with respect to all movable classes.

68 Chapter 4

Example 4.4.12. Let π : X → Q be the blow up of a smooth quadric Q in P4 along the intersection of two divisors A and B with A ∈ |OQ(1)| and B ∈ |OQ(2)|. Recall 1 that NR(Q) is generated by OQ(1) that is the hyperplane sections of Q. An element 4 HQ ∈ |OQ(1)| is of the form H Q for an element H ∈ |OP (1)| and thus HQ is a quadric H N Q H C in . The space 1,R( ) is generated by the lines on Q. Write for the blow up locus of π. Then C is a curve of degree 4 in P4. A computation shows that the Mori cone is given by a fibre f in the exceptional divisor and by π∗l − 2f. This can be seen as follows. Take a curve C′ in X and suppose at first that C′ is not contained in E or the proper ∗ ′ ∗ ′ transform of A or B. Then we can write π π∗C = π (al) where a is the degree of π∗C 4 ′ ∗ ′ as a curve in P . Thus C = aπ l − kf with k ≤ a since π∗C is not contained in A or B. Note that if we take a line l in A, then l will intersect C in two points. Thus the proper transform of l is π∗l − 2f. Both extremal rays of the Mori cone correspond to divisorial contractions. The contraction of R+f is the map π. The exceptional divisor of π is denoted by E. The proper transform of A is the exceptional divisor of the contraction associated to ∗ ∗ R+(π l − 2f). The exceptional divisor of this map has class π HQ − E where HQ is ∗ a hyperplane section. The contraction associated to R+(π l − 2f) contracts the proper transform of A to an ordinary double point or a double cdV point depending on whether A is smooth or not. Now using Proposition (1.2.17) we see easily that the cone of movable curves is spanned ∗ ∗ by π l and π l − f. By Proposition (4.2.9) and the fact that TQ is semistable, we know ∗ ∗ that TX is π l-semistable. We will show that TX is semistable with respect to π l − f. Curves with class π∗l − f on X are proper transforms of lines in Q which intersect the blow up locus in one point. For a fixed point x ∈ Q, all lines in Q through x form a quadratic cone. This can be seen by computing the intersection of a projective tangent space in x with Q. Thus for each line ℓ in Q through the blow up locus and a point p ∈ ℓ there is another line through p which intersects the blow up locus. Let us denote the family of proper transforms of lines in Q which intersect the blow up locus by F . Note that an element ℓ ∈ F has the class π∗l − f. The splitting type of TX on ℓ is given by

TX ℓ = O(2) ⊕O⊕O. F Remark (4.4.6) guarantees that ℓ is contained in leaves of . Since for a point on p ∈ ℓ there exists another curve ℓ′ ∈ F through p, F has rank 2. Consider the pushforward S of a leaf of F . Then it is covered by lines through the blow up locus. We write aH for the class of S where H ∈ OQ(1). Then we will have a ≥ 2 and thus a general line through the blow up locus intersects the leaf outside the blow up locus. Proposition 4.4.13. Let φ : X′ → X be a double cover of smooth projective Fano threefolds with ρ(X′)= ρ(X) branched in a smooth divisor, then: (i) The Moving cone of X′ identiﬁes with the Moving cone of X via pullback.

(ii) If F is maximal destabilizing with respect to a movable curve φ∗h, then F is the pullback of a foliation on X.

69 The Harder-Narasimhan ﬁltration of the tangent bundle on Fano threefolds

∗ (iii) Moreover, if TX is semistable with respect to h, so is TX′ with respect to π h. Proof. If one takes ρ(X)= ρ(X′) into account, then the ﬁrst statement is an immediate consequence of the projection formula and the duality statement of [BDPP04]. Let ℓ be a movable class on X′. Then we can write ℓ = φ∗h for a movable class h on X. Suppose there is a maximal destabilizing subsheaf F of TX′ of rank f. We write i for the sheet interchanging involution. Note that by means of the diﬀerential of i, we can ∗ ∗ identify TX′ and i TX′ . Thus we can interpret i F as a subsheaf of TX′ . Since the class of ℓ is invariant under the sheet interchanging involution i, we have i∗[ℓ] = [ℓ]. Therefore, we compute ∗F 1 ∗F µℓ(i ) = f c1(i ) · ℓ 1 ∗ F = f i c1( ) · ℓ 1 F = f c1( ) · i∗ℓ 1 F = f c1( ) · ℓ = µℓ(F ). By the uniqueness of the maximal destabilizing subsheaf, it follows that F = i∗F . This shows that F is the pullback of a foliation on X. This shows (ii).

To prove (iii), let us first introduce some notation. We write D for the branch divisor ∗ 2 and B := φ (D)red. Note that in local coordinates, φ is given by (s,t,u) 7→ (s ,t,u), where B is given by the hyperplane { s =0 } . Suppose we have a maximal desta- F F ∨ 1 bilizing subsheaf X′ ⊂ TX′ . Then consider X′ ⊂ ΩX′ giving the same foliation by means of differential forms. By (ii), the foliation is the pullback of a foliation on F F ∨ 1 X, say X . We write X ⊂ ΩX for the corresponding foliation given by differential forms.

Suppose rk(FX′ ) = 2. We have to distinguish two cases: First, if D is FX invariant, ∗F ∨ then a computation in local coordinates proves that the differentials in φ X vanish on B. Thus F ∨ ∗F ∨ X′ = φ X ⊗ OX′ (B). F F ∨ ∗F ∨ Secondly, if X is transversal to D, then X′ = φ X . Recall the canonical bundle ∗ formula [−KX′ ] = [−φ KX ] − [B], see for example [BHPVdV04, Lemma 1.17], and the fact that the sheaf of differential forms corresponding to a foliation F can be identified ∗ with (TX /F ) . Thus we compute

c1(FX′ ) = −KX′ + c1(NX′ )

∗ F ∨ φ c1( X ) = −K ′ + X φ∗c (F ∨)+ B 1 X ∗ ∗ φ KX + φ c1(FX ) = K ′ + X φ∗K + φ∗c (F )+ B X 1 X φ∗c (F ) − B = 1 X , φ∗c (F ) 1 X 70 Chapter 4 where the lower line in the computation refers to the case where D is F transversal and the upper line refers to the case where D is FX -invariant. If the rank of FX′ is 1, then in case that FX is transversal to D, one can argue as above. ∗ In case that D is FX invariant, then lifting vector ﬁelds yield that c1(FX′ )= φ c1(FX ). ∗ Now it is easy to verify that if FX′ destabilizes TX′ with respect to φ h, then a fortiori FX destabilizes TX with respect to h.

Example 4.4.14. X is a double cover of V7 whose branch locus is a divisor B ∈ | − KV7 | such that a) B ∩ D is smooth or b) B ∩ D is reduced but not smooth, where D is the 3 exceptional divisor of the π : V7 → P . let us write σ for the covering map. In our special example, we immediately see that the Moving cone of X is spanned by the pullback σ∗π∗l, where l is the class of a line in P3. The other extremal ray corresponds to the extremal 2 ray of the Mori-ﬁbration V7 → P .

Since TV7 is semistable with respect to l, we obtain one destabilizing chamber corresponding to the Mori ﬁbration on X.

4.4.3. Picard number 2

Remark 4.4.15. Note that the classiﬁcation of Fano threefolds [IP99, Appendix] [MM83, Theorem 1.7 and Section 5] implies that all Fano threefolds of Picard number 2 that admit an extremal contraction onto a singular variety, i.e. a contraction of type E3, E4, E5, are treated in subsection (4.4.2), namely Examples (4.4.10), (4.4.11), (4.4.12), (4.4.14) and (4.4.7).

Remark 4.4.16. Note that there exists no contraction of type E1b on a Fano threefold X of Picard number 2. This can be seen as follows. Suppose there exists a contraction ∼ 1 1 of type E1b on X with exceptional divisor D = P × P . Then the classes of a horizontal and a vertical ﬁbre of D would span the Mori cone of X. This would imply that each curve in X intersects D negatively, which is absurd.

Proposition 4.4.17. On a Fano threefold with Picard number 2 the only possible term occurring in the Harder-Narasimhan ﬁltration with respect to any movable class is a relative tangent sheaf of a Mori ﬁbration.

Proof. Let X be a Fano threefold of Picard number 2. The Moving cone of X is spanned by exactly two extremal rays. We remind the reader that the Moving cone is cut out of the Mori cone by all exceptional divisors on X, see Proposition (1.2.17). Let C be a movable class on X. Let F be the maximal destabilizing subsheaf of TX with respect to C. Then by Remark (4.4.3), F destabilizes TX with respect to an extremal ray of the Moving cone, say R+d.

(i) If R+d corresponds to a Mori fibration with one dimensional fibres, say ϕ : X → P2, then Proposition (4.3.8) ensures that F is the relative tangent sheaf of the Mori fibration associated to R+d. Indeed, if rk(F ) = 1 this is statement (i) in Proposition (4.3.8). Note also that a second term G in the Harder-Narasimhan filtration with respect to C is impossible: By statement (iii) in Proposition (4.3.8),

71 The Harder-Narasimhan ﬁltration of the tangent bundle on Fano threefolds

G would be the pullback of the relative tangent sheaf of a Mori ﬁbration on P2 which is absurd. Note also that rk(F ) = 2 is impossible since it would also force F to be the pullback of a Mori ﬁbration on P2 by Proposition (4.3.8) (ii).

(ii) If R+d corresponds to a Mori fibration with two dimensional fibres, i.e. it corresponds to a del Pezzo fibration, then Proposition (4.3.11) ensures that F is the relative tangent sheaf of the del Pezzo fibration.

(iii) The last possibility is that R+d lies in the interior of the Mori cone. Then by Proposition (1.2.17), there exists a divisorial contraction φ : X → Y whose exceptional divisor intersects all classes on R+d with zero. Since we have treated all cases of Fano threefolds of Picard number 2 admitting a divisorial contraction onto a singular variety separately in the previous examples, see Remark (4.4.15), and since a contraction of type E1b is impossible, we might assume that Y is a Fano threefold of Picard number 1. These are known to have semistable tangent bundle, see [PW95, Theorem 3, Proposition 2.2] or [Hwa98]. Thus Proposition (4.2.9) implies that TX is also semistable with respect to d. Thus actually F does not destabilize TX with respect to d.

Remark 4.4.18. Let X be a Fano threefold with ρ(X) = 2. Let r be a class on an extremal ray of the Moving cone. Suppose r is contained in the interior of the Mori cone ∼ of X. We claim that TX is semistable with respect to r unless X = PP2 (O⊕O(2)). Indeed: Since R+r lies in the interior of NE(X) there exists a divisorial contraction ϕ : X → Y , whose exceptional divisor intersects each class on R+r with zero. If Y is a Fano threefold, then TX is semistable with respect to r, as we have seen in part (iii) of the previous proof. The Fano threefolds admitting a contraction on a singular variety have been considered in subsection (4.4.2), namely Examples (4.4.10), (4.4.11), (4.4.12), (4.4.14) and (4.4.7). As we have seen there, P(O ⊕ O(2)) is the only Fano threefold of Picard number 2, such that TX is not semistable with respect to an extremal ray of the Moving cone lying in the interior of NE(X).

4.4.4. Picard number 3

Proposition 4.4.19. Let X be a Fano manifold with Picard number 3. Then the only possible terms occurring in the Harder-Narasimhan ﬁltration of the tangent bundle associated to any movable class are relative tangent sheaves of not necessarily elementary Mori ﬁbrations.

Proof. Step 1. Preliminary remarks and assumptions. Since we have treated the only Fano threefold of Picard number 3 admitting two extremal contractions of type E1b, E3, E4, E5 separately in Example (4.4.7), see also Proposition (4.1.16), we assume that there exists at most one exceptional divisor of type E1b, E3, E4, E5 on X. The Moving cone is a convex cone spanned by a ﬁnite number of extremal rays. Each two dimensional face of the cone is spanned by exactly two extremal rays. On the other

72 Chapter 4 hand the Moving cone is cut out of the Mori cone by the exceptional divisors of extremal divisorial contractions, see Proposition (1.2.17). In order to determine the sheaves in the Harder-Narasimhan ﬁltration with respect to any movable class, we only have to compute the ﬁltration with respect to classes from the boundary of the Moving cone by Proposition (3.3.11).

Step 2. Setup of the proof. Let us consider an extremal face F = hd1,d2iR+ of the Moving cone of X and a class c ∈ F . Let F be the maximal destabilizing subsheaf with respect to c. Note that if the rank of F is one, there might exist a second term G in the Harder-Narasimhan ﬁltration with respect to c. Simply because F is the maximal destabilizing subsheaf of TX with respect to c, we have µc(F ) >µc(TX ). Remark (4.4.3) implies µ(F ) > µ(TX ) with respect to d1 or d2. Suppose there exists a second term in the Harder-Narasimhan ﬁltration of TX with respect to c. Then by Lemma (4.3.1) we also have µc(G ) > µc(TX ). Thus by Remark (4.4.3), the sheaf G destabilizes TX with respect to d1 or d2, too.

Step 3. There are several possibilities, how the face F lies in the Mori cone. We consider all possible cases and prove the Proposition for each case separately.

Case 1: F lies on the boundary of the Mori cone. (i) Suppose R+d1 and R+d2 correspond to Mori fibrations. Then the contractions associated to R+d1 and R+d2 have one dimensional fibres by Lemma (4.1.17). Denote the contraction associated to R+d1 by η : X → S, where S is a del Pezzo surface. This case is directly finished with Proposition (4.3.8). Nevertheless, let us exemplary demonstrate this in detail. As remarked in Step 2, F destabilizes TX on at least one F F extremal ray R+d1 or R+d2, say for instance µd1 ( ) >µd1 (TX ). If rk( ) = 1, then by Proposition (4.3.8) (i), F is the relative tangent sheaf of the Mori fibration η. In this case, there might exist a second term G in the Harder-Narasimhan filtration of TX with respect to c. Then Proposition (4.3.8) (iii) yields that G is the pullback of the relative tangent sheaf of a Mori fibration on S. If rk(F ) = 2, then Proposition (4.3.8) (ii) yields that F is the pullback of a Mori fibration on S.

E⊥

Figure 4.5: Case 1 (ii). This ﬁgure shows a slice through the Mori cone. The hatched area is the Moving cone.

(ii) Suppose R+d1 corresponds to a Mori ﬁbration and R+d2 is not an extremal ray of the Mori cone, see Figure (4.5). In this case there exists an extremal face

73 The Harder-Narasimhan ﬁltration of the tangent bundle on Fano threefolds

Fmori = hd1,r2iR+ of the Mori cone, such that F ⊂ Fmori. The extremal ray R+r2 corresponds to a divisorial contraction, say φ : X → Y . Note that by Proposition (4.1.12) (i) Y is smooth and by (ii) we know that R+φ∗d2 is an extremal ray of the Mori cone of Y corresponding to a del Pezzo fibration. The exceptional divisor E of φ intersects d2 with zero, see also Figure (4.5). If F or G destabilize TX with respect to d1, then, as in Case 1 (i), Proposition (4.3.8) implies that F and G are relative tangent sheaves of not necessarily elementary Mori fibrations. Let us assume that F destabilizes TX with respect to d2. Then Lemma (4.4.5) implies that φ⋆F , see Notation (4.4.4) for the definition of φ⋆F , destabilizes TY with respect to φ∗d2. Now we can apply Proposition (4.3.11) to conclude that φ⋆F is the relative tangent sheaf of the del Pezzo fibration associated to R+φ∗d2. Consequently, F is the relative tangent sheaf of the composition of φ and the del Pezzo fibration. The same argumentation holds if G destabilizes TX with respect to d2.

Figure 4.6: This ﬁgure illustrates the situation in Case 1 (iii).

(iii) We assume that both R+d1 and R+d2 are not extremal rays of the Mori cone,

see Figure (4.6). Then there exists an extremal face Fmori := hr1,r2iR+ such that F ⊂ Fmori. The rays R+r1 and R+r2 correspond to extremal divisorial contractions, say φ1 : X → Y1 and φ2 : X → Y2 respectively. By Proposition (4.1.12) (i), Y1 and Y2 are smooth and by (ii) we know that R+φ1∗(d1) and R+φ2∗(d2) are extremal rays of the Mori cone of Y1 and Y2 respectively corresponding to del Pezzo fibration. Now if F destabilizes TX with respect to d1 then the same is true for φ1⋆(F ) ⊂ TY with respect to φ∗d1, see Lemma (4.4.5). Thus φ1⋆(F ) is the relative tangent sheaf of the del Pezzo fibration associated to R+φ1∗(d1) by Proposition (4.3.11). The same argumentation holds if F destabilizes TX with respect to d2. Thus we have shown that in this case F is the relative tangent sheaf of a non elementary del Pezzo fibration. Since F has rank 2, it is the only term in the Harder-Narasimhan filtration with respect to a class of the face F .

Case 2: F does not lie on the boundary of the Mori cone. In this case there exists a divisorial contraction ψ : X → Y whose exceptional divisor E cuts F out of the Mori cone, i.e F ⊂ E⊥, see Proposition (1.2.17). If ψ is of type E1a or E2, then Y is a Fano threefold. Since ρ(Y ) = 2 and since Y is Fano, we know by Proposition (4.4.17) that each term in the Harder-Narasimhan ﬁltration with respect to any movable class on Y is the relative tangent sheaf of a Mori ﬁbration. Since

74 Chapter 4 any class d ∈ F intersects the exceptional divisor of ψ with zero, the Harder-Narasimhan filtration of TX with respect to d is the pullback of the Harder-Narasimhan filtration of TY with respect to ψ∗d, see Proposition (4.2.9). Thus if ψ is of type E1a or E2 each term in the Harder-Narasimhan filtration with respect to classes in F is the relative tangent sheaf of a non elementary Mori fibration. Thus in the sequel, we can assume that ψ is of type E1b, E3, E4 or E5.

E⊥ d1 d2

Figure 4.7: This ﬁgure illustrates the situation in Case 2 (i).

(i) Suppose d1 and d2 lie in the interior of the Mori cone, see Figure (4.7). Then there exists another divisorial contraction φ : X → Z whose exceptional divisor D ⊥ ⊥ ⊥ cuts d1 out of E , i.e. R+d1 = E ∩ D . Since we assume that there exists only one extremal contraction of type E1b, E3,...,E5, see Step 1, the contraction φ is of type E1a or E2 meaning that Z is a Fano threefold. By Proposition (4.1.11) R+φ∗(d1) is an extremal ray of the Moving cone of Z lying in the interior of NE(Z). Thus by Remark (4.4.18) we know that unless Z =∼ P(O ⊕ O(2)), the tangent sheaf TZ is semistable with respect to φ∗d1. In case that TZ is semistable with respect to φ∗d1, Proposition (4.2.9) yields that TX is semistable with respect to d1. In this case neither F nor G can destabilize TX with respect to d1. ∼ Let us assume that Z = P(O ⊕ O(2)) and F destabilizes TX with respect to d1. After scaling, we can assume that d1 is the pullback of the class g as in example 5 (4.4.7). As noted in Remark (4.4.8) we have µg(TY ) = 3 and since d1 · D = 0, we 5 F F 5 have µd1 (TX )= 3 . Since destabilizes TX with respect to d1, we have µd1 ( ) > 3 . F 5 Using Lemma (4.4.5), we see that µg(φ⋆ ) > 3 . On the other hand, the relative tangent sheaf of the Mori ﬁbration on Z is maximal destabilizing with respect to this class. Since µg(TZ/P2 ) = 2, we obtain 5 2 ≥ µ (φ F ) > . g ⋆ 3

The inequalities imply µg(φ⋆F ) = 2 as the only possible value for µg(φ⋆F ). Since the maximal destabilizing sheaf is unique, we obtain φ⋆F = TZ/P2 and thus F is the pullback of the relative tangent sheaf of the Mori ﬁbration. The same argumentation holds if F destabilizes with respect to d2. A second term G in the Harder-Narasimhan ﬁltration with respect to a class c ∈ F is impossible. Indeed, G would destabilize TX with respect to d1 or d2 and we can argue as above. (ii) Suppose that d2 lies on the boundary of the Mori cone and d1 lies in the interior of the Mori cone, see Figure (4.8). We can assume that F and G destabilizes TX

75 The Harder-Narasimhan ﬁltration of the tangent bundle on Fano threefolds

d1 d2 E⊥

Figure 4.8: This ﬁgure illustrates the situation in Case 2 (ii).

with respect to d2, since the case that F destabilizes TX with respect to d1 is treated in Case 2 (i). Clearly, there exists another two dimensional face F ′ of the Moving cone containing ′ d2. If F lies on the boundary or of the Mori cone or R+d2 itself is an extremal ray of the Mori cone, we can argue exactly as in Case 1 (i)or(ii), in order to prove the Proposition. Thus suppose F ′ does not lie on the boundary of the Mori cone. Then there exists another divisorial contraction φ : X → Z with exceptional divisor D such that D · d2 = 0. Then Z is Fano by our assumptions. If the contraction ψ is of type E3, E4, E5, then R+φ∗d corresponds to a del Pezzo ﬁbration by Proposition (4.1.12) (iii). In this case ψ⋆F is the relative tangent sheaf of the del Pezzo ﬁbration by Proposition (4.3.11) and we are done.

If ψ is of type E1b, we can directly consider the face hψ∗d1, ψ∗d2iR+ of the Moving cone on Y . By Proposition (4.2.9) and since F is maximal destabilizing with respect F to c ∈ hd1,d2iR+ , we see that ψ⋆( ) is maximal destabilizing with respect to ψ∗c and it destabilizes TY with respect to ψ∗d2. Note that R+ψ∗d2 corresponds to a Mori fibration. If R+ψ∗d2 corresponds to a del Pezzo fibration, then we are done by Proposition (4.3.11). Otherwise R+ψ∗d2 correspond to a conic bundle ρ : X → S. We claim that S ∼= P2. This can be seen as follows. The contraction ρ ◦ ψ is same as the ′ composition of φ and the Mori fibration ρ : Z → S associated to R+φ∗d2. Now since Z is Fano of Picard number 2, we have S ∼= P2. Then ψ⋆(F ) is the relative tangent sheaf TY/P2 by Proposition (4.3.8)(i) and (ii) and a second term in the Harder-Narasimhan filtration is impossible, since it would be the relative tangent sheaf of a Mori fibration on P2 by Proposition (4.3.8) (iii).

4.4.5. Picard number ≥ 4

Remark 4.4.20. On a Fano threefold with Picard number greater than or equal to 4 which is not isomorphic to P1 × S, where S denotes a del Pezzo surface of degree i with 1 ≤ i ≤ 7, there exists no elementary Mori fibration. This is due to the facts that every Fano conic bundle over a del Pezzo surface of Picard number greater or equal 3 is trivial [MM86, Theorem 4.20] and that there exits no Mori fibration with two dimensional fibres, see Lemma (4.1.17).

76 Chapter 4

Remark 4.4.21. Let X ∼= P1 × S be a trivial Fano conic bundle over a del Pezzo surface of degree 1 ≤ i ≤ 7. Suppose on X exists a divisorial contraction with exceptional divisor D. Let ℓ be a fibre of the canonical projection π : X → S. We claim that D · ℓ = 0. Assume on the contrary that D intersects ℓ. Then we would obtain a finite surjection D → S. This in turn would yield an injection N 1(S) → N 1(D). This is impossible for reasons of the rank of the Néron-Severi group since ρ(S) ≥ 3 and ρ(D) ≤ 2. Thus D does not intersect a general fibre of π. Lemma 4.4.22. Let X ∼= P1 ×S be a trivial Fano conic bundle over a del Pezzo surface of degree i with 1 ≤ i ≤ 7. Then on X exists no divisorial contraction of type E1b, E2,...,E5. Proof. Let π : P1 ×S → S be the projection map. As we have seen in Remark (4.4.21), an exceptional divisor D does not intersect a fibre of π. Since all fibres of π are irreducible, ∗ D = π ℓ for a curve ℓ ∈ S which immediately excludes contractions of type E2,...,E5. If ∗ ∼ 1 1 D = π ℓ = P × P is of type E1b, we could contract the vertical and the horizontal fibres in D, which is impossible. Remark 4.4.23. If π : X ∼= P1 × S → S is a trivial Fano conic bundle with S a del Pezzo surface of degree i with 1 ≤ i ≤ 7, then X has exactly one extremal ray corresponding to a Mori fibration, namely the extremal ray spanned by a fibre of π. Example 4.4.24. Suppose X ∼= P1 × S where S is the del Pezzo surface of degree 7, i.e. S is isomorphic to P2 blown up in two points. Thus the Picard number of X is 4. We will prove in this Example that the only possible terms occurring in the Harder- Narasimhan filtration with respect to any movable class is the relative tangent sheaf of a not necessarily elementary Mori fibration. By Proposition (3.3.11), we only have to determine the subsheaves occurring in the Harder-Narasimhan filtration with respect to classes c on the two dimensional faces of Mov(X). Thus let F = hd1,d2iR+ be a two dimensional face of the Moving cone. We claim that there exists a divisorial contraction φ : X → Y with exceptional divisor D such that F ⊂ D⊥. Indeed, this follows from Remark (4.4.21) and Remark (4.4.23) and the fact that the Moving cone is cut out of the Mori cone by the exceptional divisors of extremal contractions, see Proposition (1.2.17). By Proposition (4.2.9) the Harder-Narasimhan filtration with respect to c ∈ F is the pullback of the Harder-Narasimhan filtration of TY with respect to φ∗c. By Lemma (4.4.22), φ is of type E1a and thus Y is a Fano manifold of Picard number 3. Proposition (4.4.19) yields that all possible terms in the Harder-Narasimhan filtration with respect to φ∗c are relative tangent sheaves of not necessarily elementary Mori fibration. In the proof of the next Proposition we will need two statements. Lemma 4.4.25. Let X be a Fano threefold of Picard number 4. Consider a two dimensional face hr1,r2iR+ of NE(X). Suppose φ1 : X → Y1 and φ2 : X → Y2 are divisorial contraction associated to R+r1 and R+r2 with exceptional divisors D1 and D2, respectively. Suppose furthermore that φ1 is of type E1b and φ2 is of type E1a. Finally, we assume that ⊥ ⊥ d is a movable class in hr1,r2iR+ , such that R+d ∈ D1 ∩D2 . Then R+φ1∗d is an extremal ray of NE(Y1) corresponding to a Mori fibration Y1 → S with S a del Pezzo surface.

77 The Harder-Narasimhan ﬁltration of the tangent bundle on Fano threefolds

Proof. Proposition (4.1.12) (ii) says that R+φi∗d spans an extremal ray of the Mori cone NE(Yi) for i =1, 2. Since d is movable, R+φi∗d corresponds to a Mori fibration for i =1, 2. Thus the only statement to prove is that φ1∗d corresponds to a Mori fibration over a del Pezzo surface. To this end consider the contraction ψ : Y2 → S associated to R+φ2∗d. This fibration has one dimensional fibres by Lemma (4.1.17), hence it corresponds to a fibration over a del Pezzo surface S. Now the composition of φ2 and ψ is the same as the composition of φ1 and the contraction associated to φ1∗d. Thus the Lemma is proved.

Lemma 4.4.26. Let X be Fano threefold of Picard number 4. Let Fmori be a three dimensional face of NE(X). Let Fmov = hd1,d2iR+ be a two dimensional face of the Moving cone of X contained in Fmori which is cut out by an exceptional divisor E of an extremal ⊥ contraction ϕ associated to an extremal ray of Fmori, i.e. Fmov ⊂ E . Suppose that d1 is contained in the relative interior of Fmori. Then

(i) there exists a divisorial contraction ψ1 : X → Y of an extremal ray in Fmori with exceptional divisor D1 such that D1 · d1 =0 and D1 =6 E.

(ii) Suppose Y as in (i) is a Fano threefold. Then there exists a divisorial contraction ψ2 : Y → Z with exceptional divisor D2 such that D2 · ψ1∗d1 =0 and ψ2∗ψ1∗d1 spans an extremal ray of NE(Z) corresponding to a del Pezzo ﬁbration.

Proof. Statement (i) follows directly from the description of the Moving cone given in Proposition (1.2.17), see also Remark (4.4.2) (iii). Let us prove (ii). Let ψ1 : X → Y be the divisorial contraction with exceptional divisor D1 such that D1 · d1 = 0. Note that the exceptional divisor D1 cuts out another two ′ ′ ⊥ dimensional face Fmov = hd1,d3iR+ out of the Mori cone, i.e. Fmov ⊂ D1 . We can easily see that hψ1∗d1, ψ1∗d3iR+ is an extremal face of the Moving cone lying on a two dimensional face of NE(Y ): Indeed, by Proposition (4.1.11) both ψ1∗d1 and ψ1∗d3 are extremal rays of the Moving cone. To see that ψ1∗d1 and ψ1∗d3 lie on a two dimensional face of NE(Y ), we take a nef divisor N which cuts out the face Fmori. Then ψ1∗N is nef by Lemma (4.1.10) and ψ1∗N · ψ1∗di = 0 for i =1, 3.

This shows that that hψ1∗d1, ψ1∗d3iR+ is an extremal face of the Moving cone contained in a two dimensional face of the Mori cone. Note that ψ1∗d1 itself is not an extremal ray of NE(Y ). This, together with the description of the Moving cone in Proposition (1.2.17), implies that there exists an extremal divisorial contraction ψ2 : Y → Z whose exceptional divisor D2 intersects ψ1∗d1 with zero. Finally, Proposition (4.1.12) (ii) veriﬁes that R+ψ2∗ψ1∗d1 is an extremal ray of the Mori cone of Z whose contraction is a del Pezzo ﬁbration.

Proposition 4.4.27. Let X be a Fano threefold with ρ(X) ≥ 4. Then the only possible terms occurring in the Harder-Narasimhan ﬁltration of the tangent bundle associated to a movable class are relative tangent sheaves of not necessarily elementary Mori ﬁbrations.

Proof. We suppose that ρ(X) = 4. We assume that X is not the trivial conic bundle since this is treated in Example (4.4.24). Thus all contractions associated to extremal rays of NE(X) are divisorial, see Remark (4.4.20). Furthermore, we assume that X admits at most one extremal contraction of type E1b, E3,...,E5 since the only Fano

78 Chapter 4 threefold of Picard number 4 admitting two extremal contractions of these types has been treated in Example (4.4.9), see also Proposition (4.1.16).

Step 1. Let F be a face of the Moving cone contained in a three dimensional face Fmori of the Mori cone. We claim the following: If an extremal contraction associated to an extremal ray in Fmori is of type E2,...,E5, then dim F < 3. Suppose on the contrary that dim F = 3 and there is an extremal ray of Fmori corresponding to a contraction of type E2,...,E5. Denote this contraction by ϕ : X → Y and its exceptional divisor by D. For each extremal ray R+r of F , we ﬁnd a covering family of irreducible curves whose numerical class lie on R+r, see [Ara10, Corollary 1.2]. Since dim F = 3, we ﬁnd a covering family of irreducible curves whose class lie in F , such that the general curve of this family intersects the exceptional divisor positively. The contraction associated to the face Fmori will therefore contract X to a point. This is impossible since the relative Picard number of the contraction of Fmori is 3 whereas the contraction to a point has relative Picard number 4.

Step 2. Let Fmori be a three dimensional face of the Mori cone. Let hr1,r2iR+ be a two dimensional subface of Fmori. We claim that Mov(X) ∩hr1,r2iR+ consists of at most one extremal ray of the Moving cone.

Suppose Mov(X) ∩hr1,r2iR+ is two dimensional. Note that by our assumptions R+r1 and R+r2 correspond to divisorial contractions with one contraction being of type E1a, say φ : X → Y . Let d be a movable curve in Mov(X) ∩hr1,r2iR+ . Then by Proposition (4.1.12) (ii), R+φ∗d correspond to a del Pezzo fibration. Since Y is Fano, this contradicts that each Mori fibration on a Fano threefold of Picard number greater than or equal to 3 has one dimensional fibres, see Lemma (4.1.17).

Step 3. By Proposition (3.3.11), we only have to determine the sheaves occurring in the Harder-Narasimhan ﬁltration of TX with respect to movable classes lying on two dimensional faces of the Moving cone in order to prove the Proposition. Thus let

Fmov = hd1,d2iR+ be a two dimensional face of the Moving cone.

(i) Suppose Fmov does not lie on the boundary of the Mori cone. Recall that by Proposition (1.2.17) the Moving cone is cut out of the Mori cone by the exceptional divisors of extremal divisorial contractions. Since Fmov is a two dimensional face of the Moving cone and it does not lie on the boundary of the Mori cone, there exist two extremal divisorial contractions ϕ1,ϕ2 with exceptional divisors D1 and ⊥ D2 such that Fmov ⊂ Di for i = 1, 2. Our assumptions imply that one of the extremal contractions ϕ1,ϕ2 is of type E1a, say ϕ1 : X → Y . Then Y is a Fano threefold of Picard number 3. By Proposition (4.2.9), the Harder-Narasimhan filtration with respect to a class c ∈ Fmov is the pullback of the Harder-Narasimhan filtration of TY with respect to ϕ1∗c. Since the only terms occurring in the Harder-Narasimhan filtration with respect to ϕ1∗c are relative tangent sheaves of not necessarily elementary Mori fibrations by Proposition (4.4.19), this is true for the Harder-Narasimhan filtration of TX with respect to c.

79 The Harder-Narasimhan ﬁltration of the tangent bundle on Fano threefolds

(ii) Suppose Fmov lies on the boundary of the Mori cone. Then there exists a three dimensional face Fmori of the Mori cone, such that Fmov ⊂ Fmori. By Step 2, Fmov is not contained in a two dimensional subface of Fmori. Thus there exists a divisorial ⊥ contraction ϕ : X → W with exceptional divisor E such that Fmov ⊂ E , see also Remark (4.4.2). a) Suppose there exists an extremal ray of type E2,...,E5 on Fmori. Then by Step 1, the face Fmov is at most two dimensional and it does not lie on a two dimensional face of the Mori cone by Step 2. Thus there exists another divisorial contraction ⊥ ψ : X → V with exceptional divisor D and such that Fmov ⊂ D . Moreover, by our assumptions V is a Fano threefold. Then we conclude as in (i) that the only terms occurring in the Harder-Narasimhan filtration with respect to a class on Fmov are relative tangent sheaves of a Mori fibration. b) Since we have treated a possible extremal contraction of type E2,...,E5 of Fmori in a), we can assume that ϕ : X → W is of type E1. If ϕ is of type E1a, we can argue as in (i). Thus we assume that ϕ is of type E1b. Let F be the maximal destabilizing subsheaf of TX with respect to c ∈ Fmov. Then by Remark (4.4.3), F destabilizes TX with respect to d1 or d2, say d1. By Step 2 there exists another extremal contraction ψ1 : X → M on a Fano threefold M with exceptional divisor ⊥ ⊥ D1 such that R+d1 = E ∩ D1 ∩ Fmori, see also Remark (4.4.2). We distinguish two cases: First, if d1 lies on the relative boundary of Fmori, then Lemma (4.4.25) implies that R+ϕ∗d1 is an extremal ray of the Mori cone of W corresponding to a Mori fibration η : W → S with one dimensional fibres over a del Pezzo surface S. Since ϕ⋆(F ) destabilizes TW with respect to ϕ∗d1, see Lemma (4.4.5), and moreover ϕ⋆(F ) is the maximal destabilizing subsheaf of TW with respect to ϕ∗c, see Proposition (4.2.9), we conclude that ϕ⋆(F ) is the relative tangent sheaf of a not necessarily elementary Mori fibration by Proposition (4.3.8) (i) and (ii). A second term G in the Harder-Narasimhan filtration with respect to ϕ∗c is then the pullback of the relative tangent sheaf of a Mori fibration on S by Proposition (4.3.8) (iii). Second, suppose that d1 lies in the relative interior of Fmori. Consider the extremal contraction ψ1 : X → M on the Fano threefold M with exceptional divisor D1 ⊥ ⊥ such that R+d1 = E ∩ D1 ∩ Fmori. Then by Lemma (4.4.26) (i) there exists a divisorial contraction ψ2 : M → Z whose exceptional divisor intersects ψ1∗d1 with zero. In addition, Lemma (4.4.26) (ii) ensures that ψ2∗ψ1∗(d1) is an extremal ray corresponding to a del Pezzo fibration. Lemma (4.4.5) yields that (ψ2 ◦ ψ1)⋆(F ) destabilizes TZ with respect to ψ2∗ψ1∗(d1). Now since R+ψ2∗ψ1∗(d1) corresponds to a del Pezzo fibration, we can apply Proposition (4.3.11). This shows that (ψ2 ◦ ψ1)⋆(F ) is the relative tangent sheaf of the del Pezzo fibration associated to R+ψ2∗ψ1∗(d1).

Summing up, we have proved the Proposition for Fano manifolds with ρ(X) < 5. Suppose ρ(X) ≥ 5 and suppose X is not a trivial conic bundle. We can assume that there exists at most one extremal contraction of type E1b, E3, E4, E5 on X, see Proposition (4.1.16). In fact, besides the trivial conic bundles, there are only two Fano threefolds with ρ(X) ≥ 5, namely one with Picard number 5 and one with Picard number 6, see [IP99, Appendix].

80 Chapter 4

Let Fmori be a three dimensional face of the Mori cone of X. Let Fmov be a face of the Moving cone inside Fmori. We claim that dim Fmov < 3. All contractions of extremal rays of Fmori are divisorial and by our assumptions we find an extremal ray in Fmori of type ′ E1a or E2. Thus let ρ : X → X be the contraction of an extremal ray in Fmori of type ′ ′ E1a or E2 with exceptional divisor D . Keep in mind that the Picard number of X is greater than or equal to 4. Suppose on the contrary that dim Fmov = 3. Then exactly as in Step 1, we can find a covering family of curves whose numerical class lie in Fmov and intersects D′ positively. Let us denote the class of a curve in the family by d. With the ′ help of Lemma (4.1.10) and Proposition (4.1.11) one proves that d := ρ∗d lies on a two dimensional face of the Moving cone of X′ which itself lies on a two dimensional face F ′ of the Mori cone of X′. Then the contraction η : X′ → T associated to the face F ′ has two dimensional fibres and thus T is a curve and consequently T ∼= P1. This is impossible, since the Picard number of T is 1 and the relative Picard number of η is 2. This shows that we need at least two exceptional divisors to cut a two dimensional face of the Moving cone out of the Mori cone. One of these exceptional divisors correspond to a contraction onto a Fano threefold by our assumptions. Now we can argue inductively by Proposition (4.2.9). Finally, if X is a trivial conic bundle, we can easily prove the Proposition since by Lemma (4.4.22) all divisorial contractions are of type E1a and by Remark (4.4.23) there is only one Mori fibration on X.

Now we have essentially proved Theorem (4.1).

Proof of Theorem (4.1). We have explicitly determined all terms in the Harder- Narasimhan ﬁltration with respect to any movable curve in Proposition (4.4.17), (4.4.19) and (4.4.27). The remaining assertions follow directly from Theorem (3.3.4) and Proposition (3.3.5).

1 Example 4.4.28. Let us consider the Fano threefold X = P × F1. Let us consider the following diagram π1 π2 P1 o P1 × P1 / P1 (4.3) O ϕ

p2 X / F1

p1 P1 P2

Let F be the divisor class of a fibre of p1 and E be the class of the pullback of the exceptional divisor via p2. Furthermore let H be the pullback of the ruling structure of F1. Then the Néron-Severi space is generated by the divisors F , E and H. ∼ Denote by l ⊂ X the class of a line of the ruling in F = F1 and let e be the class of the exceptional divisor in F . Furthermore we write h for the class of a fibre of p2. Then the space of cycles modulo numerical equivalence is generated by these cycles that is N1,R(X)= hh, l, ei. The intersection product is given in the following table.

81 The Harder-Narasimhan ﬁltration of the tangent bundle on Fano threefolds

H E F h 0 0 1 e 1 −1 0 l 0 1 0 Note that the Mori cone is given by

NE(X)= hl, h, ei.

By Proposition (1.2.17), we can easily compute the Moving cone. It is given by

Mov(X)= hl, h, l + ei.

The cone is sketched in Figure 4.9.

l + e E⊥

l h

Figure 4.9: A slice through Mori cone of X. The hatched area corresponds to the cone of moving curves of X.

With the notations introduced in diagram (4.3), we have the following Mori ﬁbrations:

The contraction of the extremal ray l is given by ϕ. We denote the relative tangent sheaf of ϕ by Fl.

The contraction of the extremal ray h is given by p2. We denote the relative tangent sheaf of p2 by Fh.

The contraction of the extremal face hl, hiR+ is given by π2 ◦ ϕ. The class of a ﬁbre of this contraction is H. We denote the relative tangent sheaf by Fh+l.

The contraction of the extrema face he, liR+ is given by π1 ◦ ϕ. The ﬁbre of this contraction is given by F . We denote the relative tangent sheaf by Fe+l.

Note that there is another contraction of an extremal Face, namely the contraction of he, hi. The foliation induced by the relative tangent sheaf is the same as the sheaf given by Fh.

82 Chapter 4

e + l

0 ⊂ Fe+l ⊂ TX

0 ⊂ Fl ⊂ Fe+l ⊂ TX 0 ⊂ Fh ⊂ TX

F F 0 ⊂ Fl ⊂ Fh+l ⊂ TX 0 ⊂ h ⊂ h+l ⊂ TX

l h

Figure 4.10: The destabilizing chambers of maximal dimension of X

A computation of the ﬁrst Chern classes of the relative tangent sheaves of the Mori ﬁbration and the canonical divisor gives:

c1(TX ) =3H +2F +2E c1(Fl) = H +2E c1(Fh) =2F c1(Fl+h) = H +2F +2E c1(Fl+e) = 3H +2E

Now one can compute the chamber structure of X. We omit the computation here. The result is shown in Figures (4.4.28) and (4.4.28). Note that the semistable chamber consists 2 of one ray. This ray is given by the cycle (−KX ) that is X is exactly semistable with respect to −KX .

e + l

semistable chamber

0 ⊂ Fl ⊂ TX

0 ⊂ Fl+h ⊂ TX

l h

Figure 4.11: The lower dimensional destabilizing chambers and the semistable chamber

83 The Harder-Narasimhan ﬁltration of the tangent bundle on Fano threefolds

4.5. Another method to compute the Harder-Narasimhan ﬁltra- tion of TX

In this section we demonstrate another way, to determine the Harder-Narasimhan ﬁltration of the tangent bundle. The key observation, which was already mentioned in Chapter 2, is that the terms in the Harder-Narasimhan ﬁltration are invariant under the automorphisms contained in the connected component of the identity. After proving the main technical result in the following Proposition, we will illustrate, how it can by applied in the study of special examples.

Let X be projective manifold and F ⊂ TX . Let σ be an element from the group of automorphisms on X. By means of the diﬀerential of σ, we interpret σ∗F as a subsheaf of TX .

Proposition 4.5.1. Let X be projective manifold and σ ∈ Aut0(X). Let F be a term ∗ of the Harder-Narasimhan ﬁltration of TX . Then σ (F ) = F . In particular, if F is a foliation, then σ maps each leaf of F to another leaf of F .

Proof. Let 0 ⊂ F1 ⊂ . . . ⊂ Fk = TX be the Harder-Narasimhan ﬁltration of TX with respect to a curve C. Then exactly as in ∗ the proof of Proposition (2.1.9), we see that µC (σ (F )) = µC (F ) and therefore

∗ ∗ 0 ⊂ σ (F1) ⊂ . . . ⊂ σ (Fk)= TX satisfies the defining properties of the Harder-Narasimhan filtration. By the uniqueness, ∗ we conclude σ (Fk)= Fk.

Note that the universal property of the blow up implies the following remark.

Remark 4.5.2. Let X˜ → X be the blow up of a projective manifold with center C. Then each σ ∈ Aut(X) with σ(C)= C lifts to an automorphism on X˜.

We demonstrate how to compute the destabilizing chambers on a Fano threefold.

Example 4.5.3 (The blow up of P3 in a line). Let X be the blow up of P3 in a line ℓ. Let π be the projection map π : X → P3. Recall that the N´eron-Severi group of X is generated by the pullback of a hyperplane class H in P3 via π and the exceptional divisor E. Let π∗l be the pullback of a line in P3 and let f ⊂ E be a ﬁber over ℓ. The intersection product is given by

π∗H E π∗l 1 0 f 0 −1 Furthermore π∗H3 =1, π∗H2.E =0, π∗H.E2 = −1, E3 = −2.

84 Chapter 4

The Mori cone is given by ∗ NE(X)= hπ l − f, fiR+ . ∗ Denote the extremal contraction associated to R+(π l − f) by ϕ. One easily sees that ϕ is a del Pezzo fibration. More precisely, the fibres of ϕ : X → P1 are the proper transforms of planes in P2 containing the blow up locus. An easy computation shows that a divisor class aπ∗H + bE is ample iff a + b > 0 and b < 0. Another computation yields that the relative tangent bundle TX/P1 of ϕ destabilizes TX with respect to the complete intersection (aπ∗(H)+ bE)2 if and only if −2a2 − 10ab − 12b2 > 0. To show that this is the only term in the Harder-Narasimhan filtration we use Proposition (4.5.1).

Let p ∈ X and p∈ / E. Let us determine automorphisms on X which stabilize p. To this end, we consider automorphisms σ of P3 with σ(ℓ) = ℓ and σ(p) = p. After a projective transformation, we might assume that ℓ is the line given by { (0:0: a : b) ∈ P3 a, b ∈ C } and p =(0:1:0:0). Then all these automorphisms can be represented by elements in P(Gl4(C)) of the form

∗ 0 0 0 ∗ ∗ 0 0  ∗ 0 ∗ ∗   ∗ 0 ∗ ∗      Now it is easy to verify that each curve through p can be moved by automorphisms of X leaving p fixed. Thus by Proposition (2.1.9) there is no maximal destabilizing subsheaf of rank one. Furthermore, each two dimensional subvariety through p except the proper transform of a plane containing ℓ can be moved. Thus by Proposition (4.5.1), we conclude that the only possible term in the Harder-Narasimhan filtration of TX is the relative tangent sheaf of the Mori fibration.

Chapter 5 Prospects and open questions

5.1. A generalization of Miyaoka’s uniruledness criterion

Deﬁnition 5.1.1. Let E be a vector bundle on a n-dimensional manifold X. Then E is called generically nef if for all ample line bundles H1,...,Hn−1 on X, for all mi suﬃciently large and for all general curves C ∈|m1H1 ∩ . . . ∩ mn−1Hn−1|, the restricted bundle E C is nef.

With this vocabulary, Miyaoka’s criterion [Miy87, Theorem 8.5] reads as follows. Theorem 5.1.2. The cotangent bundle of a projective manifold is generically nef unless X is uniruled. Note that this means that if for a projective manifold X there exists a complete 1 intersection curve C such that ΩX C is not nef, then X is uniruled. By a Theorem of Hartshorne the statement Ω1 is not nef is equivalent to the existence of a quotient Q X C C of Ω1 such that c (Q ) · C < 0. It is natural to ask if one can generalize Miyaoka’s X C 1 C criterion of uniruledness. Since uniruledness is a birational property one might hope that one can replace the complete intersection curve which is involved in Miyaoka’s criterion of uniruledness by a movable curve. This is recently done in [CP07]. They prove the following. Theorem 5.1.3 ([CP07, Theorem 1.4]). Let X be a projective manifold and C ∈ Mov(X) ′ a movable curve of the form C = π∗(H1 ∩ . . . ∩ Hn−1) with π : X → X a modification of ′ X and Hj very ample on X for all j =1,...,n − 1. If there exists a torsion free quotient sheaf 1 ΩX → Q → 0 such that c1(Q) · C < 0, then X is uniruled. On the other hand, Theorem (1.1.12) by Kebekus-SoláConde-Toma is also a generalization of Miyaoka’s uniruledness criterion. More specifically Theorem (1.1.12) says that if 1 there exists a complete intersection curve C such that ΩX C is not nef, then there exists a subsheaf F ⊂ T which induces a foliation with algebraic and rationally connected leaves, X see also [KSC06, Part 1, Section 2] for a detailed discussion and an explicit formulation of this result. In other words, we not only know that X is uniruled, but also know that the covering family of rational curves are tangent to F . In this sense Theorem (1.1.12) gives an effective version of Miyaoka’s uniruledness criterion. It remains open if there is an effective version of a uniruledness criterion using moving curves instead of complete intersection curves. Prospects and open questions

5.2. Maximal rationally connected foliations on higher dimensional manifolds

In Chapter 2 we have proven that on a uniruled surface there always exists a polarisation such that the Harder-Narasimhan filtration of the tangent bundle with respect to the polarisation yields the maximal rationally connected quotient. Of course, it is natural to study the question for higher dimensional uniruled manifolds. Let X be a uniruled manifold of any dimension. We can ask whether there exists a complete intersection curve on X such that the Harder-Narasimhan filtration with respect to this curve yields the maximal rationally connected quotient of X. Unfortunately, this question is not easy to answer. But if one asks for the existence of a movable curve such that the foliation associated to the fibres of the maximal rationally connected fibration appears in the Harder-Narasimhan filtration, then the answer is positive. This has been shown in [SCT08].

Let 0= F0 ⊂ F1 ⊂ . . . ⊂ Fk = TX be the Harder-Narasimhan ﬁltration with respect to a movable class c. Let i ∈{ 1,...,k }. Recall that we call Fi a positive term in the Harder-Narasimhan ﬁltration with respect to c if the slope of Fi/Fi−1 is positive. Note that if µc(F1) > 0, then there exists a maximal number

s := max { i 1 ≤ i ≤ k,µc(Fi/Fi−1) > 0) } .

Then Fs is called the highest positive term in the Harder-Narasimhan ﬁltration of TX with respect to c.

Theorem 5.2.1 ([SCT08, Theorem 1.1]). Let X be a projective uniruled manifold. Then there exists a movable class C such that the relative tangent sheaf of the maximal rationally connected quotient appears as the highest positive term in the Harder-Narasimhan ﬁltration with respect to C.

Sketch of the proof. Let C be a movable curve and denote the highest positive term in the Harder-Narasimhan ﬁltration of TX with respect to C by F . We claim that if G ⊂ TX G G F is a subsheaf such that C is locally free and ample, then ⊂ . Indeed, the exact sequence

0 → F ∩ G → F ⊕ G → F + G → 0, G F F G G F G G G F yields c1( + / ) = c1( / ∩ ). Since C is ample, c1( / ∩ ) · C > 0 unless G ⊂ F . Thus if G 6⊂ F , we have µ (G + F /F ) > 0. Since F is the highest positive C term in the Harder-Narasimhan ﬁltration, we conclude G ⊂ F .

Let Q be the foliation associated to the relative tangent sheaf of the maximal rationally connected quotient ϕ : X → Y . Then we take a very free rational curve ℓ in a general Q Q smooth ﬁbre F of ϕ, i.e. TF ℓ is ample. Then it follows that ℓ is ample and thus is contained in the highest positive term of the Harder-Narasimhan ﬁltration of T with X

88 Chapter 5 respect to ℓ.

Finally, it remains to prove that Q is indeed the highest positive term in the Harder- Narasimhan ﬁltration. This is shown in [SCT08, Lemma 2.9.] and we don’t recall the proof here.

Remark 5.2.2. In the proof of the Theorem above given in [SCT08], the authors consider the Harder-Narasimhan filtration with respect to curves of the form C + C′, where C is a complete intersection and C′ is a sum of many very free rational curves in general fibres of the maximal rationally connected fibration. This allows them to show that there exists a curve in the interior of the Moving cone with the properties stated in the Theorem above.

Compared to [SCT08], we have considered the Harder-Narasimhan ﬁltration with respect to a curve ℓ which possibly lie on the boundary of the Moving cone. Nevertheless one can show that the above Theorem still holds if we perturb ℓ slightly.

Corollary 5.2.3. Let X be a uniruled projective manifold. Then there exists a destabilizing chamber of maximal dimension such that the highest positive term in the Harder- Narasimhan ﬁltration with respect to the chamber is given by the foliation associated to the maximal rationally connected quotient.

Proof. Let ∆ℓ be the destabilizing chamber with respect to ℓ, where ℓ is a very free rational curve in a general fibre F of the maximal rationally connected quotient, i.e. TF ℓ is ample. Denote the highest positive term in the Harder-Narasimhan filtration with respect to ∆ ℓ by F . By the Theorem above, F is the foliation associated to the maximal rationally connected quotient. If dim ∆ℓ is maximal, there is nothing to prove. Otherwise there exists a neighboring destabilizing chamber Θ of ∆ℓ by Lemma (3.3.3). Denote the highest positive term of the Harder-Narasimhan filtration with respect to Θ by G . We have to prove that G = F . Then again by using the exact sequence

0 → F ∩ G → F ⊕ G → F + G → 0, we see that c1(F + G /G ) = c1(F /F ∩ G ). Using that F |ℓ is ample, we see that c1(F /F ∩ G ) · ℓ > 0 unless F ⊂ G . So if F 6⊂ G then c1(F + G /G ) · ℓ > 0. Since Θ is a neighboring chamber, we see that this inequality also holds for slight perturbations of ℓ which lie in Θ. This contradicts that G is the highest positive term of the Harder- Narasimhan ﬁltration with respect to Θ. Thus F ⊂ G . Since G contains the relative tangent sheaf of the maximal rationally connected quotient, we can apply [SCT08, Lemma 2.9] to see that G = F . The Corollary is proved.

Suppose ℓ is a very free rational curve in a general fibre of the maximal rationally connected quotient or suppose ℓ is an element in the destabilizing chamber Θ as in the proof of the Corollary above. We have seen that with respect to ℓ, the highest positive term in the Harder-Narasimhan filtration is given by the relative tangent sheaf of the maximal rationally connected quotient. One might conjecture that one can approximate the class ℓ in the Néron-Severi vector space of 1-cycles by complete intersection curves, but this is not obvious.

89 Prospects and open questions

5.3. The Harder-Narasimhan ﬁltration of TX on higher dimensional Fano manifolds

Of course, one might conjecture that the only maximal destabilizing subsheaf or even any sheaf occurring in the Harder-Narasimhan filtration of the tangent bundle of a Fano manifold with respect to any movable curve is the relative tangent sheaf of a Mori fibration, but this seems very hard to prove in general. Note that if one is able to give a conceptual proof of this conjecture, it would in particular imply that the tangent bundle of Fano manifolds with Picard number 1 is semistable. This is a challenging problem on its own and some effort has been made to prove it. Nevertheless the semistability of the tangent bundle is only known for low dimensional Fano manifolds of Picard number 1, see [Hwa98] and [PW95].

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