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

Transcription

1 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

2 Dekan: Prof. Dr. Kay Königsmann 1. Gutachter: Prof. Dr. Stefan Kebekus 2. Gutachter: Prof. Dr. Thomas Peternell Datum der Promotion:

3 Contents Introduction 1 1 Preliminaries Rationally connected foliations Intersection numbers, cones and extremal contraction Minimal rational curves Rationally connected foliations on surfaces Rationally Connected Foliations on Surfaces and the MRC-quotient The slope with respect to movable curves The Harder-Narasimhan filtration Complete intersection curves versus movable curves Destabilizing chambers on projective manifolds The Harder-Narasimhan filtration of the tangent bundle on Fano threefolds Preliminary Results for Fano threefolds Pullback of Foliations Minimal rational curves and Mori fibrations The Harder-Narasimhan filtration on Fano threefolds Preliminary remarks Examples of Harder-Narasimhan filtration of special Fano threefolds Picard number Picard number Picard number Another method to compute the Harder-Narasimhan filtration of T X Prospects and open questions A generalization of Miyaoka s uniruledness criterion Maximal rationally connected foliations on higher dimensional manifolds The Harder-Narasimhan filtration of T X on higher dimensional Fano manifolds

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5 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 H 1,...,H n 1 such that the class of C equals the intersections H 1... H n 1. Then there exists a unique filtration 0 = F 0 F 1... F k = T X 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 T X with respect to C. Kebekus-Solá Conde-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].

6 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 T X 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

7 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 perspective. 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 D 1 and D 2 are numerically equivalent, i.e. we have D 1 C = D 2 C for all curves C X, then we write D 1 D 2. We will use an analogous notation for 1-cycles. N 1,R (X) denotes the space of 1-cycles on X with coefficients in R modulo numerically equivalence. NR 1 (X) denotes the space of divisors on X with coefficients in R modulo numerically equivalence. N 1 R (X), N 1,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). We view N 1 R (X) and N 1,R(X) with the standard euclidean topology. For a divisor D on X, we define D := {c N 1,R (X) D c = 0 }. For a subset S R d, the linear span of S is denoted by S. Moreover, we set { n } S conv := a i s i ai = 1, s i S i=1 3

8 and { n S R+ := a i s i i=1 a i R +, s i S }. A subset S R d 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 R d is the dimension of C. If we have a cone C = R + d i R d, then the relative interior of C is the interior of C in the vector space Rd i. For example, if C is finitely generated, say C := R + e 1 + R + e 2 R 3, with e 1,e 2 linearly independent, then the relative interior is given by { λ 1 e 1 + λe 2 λ 1, λ 2 > 0 }. Analogously we define the relative boundary of C. The tangent sheaf of a projective manifold X is denoted by T X. When we refer to T X as the tangent bundle, then it will always be mentioned explicitly. We write Ω 1 X for the sheaf of 1-forms on X. If X Y is a proper morphism then the relative tangent sheaf is denoted by T X/Y. The class of a canonical divisor of a manifold X is denoted by K X. If X Y is a proper morphism then we denote the relative canonical divisor by K X/Y. 4

9 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 Köln 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. 5

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11 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 Rationally connected foliations Let X be a projective manifold and let F E be coherent sheaves of O X -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 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 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 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 Let F T X be a foliation. Then for an open set U X where F is a subbundle of T X we can apply the classical Frobenius theorem [War83, Theorem 1.60]. It follows, that through any point in U passes a unique leave. Definition Let F T X 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.

12 Preliminaries Proposition Let F T X be a foliation and denote by S the singular locus of T X /F. Then the following conditions are equivalent: (i) T X /F is torsion free. (ii) For any open set U X and any v T X (U) such that v U\S F(U\S) it follows that v F(U). Proof. Let us prove the direction (ii) (i) first. Since T X /F is locally free outside S, we only have to prove that T X /F is torsion free in p S. Let U be an open set around p and v T X (U). Suppose there exists a nonzero element f O X (U) such that fv F(U). We have to show that v F(U). Because T X /F is locally free outside S, we have v U\S F(U\S). By our assumptions we can extend vector fields in F, thus it follows v F(U). To prove the other direction, we take a vector field v T X (U) such that v F(U\S). After possibly diminishing U, we find a non zero element f O X (U) vanishing on S. Then, since F is coherent, f n v F(U) for some n 1. As T X /F is torsion free, we have v F(U). Proposition 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. (i) If there is a Zariski open set U X such that F U = G U then F = G. (ii) Suppose F is saturated in T X, then integrability is an open property, i.e. F T X is involutive on a Zariski open set if and only if it is involutive on X. (iii) Suppose we have a foliation F U T U, which is defined only on a Zariski open set U X. Then F U 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 is zero on U. But since T X /F is torsion free, a morphism α Hom( 2 F, T X /F) is zero if it is zero on a Zariski open set. This proves (ii). To prove (iii), note that we only have to find a saturated extension of F U in T X, 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 F U as a coherent sheaf [Har77, Ex ]. Remark Let ϕ : X Y be a fibration. By the foliation associated to the fibration ϕ, we mean simply the relative tangent sheaf T X/Y of ϕ. 8

13 Chapter 1 Definition 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 µ H (F) := c 1(F) H n 1. 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 µ H (G ) µ H (F). If there exists a nonzero subsheaf G F such that µ H (G ) > µ H (F), we will call G a destabilizing subsheaf of F. Theorem ([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 = F 0 F 1... F k = F of F depending on H, the Harder-Narasimhan filtration, with the following properties: (i) The quotients G i := F i /F i 1 are torsion free and semistable. (ii) The slopes of the quotients satisfy µ H (G 1 ) >... > µ H (G k ). Definition Let F be a coherent torsion free sheaf on a smooth projective variety with Harder-Narasimhan filtration 0 = F 0... F k = F with respect to an ample line bundle H. If the slope of the quotient F i /F i 1 is positive with respect to H, then F i is called positive with respect to H. Note that if µ H (F 1 ) > 0, then there exists a maximal number s := max {i 1 i k, µ H (F i /F i 1 ) > 0) }. Then F s is called the highest positive term in the Harder-Narasimhan filtration of T X 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 ([KSCT07, Theorem 1]). Let X be a smooth projective variety and let 0 = F 0 F 1... F k = T X be the Harder-Narasimhan filtration of the tangent bundle with respect to a polarisation H. Write µ i := µ H (F i /F i 1 ) for the slopes of the quotients. Assume µ 1 > 0 and set m := max {i N µ i > 0}. Then each F i 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

14 Preliminaries Let X be a smooth projective variety and assume the conditions of Theorem (1.1.12) are fulfilled. Thus we obtain foliations F 1,...,F m with algebraic and rationally connected leaves. By setting q i : X Im(q i ) Chow(X) x F i -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 Kollár-Miyaoka-Mori [KMM92], see also [Kol96, Chapter IV, Theorem 5.2]. Theorem ([KMM92, Theorem 2.7.]). Let X be a smooth projective variety. There exists a variety Z and a rational map φ : X 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 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. Notation We write N 1 (X) for the Néron-Severi group and N 1 Q (X) = N1 (X) Z Q (resp. N 1 R (X) = N1 (X) Z R) for the finite dimensional vector space of Q-divisors (resp. R-divisors) modulo numerical equivalence on X. We write N 1 (X) for the space of 1- cycles modulo numerical equivalence and we write N 1,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. Definition The dimension of NR 1 (X) is called the Picard number of X. We denote the Picard number of X by ρ(x). 10

15 Chapter 1 We have a perfect pairing coming from the intersection product. NR 1(X) N 1,R(X) R (D, C) 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 Y with codim Y (Y \im(φ) 2). Then extending the usual pullback and pushforward on the Néron-Severi group linearly to numerical classes of R- divisors yield an injective map and a surjective map φ : N 1 R (Y ) N1 R (X), φ : NR 1 (X) N1 R (Y ). These maps have the property that φ φ = id N 1 R (Y ). Definition The numerical pullback of curves φ 1 : N 1,R (Y ) N 1,R (X) is defined as the dual linear map of the pushforward φ : NR 1(X) N1 R (Y ). Analogously, we define the numerical pushforward φ 1 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. Remark (Projection formula). If C N 1,R (Y ) and D NR 1 (X), then we have φ C D = C φ D. If C N 1,R (X) and D NR 1(Y ), then we have C φ D = φ C D. Definition (Cone of curves, Mori cone). The cone of curves NE(X) N 1,R (X) is the convex cone generated by all effective 1-cycles on X. Its closure is called the Mori cone of X. NE(X) N 1,R (X) Notation (Ample cone, pseudoeffective cone). We denote the cone of all ample R-divisors by Amp R (X) N 1 R (X). The cone of pseudoeffective divisors is denoted by Eff(X). 11

16 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 Let C be a closed convex cone in R n. 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 Let D be a divisor on X and let C N 1,R (X) be a subset. Then we set D 0 := {C N 1,R (X) D C 0 } D := {C N 1,R (X) D C = 0 } C D 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 ([KM98, Theorem 3.7]). Let X be a Q-factorial projective variety with only terminal singularities. Then: (i) There are countably many rational curves C i X such that NE(X) = NE(X) K 0 X + R + [C i ]. (ii) The [C i ] are locally discrete classes of rational curves in the half space NE(X) K <0 X. (iii) Let F NE(X) be a K X -negative extremal face. Then there is a unique morphism ϕ F : X Z to a projective variety such that ϕ F O X = O Z and an irreducible curve C X is mapped to a point by ϕ F 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 K X -negative side of K X there exists an extremal contraction, contracting exactly those curves on the variety, whose numerical classes lie on the extremal ray (or face). Definition Let F be a K X -negative face of the Mori cone. Then the unique map ϕ F as in Theorem (1.2.9) (iii) is called the extremal contraction of F. Fact ([KM98, Proposition 2.5]). Suppose ϕ : X Y is the extremal contraction of an K X -negative extremal ray. Then ϕ is one of the following. The contraction ϕ is birational and codim X (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 dimy < dim X. In this case we call the contraction a Mori fibration. 12

17 Chapter 1 Furthermore, if X is a smooth threefold, then there exists no small contractions. Definition 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 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 A del Pezzo surface is isomorphic to P 2, P 1 P 1 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 N 1,R (X), which will be important for our investigations, the Moving cone of X. Definition A class C N 1,R (X) is called movable if it can represented by an irreducible curve [C] = [C t0 ] such that C t0 is a member of an algebraic family (C t ) t T, which covers X. We write Mov(X) for the closure of the cone generated by all movable classes in N 1,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 ([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 N 1,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 ([Bar07, Theorem 1.1]). Let X be a Fano 3-fold with exceptional divisors E 1,...E k. Then the Moving cone of X is given by { C NE(X) C Ei 0, i = 1,...k }. 13

18 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 : P 1 X be a rational curve. By a well known Theorem of Grothendieck, every locally free sheaf on P 1 splits into a direct sum of line bundles. Thus we may write f T X = O(a1 ) O(a n ). Definition A rational curve f is said to be free if f T X is generated by its global sections and it is said to be very free if f T X O( 1) is generated by its global sections Definition A free rational curve f is called minimal, if f T X = O(2) O(1) d O n d 1. 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. Remark If f is a free rational curve, then the image cycle f P 1 is movable. Let f be a free rational curve. Considering the differential of f and taking into account that T P 1 = O(2) yield that a i 2 for some i = 1,...,n. This shows the following Remark. Remark Let f : P 1 X be a free rational curve. Then K X f P 1 2. If f is moreover minimal and f an embedding, then we can say even more. Remark Let f : P 1 X be a minimal rational curve and f be an isomorphism. The differential df : T P 1 f T X. 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 T P 1 and the unique O(2)-term in f T X. An important property of a free rational curve is that a general deformation of this curve avoids subsets of codimension 2. Proposition ([Kol96, Proposition 3.7.]). Let X be a smooth projective variety. Let f : P 1 X be a free rational curve and let Z X be a subvariety of codimension at least 2. Then g(p 1 ) Z = for a general deformation g of f. 14

19 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 T X 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 Rationally Connected Foliations on Surfaces and the MRCquotient Convention 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 Let X be a smooth projective surface. If F T X is a destabilizing subsheaf with respect to a polarisation such that T X /F is torsion-free, then the Harder- Narasimhan filtration is given by 0 F T X. Proof. Let H be a polarisation and F a destabilizing subsheaf of T X with respect to H. Consider the exact sequence 0 F T X T X /F 0.

20 Rationally connected foliations on surfaces Using that the rank and the first Chern class are additive in short exact sequences, we obtain µ H (T X ) = 1 2 µ H(T X /F) µ H(F). Since µ H (F) > µ H (T X ), we therefore have µ H (F) > µ H (T X /F). Since F and T X /F are torsion free and of rank 1, they are semistable. Thus 0 F T X 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 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 filtration. Let H NR 1 (X) be an ample bundle. To abbreviate notation, let us write HNF H (T X ) for the Harder-Narasimhan filtration of T X with respect to H. Now we consider the ample line bundles which induce the same filtration. Definition Let H be an ample R-divisor. Then we call H := {H Amp R (X) HNF H (T X ) = HNF H (T X )}. 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. Lemma Let H NR 1(X) be an ample bundle such that T X is not semistable with respect to H. Let F be the maximal destabilizing subsheaf of T X with respect to H. Then the destabilizing chamber is given by { H = H Amp R (X) ( c 1 (F) 1 2 c 1(T X ) ) } H > 0. Proof. By Lemma (2.1.2) the condition (c 1 (F) 1 2 c 1(T X )) H > 0 ensures that for all polarisations in H we get the same Harder-Narasimhan filtration, namely 0 F T X. Concerning the structure of these chambers we prove the following Proposition. Proposition Let X be a smooth projective surface. We have: 16

21 Chapter 2 (i) The destabilizing chambers are convex cones in Amp R (X). (ii) The semistable chamber is closed in Amp R (X). (iii) A destabilizing chamber different from the semistable chamber is open in Amp R (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 Amp R (X) such that T X is semistable with respect to both H and H. Let F T X be a coherent subsheaf of rank 1. Then µ H+H (F) = µ H (F) + µ H (F) µ H (T X ) + µ H (T X ) = µ H+H (T X ), which shows that T X 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 T X the condition ( c1 (F) 1 2 c 1(T X ) ) H > 0 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 T X 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 Let X be a smooth projective surface. Let l be a line segment in Amp R (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 Let X be a projective manifold. Then we write Aut(X) for the group of automorphisms on X and Aut 0 (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 T X. By means of the differential of σ, we can identify T X and σ T X. Thus we can interpret σ (F) as a subsheaf of T X. For instance, if p X and F := T X I p, then σ (F) is identified with T X I σ 1 (p) T X. Proposition Let X be a smooth projective surface and let σ Aut 0 (X). Let F be the maximal destabilizing subsheaf of T X 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

22 Rationally connected foliations on surfaces y 0 T 1 X/P 1 T X 0 T 2 X/P 1 T X x Figure 2.1: The ample cone of X = Σ 0 and the chamber structure. Here T 1 X/P 1 and T 2 X/P 1 denote the relative tangent bundle of the first and second projection. Proof. Let H Amp R (X) and let F be the maximal destabilizing subsheaf of T X with respect to H. We compute the slope of σ (F) T X : µ H ( σ (F) ) = H (c 1 (σ (F)) ) = H σ (c 1 (F)) = H c 1 (F) > 1 2 c 1(T X ) H. We give an explanation of the third equality. Recall that the group of automorphisms acts on the Néron-Severi group. Since N 1 (X) is discrete and Aut 0 (X) is the connected component of the identity, Aut 0 (X) acts trivially on N 1 (X), i.e. σ (c 1 (F)) = c 1 (F). We have shown that σ (F) is a destabilizing subsheaf of T X. By Lemma (2.1.2) and the uniqueness of the maximal destabilizing subsheaf of T X, we conclude that σ F = F. Example (Hirzebruch Surfaces). Let Σ n be the n-th Hirzebruch surface and let π : Σ n P 1 be the projection onto the projective line. We denote the fiber under the projection by f and the distinguished section with selfintersection n by C 0. Recall that N 1 R (Σ n) = C 0, f and a divisor D ac 0 + bf is ample if and only if a > 0 and b > an, see see [Har77, chapter V.2]. The canonical bundle is given by K Σn = 2C 0 + (2 + n)f. The relative tangent bundle of π is a natural candidate for a destabilizing subbundle. We have the sequence 0 T Σn/P 1 T Σ n π TP 1 0 Let H := xc 0 + yf be a polarisation. Then one can compute that T Σn/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 T X/P 1 T X for all polarisations. In other words we obtain only one destabilizing chamber. For n = 0 we have Σ 0 = P 1 P 1 and we get three chambers. The two destabilizing chambers correspond to the two relative tangent bundles of the projections. They are cut 18

23 Chapter 2 out by the inequalities x > y and x < y. There is a chamber of semistability, which is determined by the equation x = y. For n = 1 we see that for x > 3 y the relative tangent bundle is destabilizing. Since 2 Σ 1 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 T Σ1, 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 C 0 fixed. Thus the foliation induced by F would have singularities on a dense open subset of Σ 1 which is absurd. So the tangent bundle is semistable for x 3 y. 2 f 0 T X/P 1 T X Semistable chamber C 0 Figure 2.2: The chamber structure of X := Σ 1 Example (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 selfintersection e by C 0. Recall that N 1 R (Σ n) = C 0, f R+ and a divisor D ac 0 + bf is ample if and only if a > 0 and b > ae in case that e 0 and it is ample if a > 0, b > 1 2 ae in case e < 0. The canonical bundle is given by K Σn = 2C 0 (2g 2 e)f. Using the exact sequence 0 T X/C T X π T C 0, we compute c 1 (T X/C ) = 2C 0 + ef. Let H := ac 0 + bf be a polarisation. Then by Lemma (2.1.2) T X/C is maximally destabilizing with respect to ac 0 + bf iff c 1 (T X/C ) H > 1 2 c 1(T X ) H. A computation shows that the equality above is equivalent to 1 (2g 2 e) a + b >

24 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 ([Eck08, Section 3]). Let X = P 1 C, where C is an elliptic curve. Let p 1 : X C and p 2 : X P 1 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 E 1, E 2, E 3. Then the maximal rationally connected quotient is given by the blow up of the ruling of X. Let F be the associated foliation. A computation shows that c 1 (F) = p 2 O(2) E 1 E 2 E 3. Let L := p 1 A + p 2 B with deg C A = 3 and deg C P 1 = 4. Then one shows that L := π L 2E 1 2E 2 2E 3 is ample, but L c 1 (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 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 c 1 (T X ) A > 0. Indeed, there exists a free rational curve f : P 1 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 (T X ) = O(a 1 ) O(a 2 ) with a 1 + a 2 2. Thus K X f P 1 = a 1 + a 2 2, see also Remark (1.3.2). Write l := f P 1 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 K X 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 c 1 (T X ) H > 0. There exists a destabilizing subsheaf F of T X, since otherwise X would be rationally connected by Theorem (1.1.12). Furthermore the slope of F has to be bigger than c 1 (T X ) 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 T X l is ample. So we know that each quotient of T X 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

25 Chapter 2 know that H ε := l + εh is ample in N 1 Q (X) for any ε > 0. Observe that c 1(T X ) H ε > 0 for sufficiently small ε, say for 0 < ε < ε 0. If T X is semistable with respect to a certain polarisation H ε with 0 < ε < ε 0, the claim follows since T X has positive slope and induces a trivial foliation which gives the rationally connected quotient. If T X is not semistable for all polarisations H ε with 0 < ε < ε 0, let F ε be the maximal destabilizing subsheaf of T X with respect to H ε. Because of Corollary (2.1.7) the ray H ε stays in one destabilizing chamber. This ensures that F := F ε remains constant. Now it is clear that for sufficiently small ε both the slope of F and the slope of T X /F will be positive with respect to H ε. Therefore the Harder-Narasimhan filtration of T X with respect to H ε yields the maximal rationally connected quotient. Remark 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 Amp R (X) such that the Harder-Narasimhan filtration of T X with respect to all polarisation in U induces the MRC-quotient. 21

26

27 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 intersection c 1 (F) C for a curve C instead of the product c 1 (F) H n 1. The slope is then a linear function on N 1,R (X) whose formal properties are much more well behaved than the original definition. Moreover, we not only consider curves of the form C = H n 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 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 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 µ C (F) := c 1(F) 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 µ C (G ) µ C (F). Remark 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 µ λc+λ C (F) = λµ C(F) + λ µ C (F).

28 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 Let F be a coherent torsion free sheaf on a projective manifold. Given a movable class C Mov(X), there is a increasing filtration, such that 0 = F 0 F 1... F r = F (i) µ C (F i /F i 1 ) > µ C (F i+1 /F i ) for i = 1,...r 1, (ii) the factors F i /F i 1 are torsion free and semistable for all i = 1,...r, (iii) the sheaves F i 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 c 1 (F) C Lemma The degree is additive in short exact sequences Proof. This follows since even the first Chern class is additive in short exact sequences. Lemma 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 µ C (F) µ C (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 c 1 (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 c 1 (F /F) C = c 1 (F ) C c 1 (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

29 Chapter 3 Lemma 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 c 1 (E ) + c 1 (G ) = c 1 (F). Thus we have This yields rk(g ) rk(f) c 1(F) + rk(e ) rk(f) c 1(F) = c 1 (E ) + c 1 (G ), 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 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 µ C (F) > µ C (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 µ C (F) µ C (Im(f)) µ C (G ) and the lemma is proved. The key of the construction of the Harder-Narasimhan filtration is the following Proposition. Proposition Let X be a projective manifold, E a torsion free coherent sheaf on X and µ = µ C 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

30 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: F 1 F 2 iff F 1 F 2 and µ(f 1 ) µ(f 2 ). 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 a contradiction to µ(g ) µ(f). µ(f) > µ(g), 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 we obtain fµ(f) + gµ(g) = rk(f G )µ(f G ) + rk(f + G )µ(f + G ) 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 )). }{{}}{{}}{{} <