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

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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 Fakult¨at f¨ur Mathematik und Physik der Albert-Ludwigs-Universit¨at 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 three- folds 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 mani- folds ....................................... 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 va- rieties one can form the maximal rationally connected fibration or maximal rationally connected quotient based on a construction by Campana [Cam81], [Cam94] and Koll´ar- 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-Sol´aConde-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 N´eron-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 N´eron-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 de- composition 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 num- bers. 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 n 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+.
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