Twisted Cotangent Sheaves And A Kobayashi-Ochiai Theorem .

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FOURUTLES DENAAIER NANNALESSL’IN TITDEL’INSTITUT FOURIERAndreas HÖRINGTwisted cotangent sheaves and a Kobayashi-Ochiai theorem for foliationsTome 64, no 6 (2014), p. 2465-2480. http://aif.cedram.org/item?id AIF 2014 64 6 2465 0 Association des Annales de l’institut Fourier, 2014, tous droitsréservés.L’accès aux articles de la revue « Annales de l’institut Fourier »(http://aif.cedram.org/), implique l’accord avec les conditionsgénérales d’utilisation (http://aif.cedram.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme quece soit pour tout usage autre que l’utilisation à fin strictement personnelle du copiste est constitutive d’une infraction pénale. Toutecopie ou impression de ce fichier doit contenir la présente mentionde copyright.cedramArticle mis en ligne dans le cadre duCentre de diffusion des revues académiques de mathématiqueshttp://www.cedram.org/

Ann. Inst. Fourier, Grenoble64, 6 (2014) 2465-2480TWISTED COTANGENT SHEAVES AND AKOBAYASHI-OCHIAI THEOREM FOR FOLIATIONSby Andreas HÖRINGAbstract. — Let X be a normal projective variety, and let A be an ampleCartier divisor on X. Suppose that X is not the projective space. We prove that thetwisted cotangent sheaf ΩX A is generically nef with respect to the polarisation A.As an application we prove a Kobayashi-Ochiai theorem for foliations: if F ( TXis a foliation such that det F iF A, then iF is at most the rank of F .Résumé. — Soit X une variété projective normale et A un diviseur de Cartierample sur X. Supposons que X n’est pas l’espace projectif. Nous montrons quele faisceau cotangent tordu ΩX A est génériquement nef par rapport à la polarisation A. Comme conséquence nous obtenons un théorème de Kobayashi-Ochiaipour les feuilletages : si F ( TX est un feuilletage tel que det F iF A, alors iFest au plus le rang de F .1. IntroductionLet X be a smooth complex projective variety of dimension n. A classicaltheorem of Kobayashi and Ochiai [21] characterises the projective space asthe unique variety having an ample Cartier divisor A such that KX (n 1)A is trivial and hyperquadrics as the varieties such that KX nAis trivial. This result can be seen as the starting point of the adjunctiontheory of projective manifolds, combined with the minimal model programthis theory gives us very precise information about the relation betweenthe positivity of the canonical divisor KX and some polarisation A on X.The aim of this paper is to prove a basic result relating the positivity ofthe cotangent sheaf ΩX to a polarisation:Theorem 1.1. — Let X be a normal projective variety of dimension n,and let A be an ample Cartier divisor on X. Then one of the followingholds:Keywords: Cotangent sheaf, foliations, Kobayashi-Ochiai theorem.Math. classification: 14F10, 37F75, 14M22, 14E30, 14J40.

2466Andreas HÖRINGa) We have (X, OX (A)) ' (Pn , OPn (1)); orb) The twisted cotangent sheaf ΩX A is generically nef with respectto A.If the twisted cotangent sheaf ΩX A is not generically ample with respectto A, one of the following holds:c) There exists a normal projective variety Y of dimension at mostn 1 and a vector bundle V on Y such that X 0 : P(V ) admits abirational morphism µ : X 0 X such that OX 0 (µ A) ' OP(V ) (1);ord) (X, OX (A)) ' (Qn , OQn (1)) where Qn Pn 1 is a (not necessarilysmooth) quadric hypersurface.As an application we obtain a bound for the index of a Q-Fano distribution:Corollary 1.2. — Let X be a normal projective variety, and let A bean ample Cartier divisor on X. Let F ( TX be a subsheaf of rank r 0such that det F is Q-Cartier and det F iF A. Then we have iF 6 r.Indeed if we had iF r, then ΩX A F A would be a quotient withantiample determinant, in particular ΩX A would not be generically nefwith respect to A, in contradiction to the first part of Theorem 1.1. ThisKobayashi-Ochiai theorem for foliations generalises similar results obtainedrecently by Araujo and Druel [3, Thm. 1.1, Sect. 4]. Note that the methodof proof is quite different: while the work of Araujo and Druel is based onthe geometry of the general log-leaf, Theorem 1.1 improves a semipositivityresult for ΩX A proven in [18]. The proof of this semipositivity resultsrelies on comparing the positivity of a foliation F A along a very generalcurve C X with the positivity of a relative canonical divisor KX 0 /Y rµ A (cf. Section 3). The advantage of this technique is that we can makeweaker assumptions on the variety X or the foliation F, the disadvantage isthat we do not get any information about the singularities of the foliation F.For the description of the boundary case in Corollary 1.2 we therefore followclosely the arguments in [3, Thm. 4.11]:Theorem 1.3. — Let X be a normal projective variety, and let A bean ample Cartier divisor on X. Let F ( TX be a foliation of rank r 0such that det F is Q-Cartier and det F Q rA.Then X is a generalised cone, more precisely there exists a normal projective variety Y and an ample line bundle M on Y such that X 0 : P(M OY r ) admits a birational morphism µ : X 0 X such that µ TX 0 /Y Fand OX 0 (µ A) ' OP(M O r ) (1).YANNALES DE L’INSTITUT FOURIER

TWISTED COTANGENT SHEAVES2467This statement generalises a classical theorem of Wahl [28, 11] on ampleline bundles contained in the tangent sheaf. If X is smooth, then Corollary 1.2 and Theorem 1.3 are special cases of the characterisation of the projective space and hyperquadrics by Araujo, Druel and Kovács [5, Thm. 1.1].Vice versa, Theorem 1.1 yields a weak version of [5, Thm. 1.2], [27, Thm. 1.1]for normal varieties:Corollary 1.4. — Let X be a normal projective variety of dimension n, and let A be an ample Cartier divisor on X. Suppose that for somepositive m N we haveH 0 (X, ((TX A ) m ) ) 6 0.Then one of the following holds:a) There exists a normal projective variety Y of dimension at mostn 1 and a vector bundle V on Y such that X 0 : P(V ) admits abirational morphism µ : X 0 X such that OX 0 (µ A) ' OP(V ) (1);orb) (X, OX (A)) ' (Qn , OQn (1)) where Qn Pn 1 is a (not necessarilysmooth) quadric hypersurface.Indeed if ΩX A is generically ample, then (ΩX A) m is generically ample for every positive m N by [24, Cor. 6.1.16]. In particular its dual doesnot have any non-zero global section, in contradiction to the assumption.Thus the second part of Theorem 1.1 applies.Acknowledgements. — I would like to thank Frédéric Han for patientlyanswering my questions about Schur functors. The author was partiallysupported by the A.N.R. project “CLASS”. This work was done whilethe author was a member of the Institut de Mathématiques de Jussieu(UPMC).2. Basic resultsWe work over the complex numbers, topological notions always refer tothe Zariski topology. If V is a locally free sheaf of OX -modules on a variety X, we denote by P(V ) the projectivisation in the sense of Grothendieckand by OP(V ) (1) its tautological line bundle.We will frequently use the terminology and results of the minimal modelprogram (MMP) as explained in [23] or [9]. For some standard definitionsconcerning the adjunction theory of (quasi-)polarised varieties we referto [13, 7, 17].TOME 64 (2014), FASCICULE 6

2468Andreas HÖRINGDefinition 2.1. — Let X be a normal projective variety of dimensionn polarised by an ample Cartier divisor H. Let F be a torsion-free sheafon X. A MR-general curve C X is an intersectionD1 · · · Dn 1for general Dj mH where m 0 such that the Harder-Narasimhanfiltration of F C is the restriction of the Harder-Narasimhan filtration of Fwith respect to H.The abbreviation MR stands for Mehta-Ramanathan, alluding to thewell-known fact [25, 12] that for m N sufficiently high the HarderNarasimhan filtration commutes with restriction to a general complete intersection curve.Let X be a normal projectiv e Pr -bundle structure. Note that we haveXC0 ' P(VC ) where VC : (ϕC ) (OXC0 (ν µ A)), moreover by construction ν 0 F0with 0 an effective Q-divisor. Since we have(KXC0 /C ν ) Q ν µ KF Q rν µ A,and KXC0 /C ϕ C det VC (r 1)ν µ A, we obtain 0 Q ν µ A ϕ C det VC F0 .However by [3, Lemma 4.12,b)] no multiple of ν µ A ϕ C det VC F0 hasa global section, a contradiction.Step 3: The µ-exceptional locus. Since X is not necessarily Q-factorialthe µ-exceptional locus might have irreducible components of codimensionat least two. We claim that this is not the case, in fact the µ-exceptionallocus is equal to the divisor E : let C be a curve in X 0 such that µ(C)is a point. Then we have µ A · C 0, so E · C KX 0 /Y · C. Since µ isfinite on the ϕ-fibres and det V is ample we have ϕ det V ·C 0. Thereforewe haveE · C KX 0 /Y · C (ϕ det V (r 1)µ A) · C 0,hence C is contained in E.Set now W : (ϕ E ) (OE (µ A)), then W is a nef vector bundle of rank r.We have already seen thatE OP(V ) (1) ϕ det V ,TOME 64 (2014), FASCICULE 6

2478Andreas HÖRINGso pushing down the exact sequence0 OX 0 ( E µ A) OX 0 (µ A) OE (µ A) 0to Y we obtain an exact sequence(4.3)0 det V V W 0and det W ' OW . Thus W is a nef vector bundle with trivial determinant, moreover we have a morphism µE : E X such that OP(W ) (1) 'OE (µ E A). By Lemma 4.2 below this implies that W ' OY d . SinceOP(V ) (1) ' OX 0 (µ A) is semiample we know by [14, Cor. 4] that the exact sequence (4.3) splits, thus X is a generalised cone in the sense of [7,1.1.8]. Lemma 4.2. — Let Y be a normal projective vari

Ann. Inst. Fourier,Grenoble 64,6(2014)2465-2480 TWISTED COTANGENT SHEAVES AND A KOBAYASHI-OCHIAI THEOREM FOR FOLIATIONS by Andreas HÖRING Abstract.— Let X be a normal projective variety, and let A be an ample CartierdivisoronX.SupposethatX isno

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