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On the Frobenius integrability of certain holomorphic p forms

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Niveau: Supérieur, Doctorat, Bac+8
On the Frobenius integrability of certain holomorphic p-forms Jean-Pierre Demailly Dedicated to Professor Hans Grauert, on the occasion of his 70th birthday Abstract. The goal of this note is to exhibit the integrability properties (in the sense of the Frobenius theorem) of holomorphic p-forms with values in certain line bundles with semi-negative curvature on a compact Kahler manifold. There are in fact very strong restrictions, both on the holomorphic form and on the curvature of the semi-negative line bundle. In particular, these observations provide interesting information on the structure of projective manifolds which admit a contact structure: either they are Fano manifolds or, thanks to results of Kebekus- Peternell-Sommese-Wisniewski, they are biholomorphic to the projectivization of the cotangent bundle of another suitable projective manifold. 1. Main results Recall that a holomorphic line bundle L on a compact complex manifold is said to be pseudo-effective if c1(L) contains a closed positive (1, 1)-current T , or equivalently, if L possesses a (possibly singular) hermitian metric h such that the curvature current T = ?h(L) = ?i∂∂ logh is nonnegative. If X is projective, L is pseudo-effective if and only if c1(L) belongs to the closure of the cone generated by classes of effective divisors in H1,1R (X) (see [Dem90, 92]).

  • suitable coordinate patches

  • manifold admitting

  • ∂?? ?

  • projective algebraic

  • effective line

  • positive current

  • homogeneous complex

  • l? ?

  • manifold


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On the Frobenius integrability of certain holomorphicpforms JeanPierre Demailly
Dedicated to Professor Hans Grauert, on the occasion of his 70th birthday
Abstract.The goal of this note is to exhibit the integrability properties (in the sense of the Frobenius theorem) of holomorphicpforms with values in certain line bundleswithseminegativecurvatureonacompactK¨ahlermanifold.Therearein fact very strong restrictions, both on the holomorphic form and on the curvature of the seminegative line bundle.In particular, these observations provide interesting information on the structure of projective manifolds which admit a contact structure: eitherthey are Fano manifolds or, thanks to results of Kebekus PeternellSommeseWisniewski, they are biholomorphic to the projectivization of the cotangent bundle of another suitable projective manifold.
1. Mainresults Recall that a holomorphic line bundleLon a compact complex manifold is said to bepseudoeffectiveifc1(L) contains a closed positive (1,1)currentT, or equivalently, ifLpossesses a (possibly singular) hermitian metrichsuch that the curvature currentT= Θh(L) =i∂∂loghIfis nonnegative.Xis projective,Lis pseudoeffective if and only ifc1(L) belongs to the closure of the cone generated 1,1 by classes of effective divisors inH(XOur main result is) (see [Dem90, 92]). R Main Theorem.LetXbeacompactha¨Kmrelfina.dlosuAsthmethateer exists a pseudoeffective line bundleLonXand a nonzero holomorphic section p 01 θH(X,ΩL), where06p6n= dimX. LetSθbe the coherent subsheaf X of germs of vector fieldsξin the tangent sheafTX, such that the contractioniξθ vanishes. ThenSθis integrable, namely[Sθ,Sθ]Sθ, andLhas flat curvature along the leaves of the(possibly singular)foliation defined bySθ. Before entering into the proof, we discuss several consequences.Ifp= 0 or p=n, the result is trivial (withSθ=TXandSθ= 0, respectively).The most interesting case isp= 1. Corollary 1.In the above situation, if the line bundleLXis pseudo 0 11 L)is a non effective andθH(X,ΩXzero section, the subsheafSθdefines a holomorphic foliation of codimension1inX, that is,θ= 0. We now concentrate ourselves on the case whenXis acontact manifold, i.e. 0 11 dimX=n= 2m+ 1,m>1, and there exists a formθH(X,ΩL), X m0m1 called thecontact form, such thatθ()H(X, KXL) has no zeroes.