COMPACT COMPLEX MANIFOLDS WITH NUMERICALLY EFFECTIVE TANGENT BUNDLES
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COMPACT COMPLEX MANIFOLDS WITH NUMERICALLY EFFECTIVE TANGENT BUNDLES Jean-Pierre Demailly?, Thomas Peternell??, Michael Schneider?? ? Universite de Grenoble I ?? Universitat Bayreuth Institut Fourier, BP 74 Mathematisches Institut U.R.A. 188 du C.N.R.S. Postfach 10 12 51 38402 Saint-Martin d'Heres, France D-8580 Bayreuth, Deutschland Contents 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p. 2 1. Basic properties of nef line bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 5 1.A. Nef line bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Voir
- chern curvature
- nef vector
- line bundle
- tx nef
- compact kahler
- hold
- nef tangent
- bundles
- bundles still hold
COMPACT COMPLEX MANIFOLDS WITH NUMERICALLY EFFECTIVE TANGENT BUNDLES
Jean-Pierre Demailly⋆, Thomas Peternell⋆⋆, Michael Schneider⋆⋆ ⋆onlbIe´tdeGeerisrevinU⋆⋆htuet¨atBayrUniversi Institut Fourier, BP 74 Mathematisches Institut U.R.A. 188 du C.N.R.S. Postfach 10 12 51 38402Saint-Martind’H`eres,FranceD-8580Bayreuth,Deutschland
Contents
0. Introduction p. 2
1. Basic properties of nef line bundles p. 5 1.A. Nef line bundles p. 5 1.B. Nef vector bundles p. 11 2. Inequalities for Chern classes p. 18 3. Compact manifolds with nef tangent bundles. Structure of the Albanese map p. 22 3.A. Some examples p. 22 3.B. General properties p. 23 3.C.StructureoftheAlbanesemap(K¨ahlercase) p. 25 3.D. Numerical flatness of the Albanese map p. 32
4. Moishezon manifolds with nef tangent bundles p. 36
5. Two structure theorems p. 37 6. Surfaces with nef tangent bundles p. 40 7.K¨ahler3-foldswithneftangentbundles p. 44 References p. 45
Key words:Compact complex manifold, numerically effective vector bundle, Chern curvature tensor, Chern classes, Albanese map, Fano variety, Kodaira classification of surfaces, Mori theory, Frankel conjecture.
A.M.S. Classification:32J25, 32L15, 14J15
1
0. Introduction The compact Riemann surfaces which are not of general type are those which are of semipositive curvature. In higher dimensions the situation is much more subtle and it has been a long standing conjecture due to Frankel to characterize IPnsahtoelnctpaomycrmleahK¨wdlofinatisophtiolomivehicorph bisectional curvature. Hartshorne extended Frankel’s conjecture and weakened the assumption to the case of ample tangent bundle. In his famous paper [Mo79] Mori solved the Hartshorne conjecture by using characteristic p methods. Around the same time Siu and Yau [SY80] gave an (analytic) proof of the Frankel conjecture. Combining algebraic and analytic tools Mok [Mk88] classified all compact K¨hler a manifolds with semipositive holomorphic bisectional curvature. In algebraic geometry a powerful and flexible notion of semipositivity is numerical effectivity line bundle(”nefness”). ALon a projective manifoldX is said to benefifLC≥0 for all compact curvesC⊂X. It is clear that a line bundle with semipositive curvature is nef. The converse had been conjectured by Fujita [Fu83]. Unfortunately this is not true as we shall see in§1, example 1.7. A vector bundleEis callednumerically effective(nef) if the line bundle OE(1) on IP(E), the projective bundle of hyperplanes in the fibres ofE, is nef. Again it is clear that a vector bundleEwhich admits a metric with semipositive curvature(inthesenseofGriffiths)isnef.AcompactK¨ahlermanifoldXhaving semipositive holomorphic bisectional curvature has by definition a tangent bundle TXwith semipositive curvature. Again the converse does not hold. From an algebraic geometric point of view it is natural to consider the class of projective manifoldsX Thiswhose tangent bundle is nef. has been done by Campana and Peternell [CP91] and –in case of dimension 3– by Zheng [Zh90]. In particular, a complete classification is obtained for dimension at most three. ThepurposeofthispaperistoinvestigatecompactKa¨hlermanifoldswith nef tangent bundles. One difficulty in carrying over the algebraic definition of nefnesstotheKa¨hlercaseisthepossiblelackofcurves.Thisisovercomebythe following Definition. —LetXbe a compact complex manifold with a fixed hermitian metricω line bundle. ALoverX for everyis nef ifε >0there exists a smooth hermitian metrichεonLsuch that the curvature satisfies Θhε≥ −ε ω .
This means that the curvature ofLcan have an arbitrarily small negative part. Clearly a nef line bundleLsatisfiesLC≥0 for all curvesC⊂X, but the converse is not true. For projective algebraicX mainboth notions coincide. Our result is a structure theorem on the Albanese map. 2
Main Theorem. —LetXegnatfentnacmp¨atKbcoeadlofhtiwrelhinam e bundleTX. LetXteniafibeeloce´atfevorXof maximum irregularityq= e e q(X) =h1(XOe). Then ,X e (i)π1(X)≃Z2q. e e (ii)The Albanese mapα:X→A(X)is a smooth fibration over a q-dimensional torus with nef relative tangent bundle. (iii)The fibresFofαare Fano manifolds with nef tangent bundles. As a consequence one gets the Corollary. —an abstract group, the fundamental groupAs π1(X)is an 2q extension of a finite group byZ. It is conjectured in [CP91] that Fano manifolds (i.e. manifolds such that −K Once this isis ample) with nef tangent bundle are rational homogeneous. provedourmaintheoremclassifiesthecompactK¨ahlermanifoldswithneftangent bundleupto´etalecover.In§1 we prove basic properties of nef vector bundles. An important technical point – used over and over – is the following: Lemma. —LetEbe a nef vector bundle on a compact complex connected manifoldXand letσ∈H0(X, E⋆) Thenbe a non trivial section.σhas no zeros at all. The main result of§1 characterizes vector bundlesEwhich are numerically flat, i.e. such thatEandE⋆are nef. Theorem. —LetEemirebunflytaacllomecthonahK¨ctpafinamreldloX. ThenEadmits a filtration by vector bundles whose graded pieces are hermitian flat, i.e. given by unitary representations ofπ1(X). The proof uses two deep facts: the Uhlenbeck-Yau theorem on the existence of Hermite-Einstein metrics on stable vector bundles and Kobayashi’s flatness criterionderivedfromLu¨bke’sinequalityonChernclasses.In§2 it is shown that the Fulton-Lazarsfeld inequalities [FL83] for Chern classes of ample vector bundles stillholdfornefvectorbundlesoncompactKa¨hlermanifolds. Theorem. —LetXbedneltnafiloadK¨ahlermacompactEbe a nef vector bundle onX. ThenEis numerically semipositive, i.e.P(c(E))≥0for all positive polynomialsP. As a consequence we get the following Corollary. —IfEis nef andc1(E)n= 0,n= dimX, then all Chern polynomialsP(c(E))of degree2nvanish. 3
This will be used in the proof of our main theorem.§3 contains the proof of our main theorem cited above. One of the key points is the following Proposition. —LetXfon-erhl¨atKacmpocaebldwithTXnef. Then (i)Ifc1(X)n>0, thenXis a Fano manifold. (ii)Ifc1(X)n= 0, thenχ(OX) = 0and there exists a non zero holomorphic e e p-form,pusdoelbatie´etelatdnadinfiavecorX→Xsuch thatq(X)>0. The difficult part is (ii). Sincec1(X)n= 0 we get by the above corollary thatχ(OX this we infer the existence of a non zero) = 0. Fromp-formufor some suitable odd numberp contraction with. Byuwe get a mapS: Λp−1TX→Ω1X. The imageEconstant rank and is therefore a subbundle of Ωhas 1X. HenceE is numerically flat. The theorem in§1 implies the existence of a hermitian flat subbundleE1⊂E. IfE1ehevtrniteomefileco´etaafalrsteetdbviriluow proposition would be proved. We use a result of Tits that every subgroup of a linear group contains either a non abelian free subgroup or a solvable subgroup of finite index. The first case cannot occur since we show in [DPS92] thatπ1(X) has sub-exponential growth. The second case leads to the desired conclusion. In§4 we provedgive algebraic proofs of the following two results, stated resp. in 3.6 and 3.10 (i). Theorem. —manifold with nef tangent bundle is projective.Any Moishezon
Theorem. —LetXleahK¨ctdwol-frnhtiebcamoapTXnef andc1(X)n>0. ThenXis a Fano manifold. In§5 we show that a Mori contractionϕ:X→Yof a projective manifold XwithTXnef is smooth; moreoverY Asis smooth too. a consequence we get an algebraic proof of the crucial proposition 3.10. In§6 we classify all non algebraic surfaces with nef tangent bundles. Theorem. —The smooth non algebraic compact complex surfaces with nef tangent bundles are precisely the following: (i)Non algebraic tori; (ii)Kodaira surfaces; (iii)Hopf surfaces. InthefinalsectionwegiveaclassificationofallnonalgebraicK¨ahler3-folds. Theorem. —LetXactK¨ahlermanifodl.anbealonbrgec3aimid-isnelanopmoc ThenTXis nef if and only ifXroforosuretaiehttheotpusierovecalete´itfin formIP(E)whereEa numerically flat rank 2-bundle over a 2-dimensional torus.is 4
