We prove that the intersection of at least n su64259ciently ample general hypersurfaces in a complex abelian variety of dimension has ample cotangent bundle We also discuss analogous questions for complete intersections in the projective space Final ID: 35510 Download Pdf

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We prove that the intersection of at least n su64259ciently ample general hypersurfaces in a complex abelian variety of dimension has ample cotangent bundle We also discuss analogous questions for complete intersections in the projective space Final

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Varieties with ample cotangent bundle Olivier Debarre Abstract The aim of this article is to provide methods for constructing smooth projective complex varieties with ample cotangent bundle. We prove that the intersection of at least n/ suﬃciently ample general hypersurfaces in a complex abelian variety of dimension has ample cotangent bundle. We also discuss analogous questions for complete intersections in the projective space. Finally, we present an unpublished result of Bogomolov which states that a general linear section of small dimension of a product of

suﬃciently many smooth projective varieties with big cotangent bundle has ample cotangent bundle. 1. Introduction Projective algebraic varieties with ample cotangent bundle have many properties: the subvarieties of are all of general type; there are ﬁnitely many nonconstant rational maps from any ﬁxed projective variety to ([NS]); if is deﬁned over , any entire holomorphic mapping is constant ([De], (3.1)); if is deﬁned over a number ﬁeld , the set of -rational points of is conjectured to be ﬁnite ([M]). Although these varieties are expected to

be reasonably abundant, few concrete constructions are available. The main result of this article, proved in section 2, is that the intersection of at least n/ suﬃciently ample general hypersurfaces in an abelian variety of dimension has ample cotangent bundle. This answers positively a question of Lazarsfeld. As a corollary, we obtain results about cohomology groups of sheaves of symmetric tensors on smooth subvarieties of abelian varieties. In section 3, mostly conjectural, we discuss analogous questions for complete intersections in the projective space. Finally, we present in

section 4 an unpublished result of Bogomolov which states that a general linear section of small dimension of a product of suﬃciently many smooth projective varieties with big cotangent bundle has ample cotangent bundle. This shows in particular that the fundamental group of a smooth projective variety with ample cotangent bundle can be any group arising as the fundamental group of a smooth projective variety. We work over the complex numbers. Given a vector bundle , the projective bundle ) is the space of 1-dimensional quotients of the ﬁbers of . It is endowed with a line bundle

(1). We say that is ample (resp. nef, resp. big) if the line bundle (1) has the same property. Following [So1], we say more generally that given an integer , the vector bundle is -ample if, for some m> 0, the line bundle ) is generated by its global sections and each ﬁber of the associated map has dimension . Ampleness coincide with 0-ampleness. 2000 Mathematics Subject Classiﬁcation 14K12 (primary), 14M10, 14F10 (secondary). Keywords: Ample cotangent bundle, complete intersections.

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Olivier Debarre 2. Subvarieties of abelian varieties We study the positivity

properties of the cotangent bundle of a smooth subvariety of an abelian variety 2.1 Preliminary material Using a translation, we identify the tangent space A,x at a point of with the tangent space A, at the origin. We begin with a classical result. Proposition Let be a smooth subvariety of an abelian variety . The following properties are equivalent: (i) the cotangent bundle is -ample; (ii) for any nonzero vector in A, , the set X,x has dimension Proof. The natural surjection ( induces a diagram ( ( ( A, ( A, (1) and ( (1) = ( (1) = ( A, (1) It follows that is -ample if and only if each

ﬁber of has dimension ([So1], Corollary 1.9). The proposition follows, since the restriction of the projection ( to any ﬁber of is injective. Remarks (1) Let = dim( ) and = dim( ). Since dim( ( )) = 2 1, the proof of the proposition shows that the cotangent bundle of is (2 )-ample at best. It is always -ample, and is ( 1)-ample except if has a nonzero vector ﬁeld, which happens if and only if is stable by translation by a nonzero abelian subvariety (generated by the vector ﬁeld). (2) Many things can prevent the cotangent bundle of from being ample. Here are two

examples. Assume , where and are subvarieties of of positive dimension. For all smooth on and all , one has ,x X,x , hence the cotangent bundle of is not dim( -ample. In the Jacobian of a smooth curve , the cotangent bundle of any smooth ) is therefore exactly ( 1)-ample (although its normal bundle is ample). If is (isogenous to) a product and { ), the cotangent bundle of is at most (2 dim( dim( ))-ample, because of the commutative diagram ( reg ( A, ( reg ( In particular, if dim( dim( ) for some , the cotangent bundle of cannot be ample. We will encounter the following situation

twice: assume and are vector bundles on a projective variety that ﬁt into an exact sequence 0 (2)

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Varieties with ample cotangent bundle where is a vector space. Lemma In the situation above, if moreover rank( dim( , we have ample nef and big Proof. As in the proof of Proposition 1, is nef and big if and only if the morphism ) induced by (2), which satisﬁes (1) = (1), is generically ﬁnite. Let be the dimension of , let be the dimension of , let be the rank of , and let be the Grassmannian of vector subspaces of of dimension , with tautological quotient

bundle of rank . The dual of the exact sequence (2) induces a map such that Assume that has dimension dim( )) = 1. There exists a linear subspace of of dimension + 1 such that ) does not meet . In other words, the variety ) does not meet the special Schubert variety }} , whose class is ). We obtain ) = 0, hence 0 = ) = ), and cannot be ample by [BG], Corollary 1.2. 2.2 Nef and big cotangent bundle A characterization of subvarieties of an abelian variety whose cotangent bundle is nef and big follows easily from a result of [D1]. Proposition The cotangent bundle of a smooth subvariety of an

abelian variety is nef and big if and only if dim( ) = 2 dim( Proof. The cotangent bundle of is nef and big if and only if the morphism in (1) is generically ﬁnite onto its image ( X,x ), i.e., if the latter has dimension 2 dim( 1. The proposition follows from [D1], Theorem 2.1. The condition dim( ) = 2 dim( ) implies of course 2 dim( dim( ). The converse holds if is nondegenerate ([D1], Proposition 1.4): this means that for any quotient abelian variety , one has either ) = or dim( )) = dim( ). This property holds for example for any subvariety of a simple abelian variety. It has also

an interpretation in terms of positivity of the normal bundle of Proposition The normal bundle of a smooth subvariety of an abelian variety is nef and big if and only if is nondegenerate. Proof. The normal bundle X/A to in is nef and big if and only if the map in the diagram X/A A, A, (3) is generically ﬁnite onto its image (i.e., surjective). To each point in the image of corresponds a hyperplane in A, such that X,x for all in the image in of the ﬁber. This implies ,x for all in , hence the tangent space at the origin of the abelian variety generated by is contained in ([D2],

Lemme VIII.1.2). An abelian variety is simple if the only abelian subvarieties of are 0 and . For more about nondegenerate subvarieties, see [D2], Chap. VIII.

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Olivier Debarre Since has at most countably many abelian subvarieties, the abelian variety is independent of the very general point in the image of . Let be the corresponding quotient. The diﬀerential of is not surjective at any point of since its image is contained in the hyperplane T ). By generic smoothness, is not surjective. If is nondegenerate, is generically ﬁnite onto its image, hence is

ﬁnite and is generically ﬁnite onto its image. It follows that X/A is nef and big. Conversely, assume that X/A is nef and big. Let be a quotient of such that . The tangent spaces to along a general ﬁber of are all contained in a ﬁxed hyperplane. This ﬁber is therefore ﬁnite, hence is nondegenerate. Proposition Let be a smooth subvariety of an abelian variety , of dimension at most dim( . We have ample X/A nef and big nef and big Proof. The ﬁrst implication follows from Lemma 3 applied to the exact sequence 0 X/A 0. The second implication follows

from Propositions 4 and 5 and the fact that for a nondegenerate subvariety of , the equality dim( ) = min(2 dim( dim( )) holds ([D1], Proposition 1.4). 2.3 Ample cotangent bundle In this subsection, we prove that the intersection of suﬃciently ample general hypersurfaces in an abelian variety has ample cotangent bundle, provided that its dimension be at most dim( ). We begin by ﬁxing some notation. If is a smooth variety, a vector ﬁeld on , and a line bundle on , we deﬁne, for any section of with divisor , a section ∂s of by the requirement that for any open

set of and any trivialization , we have ∂s ))) in . We denote its zero locus by ∂H . We have an exact sequence A,L H,L A, ∂s 7 ∂ ^c where ) is considered as an element of A, ) and the cup product is the contraction A,T A, A, 2.3.1 The simple case We begin with the case of a simple abelian variety, where we get an explicit bound on how ample the hypersurfaces should be. Theorem Let ,...,L be very ample line bundles on a simple abelian variety of dimension . Consider general divisors ∈| ,...,H ∈| . If ,...,e are all > n , the cotangent bundle of is max( c, 0)

-ample. Proof. We need to prove that the ﬁbers of the map in (1) have dimension at most = max( c, 0). This means that for general in and any nonzero constant vector ﬁeld on , the dimension of the set of points in such that X,x is at most ; in other words, that dim( ∂H ∂H It is enough to treat the case n/ 2. We proceed by induction on , and assume that the variety ∂H ∂H has codimension 2 2 in for all nonzero . Let ∂, ,...,Y ∂,q be its irreducible components.

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Varieties with ample cotangent bundle Let ∂,i ) be the open set

of divisors in such that ( ∂,i red is integral of codimension 1 in ∂,i . If ∂,i ), I claim that ∂,i ∂H has codimension 2 in ∂,i . Indeed, let A,L ) deﬁne and set = ( ∂,i red . The scheme ∂H is the zero set in of the section ∂s deﬁned above. In the commutative diagram, H,L A, ∂s ∂ ^ec H,L Y, (4) the restriction is injective because generates , hence ∂s does not vanish identically on the integral scheme It follows that for ) = =1 ∂,i ), the scheme ∂H has codimension 2 in . Thus, for ( A, ), the

intersection ∂H ∂H has codimension 2 in for all nonzero constant vector ﬁeld on (note that when = 1, there is no condition on ). Lemma 12, to be proved in 2.3.4, shows that the complement of ) in has codimension at least 1. For e>n , the intersection ( A, ) is therefore not empty and the theorem follows. 2.3.2 The general case A variant of the same proof works for any abelian variety, but we lose control of the explicit lower bounds on ,...,e Theorem Let ,...,L be very ample line bundles on an abelian variety of dimension . For ,...,e large and divisible enough positive

integers and general divisors ∈| ,...,H ∈| the cotangent bundle of is max( c, 0) -ample. Let us be more precise about the condition on the . What we mean is that there exists for each ∈{ ,...,c a function such that the conclusion of the theorem holds if , e ,e ,...,e ,...,e with ,...,e >n (5) Proof. We keep the setting and notation of the proof of Theorem 7. Everything goes through except when, in diagram (4), ∂ ^ c )) = 0. In this case, let 00 be the abelian subvariety of generated by and let be its complement with respect to , so that the addition 00 is an isogeny and

00 . We have 00 , with 00 00 , and ,T ). In particular, we have an injection A,L ,L 00 ,L 00 It is however diﬃcult to identify in a manner useful for our purposes the sections of inside this tensor product. Instead, we use a trick that will unfortunately force us to lose any control of the numbers involved. The trick goes as follows. The kernel of , being ﬁnite, is contained in the group of -torsion points of 00 for some positive integer . Multiplication by factors as 00 00

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Olivier Debarre and 00 ) is some power of . Sections of that come from ,L 00 ,L 00 )

induce a morphism from to some projective space that factors through and embeds 00 We will consider sections of ee of the type , with ,L 00 ,L 00 ). If the divisor of on 00 corresponds to a degree hypersurface in )), the intersection is integral of codimension 1 in ) = ra } rY 00 Fix a basis ( 00 ,...,s 00 ) for 00 ,L 00 ) and write =1 00 , so that div( =1 ra 00 ∂H div( =1 ra 00 div( =1 ∂s ra 00 Since is integral, ∂H has codimension 2 in ) (hence has codimension 2 in ) unless, for some complex number , the section =1 λs ∂s )( ra 00 of 00 vanishes on rY 00 . In

other words, if we let rY 00 ,...,a =1 00 vanishes on rY 00 and ra ra ∂s ra ∂s ra we have ( λ, 1) rY 00 . Now we may pick any collection ( ,...,s ) we like. Fix one such that the corresponding matrix has rank 2 for all nonzero and apply a square matrix of size dim( ). The condition is now that the composition Im( rY 00 is not injective, that is, either Im( rY 00 , which imposes codim( rY 00 1 conditions on or Ker( Im( , which imposes 1 conditions on The “bad” locus for corresponds to the space of matrices that satisfy either one of these properties for some nonzero ,T ).

Since, on the one hand 00 ,L 00 > e and, on the other hand, the codimension of rY 00 is the rank of the linear map 00 ,L 00 rY 00 ,L rY 00 ), which is >e , the codimension of the “bad” locus is at least dim( ) + 2. This means that for 00 (hence ) ﬁxed, e>n , and general in ee , for any component of that spans (as a group) 00 , the intersection ∂H has codimension 2 in for all nonzero in A,T ). Since has at most countably many abelian subvarieties, there are only ﬁnitely many diﬀerent abelian subvarieties spanned by components of ∂H ∂H for ,...,H general in

,..., as runs through the nonzero elements of A,T ). Therefore, for some positive integer , any e > n , and general in e , the intersection ∂H has codimension 2 in for all nonzero A,T ). This proves our claim by induction on hence the theorem. 2.3.3 The four-dimensional case In case the ambient abelian variety has dimension 4, we can make the numerical conditions in Theorem 8 explicit.

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Varieties with ample cotangent bundle Theorem Let and be line bundles on an abelian fourfold , with ample and very ample. For , and ∈| and ∈| general, the surface has ample

cotangent bundle. Proof. I claim that for general in , the scheme ∂H is an integral surface for each nonzero vector ﬁeld on . Granting the claim for the moment and using the notation of the proof of Theorem 7, the scheme ∂H is then, for ), an integral curve that generates since its class is . The argument of the proof of Theorem 7 applies in this case to prove that ∂H ∂H is ﬁnite. Taking in ( A, ) (which is possible by Lemma 12 since 4), the intersection ∂H ∂H is ﬁnite for all nonzero vector ﬁelds , which is what we need. The

theorem therefore follows from the claim, proved in the next lemma. Lemma 10 Let be an abelian variety of dimension at least and let be an ample divisor on . For and general in , the scheme ∂H is integral for all nonzero A,T Proof. Assume to the contrary that for some smooth ∈| , we have ∂H where and are eﬀective nonzero Cartier divisors in . We follow [BD], proposition 1.6: since dim( 3, there exist by the Lefschetz Theorem divisors and on such that and . Since is eﬀective, the long exact sequence in cohomology associated with the exact sequence shows that, for

each ∈{ , either A,D = 0 or A,D = 0. The case where both A,D ) and A,D ) are zero is impossible, since we would then have a section of with divisor ∂H on . The case where both A,D ) and A,D ) are nonzero is impossible as in loc. cit. because dim( 3. So we may assume A,D = 0 and A,D ) = 0, and take eﬀective such that As in loc. cit. contains an elliptic curve such that, if is the neutral component of the kernel of the composed morphism Pic Pic the addition map is an isogeny, is tangent to , and ) = ) for some eﬀective divisor on . Pick a basis ( ,...,t ) for B,L ) and

a section of with divisor , and write =1 with ,...,s E,L ), so that ∂H is deﬁned by =1 =1 ∂s = 0 Since is contained in ∂H , for every point of the support of , we have div =1 div =1 ∂s

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Olivier Debarre Since these two divisors belong to the same linear series on , they must be equal and rank ∂s ∂s Since is irreducible, the sections ,...,s have no common zero and the morphism that they deﬁne is ramiﬁed at The vector subspace of E,L ) generated by ,...,s only depends on , not on the choice of the basis ( ,...,t ). If ,...,b

are general points of , it is also generated by ,...,s ) and rank ∂s ∂s Assume now that the conclusion of the lemma fails for general (and ). The point varies with , but remains constant for in a hypersurface of X,L ). If is in ∩{ X,L ) = ) = 0 it also satisﬁes ∂s ) = ∂s ) = 0. Since has codimension at most 3 in X,L ), this means that is not 3-jet ample and contradicts Theorem 1 of [BS]: the lemma is proved. Remarks 11 (1) Let be an abelian fourfold that contains no elliptic curves. The proof of Lemma 10 shows that for any smooth ample hypersurface in and

any nonzero A,T ), the scheme ∂H is integral. It follows that for very ample, 5, and ∈| general, the surface has ample cotangent bundle (this is only a small improvement on Theorem 7). (2) Lehavi has recently proved that on a general Jacobian fourfold (hence also on a general principaly polarized abelian fourfold), the intersection of a theta divisor with a translate by a point of order 2 is a smooth surface with ample cotangent bundle. This implies the same statement for the intersection of two general translates of general hypersurfaces in the same linear system of even degree on

a general polarized abelian fourfold. 2.3.4 Proof of the lemma We prove the lemma used in the proofs of all three theorems. Lemma 12 Let be an integral subscheme of of dimension at least and let e,n be the projective space of hypersurfaces of degree in . The codimension of the complement of ) = e,n is integral of codimension in in e,n is at least Proof. By taking hyperplane sections, we may assume that is a surface. We proceed by induction on . For = 2, this codimension is min + 2 + 2 + 2 + 1 Assume 3. Let be a component of ) of maximal dimension and let e,p be the linear subspace of e,n that

consists of cones with vertex a point . If does not meet e,p , we have codim( dim( e,p 1 = 1 + >e and the lemma is proved. We will therefore assume that meets e,p . Let } be a projection. If is general in e,p It is a pleasure to acknowledge Zak’s help with the proof of this lemma.

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Varieties with ample cotangent bundle either ) is not integral of dimension 1, the induction hypothesis yields codim codim e,p e,p codim e,n )) e,n and the lemma is proved; or the curve ) is integral of dimension 1, but is contained in the locus over which the ﬁnite morphism ) is not an

isomorphism. In the second case, if 4, the morphism is birational, has dimension at most 1, hence codim e,n codim e,p e,p codim e,n e,n contains a component of + 1 where the last inequality holds because any + 1 points in impose independent conditions on hypersurfaces of degree . The lemma is proved in this case. We are reduced to the case = 3: the curve is integral and its inverse image by , is reduced but reducible. We consider the following degeneration. Denote by an equation for ; the surface deﬁned by tx ,x ,x ,x ) = 0 is projectively equivalent to for = 0, whereas is the cone with

vertex (1 0) and base = 0). We may therefore assume that is an integral cone with vertex a point and we let } be the projection. Let be the intersection of the line pp with the plane . Pick a line in , avoiding , and consider the projections } and from (the point might be on (if ), but is not on , because p / ). The maps } 7 , and y,y 7 py are inverse one to another. Therefore, given the integral curve , we need to study the dimension of the set of curves of degree for which the curve is reducible. Using the same trick as above, we degenerate to the union of deg( ) distinct lines through some

point. At the limit, is the union of deg( ) curves isomorphic to . If is integral, the projection has the property that every irreducible component of is dominated by a unique component of , and the set of “bad” curves has codimension 1 as we saw in the case = 2. This property of the projection, begin open, carries over to . This ﬁnishes the proof of the lemma. 2.4 Cohomology of symmetric tensors Let be a smooth subvariety of an abelian variety. We are interested in the cohomology groups of the vector bundles Proposition 13 Let be an abelian variety of dimension and let be a smooth

subvariety of codimension of with ample normal bundle. For , the restriction A, X,

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Olivier Debarre is bijective for q and injective for Proof. We follow the ideas of [S]. The symmetric powers of the exact sequence 0 X/A 0 yield, for each r> 0, a long exact sequence X/A X/A By Le Potier’s vanishing theorem ([LP]; [L], Remark 7.3.6), X, X/A ) vanishes for q> and i > 0. Since is trivial, we get, by an elementary homological algebra argument ([S], Lemma, p. 176), X, Ker( )) = 0 for all c. The proposition now follows from the fact that the restriction A, X, ), hence also the

restriction A, X, ), is bijective for ([So2]). Sommese proved ([So1], Proposition (1.7)) that for any -ample vector bundle on a projective variety and any coherent sheaf on X, ) = 0 for all q>k and 0. Theorem 7 and Proposition 13 therefore imply the following. Corollary 14 Let be the intersection of suﬃciently ample general hypersurfaces in an abelian variety of dimension . We have X, = 0 for q> max c, and A, for q and A, for and 3. Subvarieties of the projective space We now study the positivity properties of the cotangent bundle of a smooth subvariety of the projective space. 3.1 Big

twisted cotangent bundle If is a smooth subvariety of of dimension , we let d, ) be the Gauss map. We denote by the universal subbundle and by the universal quotient bundle on d, ). We For the case = 0, Bogomolov gave in [B2] a very nice proof that goes as follows. Arguing as in the proof of Proposition 4, we ﬁnd that the morphism of (1) is surjective whenever . Any ﬁber of is isomorphic to its projection to , which is the zero locus of a section of A/X . It follows that when A/X is ample and c the ﬁbers of are connected, hence ( ( A, ), from which we get, for all 0, X, (

( )) ( A, ( A, )) A, To be more precise, we need condition (5) to be satisﬁed. 10

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Varieties with ample cotangent bundle have X/ 1) and a commutative diagram 0 0 X/ (1) = X/ (1) (1) +1 (1) ↓↓k (1) (1) 0 0 (6) The following result is proved as Propositions 4 and 6. Proposition 15 Let be a smooth subvariety of dimension of If is big, If and X/P 1) is ample, is nef and big. Similarly, with the same ideas, we prove an analog of Theorem 7. Theorem 16 Let be a general complete intersection in of multidegree ,...,e . If and ,...,e are all + 2 , the vector bundle is

max( c, 0) -ample. Proof. We need to prove that the ﬁbers of the composed map analogous to the map in diagram (3) have dimension at most = max( c, 0). This means that for general in and for any in , the dimension of the set of points in such that X,x is at most . Pick coordinates and write = ( ,...,t ). If is an equation of a hypersurface , we let be the hypersurface with equation =0 ∂s ∂x . With this notation, we want dim( As in the proof of Theorem 7, we proceed by induction on , assuming n/ 2. When = 1, it is clear that 2 is suﬃcient. Assume has (pure) codimension

2 2 in , with irreducible components t, ,...,Y t,m . Set = ( t,i red ; it follows from Lemma 12 that is integral of codimension 1 in for outside a closed subset of codimension 1 in Assume that this is the case. If codim 1, the section must vanish on Since the restriction Y, 1)) H, 1)) is injective, it must also vanish on . Since any distinct points of impose independent conditions on elements of 1) and the map )) 1)) is surjective, we have proved that the set of hypersurfaces in such that codim has codimension 1 in . The theorem follows. Corollary 17 Let be a general complete intersection in

of multidegree ,...,e . If and ,...,e are all + 2 , and n/ , the vector bundle (1) is big. When is a surface, (i.e., 2) results of Bogomolov ([B1], [B2]) give the much better result that ) is big. 11

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Olivier Debarre Proof. The last row of diagram (6) yields, for all positive integers , an exact sequence (1) (1) It follows from Theorem 16 that for 0, we have X, (1) = 0 for q> 1 (7) On the other hand, if , the coeﬃcient of (2 1)! in the polynomial X, (1) is ( (1) ) = =1 (1 + ( 1) )(1 1) 1) 1) 1) Since n/ 2, this is positive, so that by (7), we have, for 0, X, (1) X, (1)

αr for some α> 0. This shows that (1) is big. 3.2 Conjectures By analogy with Theorem 7, it is tempting to conjecture the following generalization of a question formulated by Schneider in [S], p. 180. Conjecture 18 The cotangent bundle of the intersection in of at least n/ general hypersur- faces of suﬃciently high degrees is ample. Ampleness can be characterized cohomologically as follows. Proposition 19 Let be a projective variety and let be an ample line bundle on . A vector bundle on is ample if and only if, for any integer , we have X, ) = 0 for all q> and Proof. Let be

an arbitrary coherent sheaf on . It has a possibly nonterminating resolution by locally free sheaves that are direct sums of powers of . Therefore, X, ) = 0 for all ∈{ ,..., dim( , all q > 0 and 0, and this implies X, ) = 0 for all q > 0 and 0. This proves that is ample ([L], Theorem 6.1.10). Conjecture 18 therefore has the following equivalent cohomological formulation. Conjecture 20 Let be as in Conjecture 18. For any integer , we have X, )( )) = 0 for all q> and Let be a smooth projective variety of dimension with ample and let be a line bundle on . It follows from [De], Theorem 14.1,

that X, ) vanishes for 0. This leads us to think that the following stronger form of Conjecture 20 might be true. Conjecture 21 Let be the intersection in of general hypersurfaces of suﬃciently high degrees and let be an integer. For , we have X, )( )) = 0 (8) except for = max c, 12

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Varieties with ample cotangent bundle Remarks 22 (1) For any smooth subvariety of of codimension , the vanishing (8) holds for q < n and + 2 by [S], Theorem 1.1, and for by Demailly’s theorem. In particular, Conjecture 21 holds for 1. (2) Under the hypotheses of Conjecture 21, one checks

that the leading coeﬃcient of the poly- nomial ( ( )) has sign ( 1) max c, . This is compatible with the conjecture. 4. Bogomolov’s construction of varieties with ample cotangent bundle We present here an old unpublished construction of Bogomolov that produces varieties with ample cotangent bundle as linear sections of products of varieties with big cotangent bundle (a diﬀerential- geometric version of this construction appeared later in [W]). Everything in this section is due to Bogomolov. Proposition 23 (Bogomolov). Let ,...,X be smooth projective varieties with big cotangent

bundle, all of dimension at least d > . Let be a general linear section of . If dim( +1)+1 2( +1) , the cotangent bundle of is ample. Proof. Since is big, there exist a proper closed subset of ( ) and an integer such that for each , the sections of ( ), i.e., the sections of , deﬁne an injective morphism ( Lemma 24 Let be a smooth subvariety of and let be a subvariety of ( . A general linear section of of dimension at most codim( satisﬁes ( Proof. Consider the variety (( t,x Λ) c, , t X,x ,x The ﬁbers of its projection to have codimension 2 , hence it does not

dominate c, ) as soon as 2 c > dim( ). This is equivalent to 2(dim( dim( )) 2 dim( codim( and the lemma is proved. Let be the (conical) inverse image of in the total space of the tangent bundle of . Let be a general linear section of and set + 1 2 dim( ). If = ( ,...,t ), with ,x , is a nonzero tangent vector to , the lemma implies that there are at least values of the index for which . If, say, is not in , there exists a section of that does not vanish at . This section induces, via the projection a section of that does not vanish at . It follows that ( ) is base-point-free and its sections

deﬁne a morphism ( We need to show that is ﬁnite . Assume to the contrary that a curve in ( ) through is contracted. Since the restriction of the projection ( to any ﬁber of is injective, and since is injective, the argument above proves that the curve ) is contracted by each projection such that The following lemma leads to a contradiction when 2 dim( ad + 1. This proves the propo- sition. Lemma 25 Let be a general linear section of a product in a projective space. If 2 dim( dim( ) + 1 , the projection is ﬁnite. I am grateful to Bogomolov for allowing me to

reproduce his construction. 13

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Olivier Debarre Proof. Let be the ambient projective space and let c, ). Set x,y,y Λ) x,y x,y Λ) General ﬁbers of the projection have codimension 2 . If Λ is general in and = ( Λ, the ﬁber of the projection at Λ, which is isomorphic to (( x,y x,y )) x,y V, x,y therefore has dimension at most 1 as soon as 2 dim( 1, or equivalently 2 dim( dim( ) + 1. When this holds, the projection is ﬁnite and the lemma is proved. Using his construction, Bogomolov exhibits smooth projective varieties with ample

cotangent bundle that are simply connected. More generally, his ideas give the following result. Proposition 26 Given any smooth projective variety , there exists a smooth projective surface with ample cotangent bundle and same fundamental group as Proof. By the Lefschetz hyperplane theorem, a suﬃciently ample 3-dimensional linear section of has same fundamental group as and ample canonical bundle. A smooth hyperplane section of with class ah satisﬁes ) = ) + ah )) This is positive for 0, hence the cotangent bundle of is big by a famous trick of Bogomolov ([B2]). Moreover, and

have isomorphic fundamental groups. Starting from a simply connected , we similarly obtain a simply connected surface with big cotangent bundle. Taking in Bo- gomolov’s construction , we produce a smooth simply connected projective surface with ample cotangent bundle. Taking in Bogomolov’s construction and , we produce a smooth projective surface with ample cotangent bundle and same fundamental group as References BS Bauer, Th., Szemberg, T., Higher order embeddings of abelian varieties, Math. Z. 224 (1997), 449–455. BD Beauville, A., Debarre, O., Une relation entre deux approches du probl`eme

de Schottky, Invent. Math. 86 (1986), 195–207. BG Bloch, S., Gieseker, D., The positivity of the Chern classes of an ample vector bundle, Invent. Math. 12 (1971), 112–117. B1 Bogomolov, F., Holomorphic tensors and vector bundles on projective varieties (in russian), Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), 1227–1287, 1439; English transl.: Math. USSR Izvestiya 13 (1979), 499–555. B2 Bogomolov, F., Holomorphic symmetric tensors on projective surfaces (in russian), Uspekhi Mat. Nauk 33 (1978), 171–172; English transl.: Russian Math. Surveys 33 (1978), 179–180. D1 Debarre, O., Fulton-Hansen and

Barth-Lefschetz theorems for subvarieties of abelian varieties, J. reine angew. Math. 467 (1995), 187–197. D2 Debarre, O., Tores et varietes abeliennes complexes , Cours Specialises , Societe mathematique de France, 1999. De Demailly, J.-P., Algebraic criteria for Kobayashi hyperbolic projective varieties and jet diﬀerentials, Algebraic geometry—Santa Cruz 1995, 285–360, Proc. Sympos. Pure Math. 62 , Part 2, Amer. Math. Soc., Providence, RI, 1997. L Lazarsfeld, R., Positivity in algebraic geometry II , Ergebnisse der

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Varieties with ample cotangent bundle LP Le Potier, J., Annulation de la cohomologie `a valeurs dans un ﬁbre vectoriel holomorphe positif de rang quelconque, Math. Ann. 218 (1975), 35–53. M Moriwaki, A., Remarks on rational points of varieties whose cotangent bundles are generated by global sections, Math. Res. Lett. (1995), 113–118. NS Noguchi, J., Sunada, T., Finiteness of the family of rational and meromorphic mappings into algebraic varieties, Amer. J. Math. 104 (1982), 887–900. S

Schneider, M., Symmetric diﬀerential forms as embedding obstructions and vanishing theorems, J. Algebraic Geom. (1992), 175–181. So1 Sommese, A., Submanifolds of abelian varieties, Math. Ann. 233 (1978), 229–256. So2 Sommese, A., Complex subspaces of homogeneous complex manifolds. I. Transplanting theorems, Duke Math. J. 46 (1979), 527–548. W Wong, B., A class of compact complex manifolds with negative tangent bundles, Complex analysis of several variables (Madison, Wis., 1982) , 217–223, Proc. Sympos. Pure Math. 41 , Amer. Math. Soc., Providence, RI, 1984. Olivier Debarre

debarre@math.u-strasbg.fr Institut de Recherche Mathematique Avancee, Universite Louis Pasteur, 7 rue Rene Descartes, F-67084 Strasbourg cedex, France 15

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