Varieties with ample cotangent bundle

15 Pages·2004·0.29 MB·English

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/2
sufficiently ample general hypersurfaces in a complex abelian variety of dimension n 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 Finally, we present an unpublished result of Bogomolov which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
ProjectivealgebraicvarietiesX withamplecotangentbundlehavemanyproperties:thesubvarieties
of X are all of general type; there are finitely many nonconstant rational maps from any fixed
projective variety to X ([NS]); if X is defined over C, any entire holomorphic mapping C → X is
constant ([De], (3.1)); if X is defined over a number field K, the set of K-rational points of X 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/2
sufficiently ample general hypersurfaces in an abelian variety of dimension n 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Finally, we present in section 4 an unpublished result of Bogomolov which states that a general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 E, the projective bundle P(E) is
the space of 1-dimensional quotients of the fibers of E. It is endowed with a line bundle O (1).
P(E)
We say that E is ample (resp. nef, resp. big) if the line bundle O (1) has the same property.
P(E)
Following [So1], we say more generally that given an integer k, the vector bundle E is k-ample if,
for some m > 0, the line bundle O (m) is generated by its global sections and each fiber of the
P(E)
associated map P(E) → PN has dimension 6 k. Ampleness coincide with 0-ampleness.
2000 Mathematics Subject Classification 14K12 (primary), 14M10, 14F10 (secondary).
Keywords: Ample cotangent bundle, complete intersections.
Olivier Debarre
2. Subvarieties of abelian varieties
We study the positivity properties of the cotangent bundle of a smooth subvariety of an abelian
variety A.
2.1 Preliminary material
Using a translation, we identify the tangent space T at a point x of A with the tangent space
A,x
T at the origin. We begin with a classical result.
A,0
Proposition 1. Let X be a smooth subvariety of an abelian variety A. The following properties
are equivalent:
(i) the cotangent bundle Ω is k-ample;
X
(ii) for any nonzero vector ξ in T , the set {x ∈ X | ξ ∈ T } has dimension 6 k.
A,0 X,x
Proof. The natural surjection (Ω )| → Ω induces a diagram
A X X
f
**
P(Ω ) (cid:31)(cid:127) g // P(Ω )| ’ P(Ω )×X p1 // P(Ω ) (1)
X XXXXXXXXXXXXAXXXXXXXXXXXXXAX,X0X++ (cid:15)(cid:15)p2 A,0
X
and
O (1) = g∗O (1) = f∗O (1)
P(ΩX) P(ΩA)|X P(ΩA,0)
It follows that Ω is k-ample if and only if each fiber of f has dimension 6 k ([So1], Corollary
X
1.9). The proposition follows, since the restriction of the projection P(Ω ) → X to any fiber of f
X
is injective.
Remarks 2. (1) Let d = dim(X) and n = dim(A). Since dim(P(Ω )) = 2d−1, the proof of the
X
proposition shows that the cotangent bundle of X is (2d−n)-ample at best. It is always d-ample,
and is (d−1)-ample except if X has a nonzero vector field, which happens if and only if X is stable
by translation by a nonzero abelian subvariety (generated by the vector field).
(2)ManythingscanpreventthecotangentbundleofX frombeingample.Herearetwoexamples.
• Assume X ⊃ X +X , where X and X are subvarieties of A of positive dimension. For all
1 2 1 2
x smooth on X and all x ∈ X , one has T ⊂ T , hence the cotangent bundle of
1 (cid:0) 1 (cid:1) 2 2 X1,x1 X,x1+x2
X is not dim(X )−1 -ample. In the Jacobian of a smooth curve C, the cotangent bundle of
2
any smooth W (C) is therefore exactly (d−1)-ample (although its normal bundle is ample).
d
• If A is (isogenous to) a product A ×A and X = X ∩(A ×{a }), the cotangent bundle of
1 2 a2 1 2
X is at most (2dim(X )−dim(A ))-ample, because of the commutative diagram
a2 1
f
P(Ω )| −→ P(Ω )
X (Xa2)reg A,0
∪ ∪
P(Ω ) −→ P(Ω )
(Xa2)reg A1,0
In particular, if dim(X ) > 1 dim(A ) for some a , the cotangent bundle of X cannot be
a2 2 1 2
ample.
We will encounter the following situation twice: assume F and G are vector bundles on a
projective variety X that fit into an exact sequence
0 → F → V ⊗O → G → 0 (2)
X
2
Varieties with ample cotangent bundle
where V is a vector space.
Lemma 3. In the situation above, if moreover rank(F) > dim(X), we have
F∗ ample ⇒ G nef and big
Proof. As in the proof of Proposition 1, G is nef and big if and only if the morphism f : P(G) →
P(V) induced by (2), which satisfies O (1) = f∗O (1), is generically finite.
P(G) P(V)
Let d be the dimension of X, let r be the dimension of V, let s be the rank of G, and let G be
the Grassmannian of vector subspaces of V∗ of dimension s, with tautological quotient bundle Q of
rank r−s > d. The dual of the exact sequence (2) induces a map γ : X → G such that γ∗Q = F∗.
Assume that f(cid:0)P(G)(cid:1) has dimension < dim(P(G)) = d+s−1. There exists a linear subspace
W∗ of V∗ of dimension r−d−s+1 such that P(W) does not meet f(cid:0)P(G)(cid:1). In other words, the
variety γ(X) does not meet the special Schubert variety {Λ ∈ G | Λ∩W∗ 6= {0}}, whose class is
c (Q). We obtain γ(X)·c (Q) = 0, hence 0 = c (γ∗Q) = c (F∗), and F∗ cannot be ample by
d d d d
[BG], Corollary 1.2.
2.2 Nef and big cotangent bundle
Acharacterizationofsubvarietiesofanabelianvarietywhosecotangentbundleisnefandbigfollows
easily from a result of [D1].
Proposition 4. The cotangent bundle of a smooth subvariety X of an abelian variety is nef and
big if and only if dim(X −X) = 2dim(X).
Proof. The cotangent bundle of X is nef and big if and only if the morphism f in (1) is generically
S
finite onto its image P(Ω ), i.e., if the latter has dimension 2dim(X)−1. The proposition
x∈X X,x
follows from [D1], Theorem 2.1.
The condition dim(X − X) = 2dim(X) implies of course 2dim(X) 6 dim(A). The converse
holds if X is nondegenerate ([D1], Proposition 1.4): this means that for any quotient abelian variety
π : A → B, one has either π(X) = B or dim(π(X)) = dim(X). This property holds for example
for any subvariety of a simple abelian variety.1 It has also an interpretation in terms of positivity
of the normal bundle of X.
Proposition 5. The normal bundle of a smooth subvariety X of an abelian variety is nef and big
if and only if X is nondegenerate.
Proof. The normal bundle N to X in A is nef and big if and only if the map f0 in the diagram
X/A
f0
**
P(N ) (cid:31)(cid:127) // P(T )| ’ P(T )×X // P(T ) (3)
X/A WWWWWWWWWWWWAWWXWWWWWWWWWAW,W0W++ (cid:15)(cid:15)p2 p1 A,0
X
is generically finite onto its image (i.e., surjective).
To each point p in the image of f0 corresponds a hyperplane H in T such that T ⊂ H for
p A,0 X,x p
all x in the image F in X of the fiber. This implies T ⊂ H for all x in F , hence the tangent
p Fp,x p p
space at the origin of the abelian variety K generated by F is contained in H ([D2], Lemme
p p p
VIII.1.2).
1An abelian variety A is simple if the only abelian subvarieties of A are 0 and A. For more about nondegenerate
subvarieties, see [D2], Chap. VIII.
3
Olivier Debarre
Since A has at most countably many abelian subvarieties, the abelian variety K is independent
p
of the very general point p in the image of f0. Let π : A → B be the corresponding quotient. The
differential of π| is not surjective at any point of F since its image is contained in the hyperplane
X p
Tπ(H ). By generic smoothness, π| is not surjective.
p X
If X is nondegenerate, π| is generically finite onto its image, hence F is finite and f0 is
X p
generically finite onto its image. It follows that N is nef and big.
X/A
Conversely, assume that N is nef and big. Let π : A → B be a quotient of X such that
X/A
π(X) 6= B. The tangent spaces to X along a general fiber of π| are all contained in a fixed
X
hyperplane. This fiber is therefore finite, hence X is nondegenerate.
Proposition 6. Let X be a smooth subvariety of an abelian variety A, of dimension at most
1 dim(A). We have
2
Ω ample ⇒ N nef and big ⇒ Ω nef and big
X X/A X
Proof. ThefirstimplicationfollowsfromLemma3appliedtotheexactsequence0 → T → T | →
X A X
N → 0.
X/A
The second implication follows from Propositions 4 and 5 and the fact that for a nondegenerate
subvariety X of A, the equality dim(X − X) = min(2dim(X),dim(A)) 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 A has ample cotangent bundle, provided that its dimension be at most 1 dim(A).
2
We begin by fixing some notation. If A is a smooth variety, ∂ a vector field on A, and L a line
bundleonA,wedefine,foranysectionsofLwithdivisorH,asection∂sofL| bytherequirement
H
∼
that for any open set U of A and any trivialization ϕ : O −→ L| , we have ∂s = ϕ(∂(ϕ−1(s)))|
U U H
in U ∩H. We denote its zero locus by H ∩∂H. We have an exact sequence
H0(A,L) −→ H0(H,L| ) −→ H1(A,O )
H A
∂s 7−→ ∂ ^ c (L)
1
where c (L) is considered as an element of H1(A,Ω ) and the cup product is the contraction
1 A
H0(A,T )⊗H1(A,Ω ) −→ H1(A,O )
A 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 7. Let L ,...,L be very ample line bundles on a simple abelian variety A of dimension
1 c
n. Consider general divisors H1 ∈ |Le11|,...,Hc ∈ |Lecc|. If e2,...,ec are all > n, the cotangent
bundle of H ∩···∩H is max(n−2c,0)-ample.
1 c
Proof. We need to prove that the fibers of the map f in (1) have dimension at most m = max(n−
2c,0). This means that for H general in |Lei| and any nonzero constant vector field ∂ on A, the
i i
dimension of the set of points x in X = H ∩···∩H such that ∂(x) ∈ T is at most m; in other
1 c X,x
words, that
dim(H ∩∂H ∩···∩H ∩∂H ) 6 m
1 1 c c
It is enough to treat the case c 6 n/2. We proceed by induction on c, and assume that the variety
Y = H ∩∂H ∩···∩H ∩∂H has codimension 2c−2 in A for all nonzero ∂. Let Y ,...,Y
∂ 1 1 c−1 c−1 ∂,1 ∂,q
be its irreducible components.
4
Varieties with ample cotangent bundle
LetU (Y )betheopensetofdivisorsH in|Le|suchthat(Y ) ∩H isintegralofcodimension
e ∂,i c ∂,i red
1 in Y . If H ∈ U (Y ), I claim that Y ∩ H ∩ ∂H has codimension 2 in Y . Indeed, let
∂,i e ∂,i ∂,i ∂,i
s ∈ H0(A,Le) define H and set Y = (Y ) . The scheme Y ∩H ∩∂H is the zero set in Y ∩H of
c ∂,i red
the section ∂s defined above. In the commutative diagram,
H0(H,Le| ) // H1(A,O ) (4)
c H A
(cid:31) //
∂s ∂ ^ ec1(Lc) ρ
(cid:29)(cid:29) (cid:2)(cid:2)
H0(Y ∩H,Le| ) // H1(Y,O )
c Y∩H Y
the restriction ρ is injective because Y generates A, hence ∂s does not vanish identically on the
integral scheme Y ∩H.
It follows that for H ∈ U (Y ) = Tq U (Y ), the scheme Y ∩H ∩∂H has codimension 2c in
e ∂ i=1 e ∂,i ∂
A. Thus, for H ∈ T U (Y ), the intersection
c [∂]∈P(ΩA,0) e ∂
H ∩∂H ∩···∩H ∩∂H
1 1 c c
has codimension 2c in A for all nonzero constant vector field ∂ on A (note that when c = 1, there
is no condition on H ). Lemma 12, to be proved in 2.3.4, shows that the complement of U (Y ) in
1 e ∂
|Le| has codimension at least e−1. For e > n, the intersection T U (Y ) is therefore not
c [∂]∈P(ΩA,0) e ∂
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 ,...,e .
2 c
Theorem 8. Let L ,...,L be very ample line bundles on an abelian variety A of dimension n. For
1 c
e2,...,ec largeanddivisibleenoughpositiveintegersandgeneraldivisorsH1 ∈ |Le11|,...,Hc ∈ |Lecc|,
the cotangent bundle of H ∩···∩H is max(n−2c,0)-ample.
1 c
Let us be more precise about the condition on the e . What we mean is that there exists for
i
each i ∈ {1,...,c−1} a function δ : Ni → N∗ such that the conclusion of the theorem holds if
i
e = e0δ (e ), e = e0δ (e ,e,)...,e = e0δ (e ,...,e )
2 2 1 1 3 3 2 1 2 c c c−1 1 c−1
with e0,...,e0 > n (5)
2 c
Proof. We keep the setting and notation of the proof of Theorem 7. Everything goes through
except when, in diagram (4), ρ(∂ ^ c (L )) = 0. In this case, let A00 be the abelian subvariety of A
1 c
generated by Y and let A0 be its complement with respect to L , so that the addition
c
π : A0×A00 → A
is an isogeny and π∗L ’ L | (cid:2) L | . We have Y = a0 + Y00, with a0 ∈ A0, Y00 ⊂ A00, and
c c A0 c A00
∂ ∈ H0(A0,T ). In particular, we have an injection
A0
H0(A,L ) ,→π∗ H0(A0,L | )⊗H0(A00,L | )
c c A0 c A00
It is however difficult to identify in a manner useful for our purposes the sections of L inside this
c
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 π, being finite, is contained in the group of r-torsion
points of A0×A00 for some positive integer r. Multiplication by r factors as
A0×A00 −π→ A −π→0 A0×A00
5
Olivier Debarre
and π0∗(Lc|A0 (cid:2)Lc|A00) is some power Lec0 of Lc. Sections of Lec0 that come from H0(A0,Lc|A0)⊗
H0(A00,L | ) induce a morphism from A to some projective space that factors through π0 and
c A00
embeds A0×A00.
We will consider sections of Lee0 of the type π0∗s, with s ∈ H0(A0,Lec|A0)⊗H0(A00,Lec|A00). If
the divisor H of s on A0×A00 corresponds to a degree e hypersurface in U (π0(Y)), the intersection
e
π0(Y)∩H is integral of codimension 1 in π0(Y) = {ra0}×rY00.
Fix a basis (s00,...,s00) for H0(A00,Le| ) and write s = Pd s0 ⊗s00, so that
1 d c A00 i=1 i i
d
X
π0(Y)∩H = π0(Y)∩div( s0(ra0)s00)
i i
i=1
d d
X X
π0(Y)∩H ∩∂H = π0(Y)∩div( s0(ra0)s00)∩div( ∂s0(ra0)s00)
i i i i
i=1 i=1
Sinceπ0(Y)∩H isintegral,π0(Y)∩H∩∂H hascodimension2inπ0(Y)(henceY∩π0−1(H)∩∂π0−1(H)
has codimension 2 in Y) unless, for some complex number λ, the section Pd (λs0 +∂s0)(ra0)s00 of
i=1 i i i
Le| vanishes on rY00. In other words, if we let
c A00
d
X
Γ = {(a ,...,a ) ∈ Cd | a s00 vanishes on rY00}
rY00 1 d i i
i=1
and
(cid:18) s0(ra0) ··· s0(ra0) (cid:19)
M = 1 d
∂ ∂s0(ra0) ··· ∂s0(ra0)
1 d
we have (λ,1)·M ∈ Γ . Now we may pick any collection (s0,...,s0) we like. Fix one such that
∂ rY00 1 d
the corresponding matrix M has rank 2 for all nonzero ∂ and apply a square matrix N of size
∂
dim(A0). The condition is now that the composition
Im(tM ) ⊂ Cd −t→N Cd −→ Cd/Γ
∂ rY00
is not injective, that is,
• either tN ·Im(tM )∩Γ 6= {0}, which imposes codim(Γ )−1 conditions on N;
∂ rY00 rY00
• or Ker(tN)∩Im(tM ) 6= {0}, which imposes d−1 conditions on N.
∂
The “bad” locus for H corresponds to the space of matrices N that satisfy either one of these
properties for some nonzero ∂ ∈ H0(A0,T ). Since, on the one hand d = h0(A00,Le| ) > e
A0 c A00
and, on the other hand, the codimension of Γ is the rank of the linear map H0(A00,Le| ) →
rY00 c A00
H0(rY00,Le| ), which is > e, the codimension of the “bad” locus is at least e−dim(A0)+2.
c rY00
This means that for A00 (hence A0) fixed, e > n, and H general in |Lee0|, for any component Y of
c
Y that spans (as a group) A00, the intersection Y ∩H∩∂H has codimension 2 in Y for all nonzero
∂
∂ in H0(A,T ).
A
Since A has at most countably many abelian subvarieties, there are only finitely many different
abeliansubvarieties spannedbycomponents of Y = H ∩∂H ∩···∩H ∩∂H forH ,...,H
∂ 1 1 c−1 c−1 1 c−1
general in |Le1|,...,|Lec−1| as ∂ runs through the nonzero elements of H0(A,T ). Therefore, for
1 c−1 A
some positive integer δ, any e > n, and H general in |Leδ|, the intersection Y ∩ H ∩ ∂H has
c ∂
codimension 2 in Y for all nonzero ∂ ∈ H0(A,T ). This proves our claim by induction on c hence
∂ A
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.
6
Varieties with ample cotangent bundle
Theorem 9. Let L and L be line bundles on an abelian fourfold A, with L ample and L very
1 2 1 2
ample. For e > 5, e > 5, and H ∈ |Le1| and H ∈ |Le2| general, the surface H ∩H has ample
1 2 1 1 2 2 1 2
cotangent bundle.
Proof. I claim that for H general in |Le1|, the scheme Y = H ∩∂H is an integral surface for
1 1 ∂ 1 1
each nonzero vector field ∂ on A. Granting the claim for the moment and using the notation of the
proof of Theorem 7, the scheme H ∩∂H ∩H is then, for H ∈ U (Y ), an integral curve that
1 1 2 2 e2 ∂
generates A since its class is e H2H . The argument of the proof of Theorem 7 applies in this case
2 1 2
to prove that H ∩∂H ∩H ∩∂H is finite. Taking H in T U (Y ) (which is possible
1 1 2 2 2 [∂]∈P(ΩA,0) e2 ∂
by Lemma 12 since e > 4), the intersection
2
H ∩∂H ∩H ∩∂H
1 1 2 2
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 A be an abelian variety of dimension at least 4 and let L be an ample divisor on
A. For e > 5 and H general in |Le|, the scheme H ∩∂H is integral for all nonzero ∂ ∈ H0(A,T ).
A
Proof. Assume to the contrary that for some smooth H ∈ |Le|, we have H ∩ ∂H = D0 + D0,
1 2
where D0 and D0 are effective nonzero Cartier divisors in H. We follow [BD], proposition 1.6: since
1 2
dim(H) > 3, there exist by the Lefschetz Theorem divisors D and D on A such that D +D ≡ H
1 2 1 2
and D | ≡ D0. Since D0 is effective, the long exact sequence in cohomology associated with the
i H i i
exact sequence
0 → O (D −H) → O (D ) → O (D0) → 0
A i A i H i
shows that, for each i ∈ {1,2}, either H0(A,D ) 6= 0 or H1(A,D −H) 6= 0. The case where both
i i
H1(A,D −H) and H1(A,D −H) are zero is impossible, since we would then have a section of Le
1 2
with divisor H ∩∂H on H. The case where both H1(A,D −H) and H1(A,D −H) are nonzero
1 2
is impossible as in loc. cit. because dim(A) > 3.
So we may assume H1(A,D −H) 6= 0 and H1(A,D −H) = 0, and take D effective such that
2 1 1
D ∩H = D0.
1 1
As in loc. cit., A contains an elliptic curve E such that, if B is the neutral component of the
kernel of the composed morphism
A −φ→H Pic0(A) → Pic0(E)
the addition map π : E ×B → A is an isogeny, ∂ is tangent to E, and π∗(D ) = p∗(D ) for some
1 1 E
effective divisor D on E. Picka basis (t ,...,t ) for H0(B,Le| ) and asection s of Le withdivisor
E 1 d B
H, and write
d
X
π∗s = s ⊗t
i i
i=1
with s ,...,s ∈ H0(E,Le| ), so that π−1(cid:0)H ∩∂H(cid:1) is defined by
1 d E
d d
X X
s ⊗t = ∂s ⊗t = 0
i i i i
i=1 i=1
Since D0 = H ∩D is contained in H ∩∂H, for every point x of the support of D , we have
1 1 E
d d
(cid:16)X (cid:17) (cid:16)X (cid:17)
div s (x)t ⊂ div ∂s (x)t ⊂ B
i i i i
i=1 i=1
7
Olivier Debarre
Since these two divisors belong to the same linear series |Le| | on B, they must be equal and
B
(cid:18) (cid:19)
s (x) ··· s (x)
rank 1 d 6 1
∂s (x) ··· ∂s (x)
1 d
Since H is irreducible, the sections s ,...,s have no common zero and the morphism ψ : E →
1 d H
Pd−1 that they define is ramified at x.
The vector subspace of H0(E,Le| ) generated by s ,...,s only depends on s, not on the choice
E 1 d
of the basis (t ,...,t ). If b ,...,b are general points of B, it is also generated by s(·+b ),...,s(·
1 d 1 d 1
+b ) and
d
(cid:18) (cid:19)
s(x+b ) ··· s(x+b )
rank 1 d 6 1
∂s(x+b ) ··· ∂s(x+b )
1 d
Assume now that the conclusion of the lemma fails for general H (and s). The point x varies with
s, but remains constant for s in a hypersurface H of H0(X,Le). If s is in
x
H0 = H ∩{t ∈ H0(X,Le) | t(x+b ) = t(x+b ) = 0}
x x 1 2
it also satisfies ∂s(x+b ) = ∂s(x+b ) = 0. Since H0 has codimension at most 3 in H0(X,Le), this
1 2 x
means that Le is not 3-jet ample and contradicts Theorem 1 of [BS]: the lemma is proved.
Remarks 11. (1) Let A be an abelian fourfold that contains no elliptic curves. The proof of
Lemma 10 shows that for any smooth ample hypersurface H in A and any nonzero ∂ ∈ H0(A,T ),
A
the scheme H ∩∂H is integral. It follows that for L very ample, e > 5, and H0 ∈ |Le| general, the
surface H ∩H0 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.2
Lemma 12. Let Y be an integral subscheme of Pn of dimension at least 2 and let V be the
e,n
projective space of hypersurfaces of degree e in Pn. The codimension of the complement V (Y) of
e
U (Y) = {F ∈ V | Y ∩F is integral of codimension 1 in Y }
e e,n
in V is at least e−1.
e,n
Proof. By taking hyperplane sections, we may assume that Y is a surface. We proceed by induction
on n. For n = 2, this codimension is
(cid:18)(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:19)
e+2 k+2 e−k+2
min − − +1 = e−1
16k6e−1 2 2 2
Assume n > 3. Let V be a component of V (Y) of maximal dimension and let C be the linear
e e,p
subspace of V that consists of cones with vertex a point p. If V does not meet C , we have
e,n e,p
(cid:18) (cid:19)
n−1+e
codim(V ) > dim(C )−1 = −1 > e−1
e,p
e
and the lemma is proved. We will therefore assume that V meets C . Let π : Pn {p} → Pn−1 be
e,p
a projection. If F is general in V ∩C ,
e,p
2It is a pleasure to acknowledge Zak’s help with the proof of this lemma.
8
Varieties with ample cotangent bundle
• either π(F)∩π(Y) is not integral of dimension 1, the induction hypothesis yields
codimVeV > codimCe,p(V ∩Ce,p)
> codimVe,n−1(Ve(π(Y))∩Ve,n−1)
> e−1
and the lemma is proved;
• or the curve π(F)∩π(Y) is integral of dimension 1, but is contained in the locus E over which
the finite morphism π| : Y → π(Y) is not an isomorphism.
Y
In the second case, if n > 4, the morphism π| is birational, E has dimension at most 1, hence
Y
codimVe,nV > codimCe,p(V ∩Ce,p)
> codimVe,n−1{F ∈ Ve,n−1 | F contains a component of E}
> e+1
where the last inequality holds because any e+1 points in Pn−1 impose independent conditions on
hypersurfaces of degree e. The lemma is proved in this case.
We are reduced to the case n = 3: the curve C = π(F) ⊂ P2 is integral and its inverse image
F ∩Y by π| : Y → P2, is reduced but reducible.
Y
We consider the following degeneration. Denote by G an equation for Y; the surface Y defined
t
by G(tx ,x ,x ,x ) = 0 is projectively equivalent to Y for t 6= 0, whereas Y is the cone with vertex
0 1 2 3 0
(1,0,0,0) and base Y ∩(x = 0). We may therefore assume that Y is an integral cone with vertex
0
a point p0 6= p and we let π0 : Y {p0} → C0 ⊂ P2 be the projection. Let O be the intersection
of the line pp0 with the plane P2. Pick a line L in P2, avoiding O, and consider the projections
C {O} → L and C0 → L from O (the point O might be on C (if F 3 p0), but is not on C0, because
p ∈/ Y). The maps
(F ∩Y) {p0} −→ (C {O})× C0
L
x 7−→ (cid:0)π(x),π0(x)(cid:1)
and
(C {O})× C0 −→ (F ∩Y) {p0}
L
(y,y0) 7−→ py∩p0y0
are inverse one to another. Therefore, given the integral curve C0, we need to study the dimension
of the set of curves C of degree e for which the curve C × C0 is reducible.
L
Usingthesametrickasabove,wedegenerateC0 totheunionC0 ofdeg(C0)distinctlinesthrough
0
some point. At the limit, C × C0 is the union of deg(C0) curves isomorphic to C. If C is integral,
L 0
the projection C× C0 → C0 has the property that every irreducible component of C0 is dominated
L 0 0 0
by a unique component of C × C0, and the set of “bad” curves has codimension e−1 as we saw
L 0
in the case n = 2. This property of the projection, begin open, carries over to C × C0 → C0. This
L
finishes the proof of the lemma.
2.4 Cohomology of symmetric tensors
Let X be a smooth subvariety of an abelian variety. We are interested in the cohomology groups of
the vector bundles SrΩ .
X
Proposition 13. Let A be an abelian variety of dimension n and let X be a smooth subvariety of
codimension c of A with ample normal bundle. For r > 0, the restriction
Hq(A,SrΩ ) −→ Hq(X,SrΩ )
A X
9
Olivier Debarre
is bijective for q < n−2c and injective for q = n−2c.3
Proof. We follow the ideas of [S]. The symmetric powers of the exact sequence 0 → N∗ →
X/A
Ω | → Ω → 0 yield, for each r > 0, a long exact sequence
A X X
0 → ∧cN∗ ⊗Sr−cΩ → ··· → N∗ ⊗Sr−1Ω → SrΩ | → SrΩ → 0
X/A A X/A A A X X
By Le Potier’s vanishing theorem ([LP]; [L], Remark 7.3.6), Hq(X,∧iN∗ ) vanishes for n−c−q >
X/A
c−i and i > 0. Since Ω is trivial, we get, by an elementary homological algebra argument ([S],
A
Lemma, p. 176),
Hq(X,Ker(SrΩ | → SrΩ )) = 0 for all q 6 n−2c.
A X X
The proposition now follows from the fact that the restriction Hq(A,O ) → Hq(X,O ), hence also
A X
the restriction Hq(A,SrΩ ) → Hq(X,SrΩ | ), is bijective for q 6 n−2c ([So2]).
A A X
Sommese proved ([So1], Proposition (1.7)) that for any k-ample vector bundle E on a projective
variety X and any coherent sheaf F on X,
Hq(X,SrE ⊗F) = 0
for all q > k and r (cid:29) 0. Theorem 7 and Proposition 13 therefore imply the following.
Corollary 14. Let X be the intersection of c sufficiently ample4 general hypersurfaces in an
abelian variety A of dimension n. We have

= 0 for q > max{n−2c,0} and r (cid:29) 0


hq(X,SrΩ ) = hq(A,SrΩ ) for q < n−2c and r > 0
X A

> hq(A,SrΩ ) for q = n−2c and r > 0.
A
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 X is a smooth subvariety of Pn of dimension d, we let γ : X → G(d,Pn) be the Gauss map.
X
We denote by S the universal subbundle and by Q the universal quotient bundle on G(d,Pn). We
3For the case q = 0, Bogomolov gave in [B2] a very nice proof that goes as follows. Arguing as in the proof of
Proposition4,wefindthatthemorphismf of(1)issurjectivewheneverX−X =A.Anyfiberoff isisomorphicto
its projection to X, which is the zero locus of a section of N . It follows that when N is ample and c<n−c,
A/X A/X
the fibers of f are connected, hence f O (r)’O (r), from which we get, for all r>0,
∗ P(ΩX) P(ΩA,0)
H0(X,SrΩ )’H0(P(Ω ),O (r))
X X P(ΩX)
’H0(P(Ω ),O (r))’H0(A,SrΩ )
A,0 P(ΩA,0) A
4To be more precise, we need condition (5) to be satisfied.
10