VARIETIES WITH AMPLE COTANGENT BUNDLE University of Michigan Harvard University Princeton University March–April 2004

VARIETIES WITH AMPLE COTANGENT BUNDLE

University of Michigan Harvard University Princeton University

March–April 2004

Olivier Debarre

1 Various notions of hyperbolicity for com- plex varieties

There exist several notions of hyperbolicity for compact complex varieties.

For a compact Riemann surface C, they all coincide. “Hyperbolic” means any of the following equivalent properties:¹

(H1) C has genus ≥2;

(H2) C carries a metric with constant negative curvature;

(H3) any holomorphic map C→C is constant;

(H4) ifC is defined over a number field K, it has finitely many K-points.

For a higher-dimensional projective complex algebraic varietyX, we may generalize these properties in several ways as follows.

1The equivalence of the first three of these properties follows from the fact that a compact Riemann surface of genus≥2 is covered by the unit diskB⊂C. The fact that smooth projective curves of genus at least 2 defined over a number field K have finitely manyK-points is a difficult theorem of Faltings.

(H1) Algebraic hyperbolicity:

(H11) all algebraic subvarieties ofX are of general type;

(H12) any map from an abelian variety toX is constant;

(H13) (Demailly) given an ample line bundle LonX, there existsε >0 such that, for every smooth projective curve C of genus g and nonconstant map f :C →X, one has

2g−2≥εdeg(f^∗L)>0 (1) (This is independent of the choice ofL.)

(H2) Metric hyperbolicity:

(H21) X carries a K¨ahler metric with negative bisectional holomorphic curvature (or negative usual sectional Riemannian curvature);

(H22) the cotangent bundle ofX is ample;

(H3) Analytic hyperbolicity: any holomorphic mapC→X is constant;

(H4) Arithmetic hyperbolicity: ifX is defined over a number field K, it has finitely many K-points.

The following diagram describes the known and conjectural implications be- tween these properties.

X carries a K¨ahler metric with negative sectional Riemannian

curvature

Ω_X ample

s{nnnnnnnnnnnnnn

nnnnnnnnnnnnnn

#+P

PP PP PP PP PP PP P

PP PP PP PP PP PP PP

All subvarieties ofX are of general type

Any holomorphic map C→X is constant

conjecturally all equivalent

&&

LL LL LL LL LL LL

xxrrrrrrrrrrrrr

88p

pp pp pp pp pp pp p ffNNNNN

NNNNNNNN

Any holomorphic map from a complex torus

toX is constant

!)K

K K K K K

K K K K KK

X is algebraically hyperbolic in Demailly’s sense

ks u}s ss ss s

s ss ss s X has finitely

many points over a number field

Here is for example how the implication on the bottom row is proved: if we have a complex torus A and a nonconstant map f : A → X, consider a smooth curve C ⊂ A on which f is not constant. If m_A : A → A is the multiplication by m, we have deg((m_A◦f|_C)^∗L) =m²deg(f|^∗_CL), which eventually violates (1).

2 Examples

(1) The product of two analytically hyperbolic varieties is analytically hyper- bolic (in particular, any (finite) product of Riemann surfaces of genus ≥2 is analytically hyperbolic). However, its cotangent bundle is not ample.

(2) Subvarieties of a complex torus. All the dotted implications in the diagram above are known to hold for a subvariety of a complex torus. This follows from the following celebrated theore.

Theorem 1 (Bloch, Ochiai, Noguchi) Let A be a complex torus. The Zariski closure of the image of a holomorphic map C→A is a (translated) subtorus of A.

In particular, a subvariety of a complex torus is analytically hyperbolic if and only if it contains no nonzero (translated) subtorus.

(3) Hypersurfaces in the projective space. The cotangent bundle of a smooth hypersurface in Pⁿ is never ample for n ≥ 3. However, it is conjectured that a very general hypersurface in Pⁿ of degree ≥ 2n −1 is analytically hyperbolic (Kobayashi). This is known (Demailly, El Goul, Siu) if the degree is large enough. It is also known that very general hypersurfaces in Pⁿ of degree ≥ 2n−1 are algebraically hyperbolic in Demailly’s sense (Clemens, Voisin, Xu), and that all their subvarieties are of general type (Pacienza).

(4) Any compact complex variety covered by a bounded domain in Cⁿ is analytically hyperbolic.

(5) Any smooth complex projective variety that is uniformized by the ball B_n inherits from the Bergman metric a metric with negative sectional Rie- mannian curvature hence has ample cotangent bundle.

It is a consequence of the Borel–Hirzebruch proportionality theorem that such a variety must satisfycⁿ⁻²₁ (c²₁−2ⁿ⁺¹_n c₂) = 0. Conversely, Yau proved, as a consequence of the Calabi conjecture, that any smooth compact complex variety of dimension n with ample canonical bundle such that cⁿ⁻²₁ (c²₁ − 2ⁿ⁺¹_n c₂) = 0 is covered by B_n.

Unfortunately, such examples are difficult to construct. Borel constructed in 1963 compact quotients of B₂ by a discontinuous group of analytic auto- morphisms. Mumford constructed in 1979 a “fake projective plane” (with

c²₁ = 3, c₂ = 1, and b₁ = 0). Hirzebruch considered in 1983 minimal desin- gularizations of certain coverings of the projective plane branched along a union of lines. Computing directly their Chern numbers, he found c²₁ = 3c2

for 3 particular configurations of lines.

(6) Mostow and Siu constructed in 1980 a compact K¨ahler surface not covered by the ball B2, with negative sectional Riemannian curvature, hence ample cotangent bundle.

(7) Bogomolov proved that in the product of sufficiently many varieties with big cotangent bundle (for instance, curves of genus at least 2), the intersec- tion of sufficiently many sufficiently ample general hypersurfaces has ample cotangent bundle. The precise result is as follows.

Proposition 2 (Bogomolov) Let X₁, . . . , X_m be smooth projective vari- eties with big cotangent bundle, all of dimension at least d >0. Let V be a general linear section of X1× · · · ×Xm. If dim(V)≤ ^d(m+1)+1_2(d+1) , the cotangent bundle of V is ample.

3 Subvarieties of abelian varieties

LetX be a smooth subvariety of an abelian varietyA. The natural surjection (Ω_A)|_X →Ω_X induces a diagram²

P(ΩX)

f

**  ^g // P(Ω_A)|_X 'P(Ω_A,0)×X _p

1 // P(Ω_A,0) X

p2

++

XX XX XX XX XX XX XX XX XX XX XX XX XX

XX (2)

and

OP(ΩX)(1) =g^∗OP(ΩA)|_X(1) = f^∗OP(ΩA,0)(1)

It follows that Ω_X is ample if and only if f is finite, i.e., if and only if, for any nonzero vector ξ inTA,0, the set

{x∈X |ξ∈T_X,x}

2Given a vector bundle E, the projective bundleP(E) is the space of 1-dimensional quotients of the fibers ofE. It is endowed with a line bundleOP(E)(1). We say thatE is ample (resp. nef, resp. big) if the line bundleOP(E)(1) has the same property.

is finite (by translation, the tangent spaceT_A,x at a pointxof Ais identified with the tangent space T_A,0 at the origin).

This is possible only if dim(X) ≤ dim(A)/2. Even when this condition is satisfied, many things can prevent Ω_X from being ample. For example, if X ⊃X₁+X₂, where X₁ and X₂ are subvarieties of Aof positive dimension, then, for all x1 smooth on X1, one has TX1,x1 ⊂ TX,x1+x2 for all x2 ∈ X2, hence the cotangent bundle of X is not ample. In the Jacobian of a smooth curveC, the cotangent bundle of any smoothW_d(C), withd≥2, is therefore not ample.

My main result is the following.

Theorem 3 Let L be a very ample line bundle on a simple abelian variety A of dimension n. Consider general divisorsH₁ ∈ |L^e1|, . . . , H_c ∈ |L^ec|, with c≥n/2 and e₂, . . . , e_c > n. The cotangent bundle of H₁∩ · · · ∩H_c is ample.

Proof. We will prove that the fibers of the mapf in (2) have dimension at most m = max(n−2c,0). This means that for H_i general in |L^ei| and any nonzero constant vector field∂ on A, the dimension of the set of points xin X =H₁∩ · · · ∩H_c such that ∂(x)∈T_X,x is at mostm; in other words, that³

dim(H1∩∂H1∩ · · · ∩Hc ∩∂Hc)≤m

It is enough to treat the case c ≤ n/2. We proceed by induction on c, and assume that the varietyY_∂ =H₁∩∂H₁∩ · · · ∩Hc−1∩∂Hc−1, with irreducible components Y∂,1, . . . , Y∂,m, has codimension 2c−2 in A for all nonzero ∂.

LetVe(Y_∂,i) be the complement in |L^e| of the set of divisors H such that (Y_∂,i)_red∩H is integral of codimension 1 inY_∂,i. IfH /∈Ve(Y_∂,i), I claim that Y∂,i∩H∩∂H has codimension 2 in Y∂,i. Indeed, lets ∈H⁰(A, L^e) define H

3For any section sof a line bundle L onA, with divisor H, the section∂s of L|_H is defined by the requirement that for any open setU ofAand any trivializationϕ:OU −→∼ L|U, we have∂s=ϕ(∂(ϕ⁻¹(s)))|H in U ∩H. We denote its zero locus byH∩∂H. We have an exact sequence

H⁰(A, L) −→ H⁰(H, L|_H) −→ H¹(A,OA)

∂s 7−→ ∂ ^ c₁(L)

wherec1(L) is considered as an element ofH¹(A,ΩA) and the cup product is the contrac- tion

H⁰(A, T_A)⊗H¹(A,Ω_A)−→H¹(A,OA)

and set Y = (Y_∂,i)_red. The scheme Y ∩H∩∂H is the zero set in Y ∩H of the section ∂s defined in note 3. In the commutative diagram,

H⁰(H, L^e|_H) //

H¹(A,OA)

ρ

∂s // ∂ ^ e c₁(L) H⁰(Y ∩H, L^e|_Y∩H) // H¹(Y,OY)

(3)

the restriction ρis injective becauseY generatesA, hence∂sdoes not vanish identically on the integral scheme Y ∩H.

It follows that forH /∈Ve(Y_∂) = Sm

i=1Ve(Y_∂,i), the scheme Y_∂∩H∩∂H has codimension 2cinA. Thus, for H_c ∈/ S

[∂]∈P(ΩA,0)Ve(Y_∂), the intersection H₁∩∂H₁∩ · · · ∩H_c ∩∂H_c

has codimension 2cinAfor all nonzero constant vector field∂ onA(note that whenc= 1, there is no condition onH₁). Lemma 4 below shows thatVe(Y_∂) has codimension at leaste−1 in|L^e|. Fore > n, the unionS

[∂]∈P(Ω_A,0)Ve(Y_∂) is therefore not the whole of |L^e| and the theorem follows.

Lemma 4 LetY be an integral subscheme ofPⁿ of dimension at least2and let Ve,n =Ve be the projective space of hypersurfaces of degree e in Pⁿ. The codimension of

Ve(Y) =Ve {F ∈Ve |Y ∩F is integral of codimension 1 in Y } in Ve is at least e−1.

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

1≤k≤e−1min

e+ 2 2

−

k+ 2 2

−

e−k+ 2 2

+ 1

=e−1

Assume n ≥3. Let V be a component of Ve(Y) of maximal dimension and letCe,pbe the linear subspace ofVethat consists of cones with vertex a point p. If V does not meet Ce,p, we have

codim(V)≥dim(Ce,p)−1 =

n−1 +e e

−1> e−1

and the lemma is proved. We will therefore assume that V meets Ce,p. Let π :Pⁿ {p} →Pⁿ⁻¹ be a projection. If F is general in V ∩Ce,p,

• either π(F)∩π(Y) is not integral of dimension 1, the induction hy- pothesis yields

codim_VeV ≥ codim_Ce,p(V ∩Ce,p)

≥ codim_Ve,n−1(Ve(π(Y))∩V^e,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 :Y → π(Y) is not an isomorphism.

In the second case, ifn≥4, the morphismπ|_Y is birational,E has dimension at most 1, hence

codim_VeV ≥ codim_Ce,p(V ∩Ce,p)

≥ codim_Ve,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 :Y →P², is reduced but reducible.

We consider the following degeneration. Denote by Gan equation for Y; the surfaceY_tdefined byG(tx₀, x₁, x₂, x₃) = 0 is projectively equivalent toY fort6= 0, whereasY₀ is the cone with vertex (1,0,0,0) and baseY∩(x₀ = 0).

We may therefore assume that Y is an integral cone with vertex 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 projectionsC {O} →L andC⁰ →LfromO (the point O might be on C (if F 3p⁰), but is not on C⁰, because p /∈Y). The maps

(F ∩Y) {p⁰} −→ (C {O})×_LC⁰ x 7−→ π(x), π⁰(x) and

(C {O})×_LC⁰ −→ (F ∩Y) {p⁰} (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 curvesC of degree efor which the curve C×LC⁰ is reducible.

Using the same trick as above, we degenerate C⁰ to the union C₀⁰ of deg(C⁰) distinct lines through some point. At the limit, C ×_L C₀⁰ is the union of deg(C⁰) curves isomorphic to C. If C is integral, the projection C ×_LC₀⁰ → C₀⁰ has the property that every irreducible component of C₀⁰ is dominated by a unique component of C×_LC₀⁰, and the set of “bad” curves has codimension e−1 as we saw in the case n = 2. This property of the projection, begin open, carries over toC×_LC⁰ →C⁰. This finishes the proof

of the lemma.

The assumption that A be simple is unnecessary, although the proof be- comes quite complicated without it. Here is an example, in case the ambiant abelian variety has dimension 4.

Theorem 5 Let L be a very ample line bundle on an abelian variety A of dimension 4. For e₁ and e₂ ≥5, and H₁ ∈ |L^e1| and H₂ ∈ |L^e2| general, the surface H₁∩H₂ has ample cotangent bundle.

Proof. I claim that for H₁ general in |L^e1|, the scheme Y_∂ = H₁ ∩∂H₁ is an integral surface for each nonzero vector field ∂ on A. Granting the claim for the moment and using the notation of the proof of Theorem 3, the scheme H₁∩∂H₁∩H₂ is then, forH₂ ∈ |L^e2| Ve2(Y_∂), an integral curve that generates Asince its class ise₂H₁²H₂. The argument of the proof of Theorem 3 applies in this case to prove thatH₁∩∂H₁∩H₂∩∂H₂ is finite. TakingH₂ outside S

[∂]∈P(Ω_A,0)Ve2(Y_∂) (which is possible by Lemma 4 since e₂ >4), the intersection

H₁∩∂H₁∩H₂∩∂H₂

is finite for all nonzero vector fields ∂, which is what we need. The theorem therefore follows from the claim, itself a consequence of the next lemma.

Lemma 6 Let A be an abelian variety of dimension at least 4 and let L be an ample divisor on A. For e≥5andH general in|L^e|, the schemeH∩∂H is integral for all nonzero ∂ ∈H⁰(A, T_A).

Proof. Assume to the contrary that for some smooth H ∈ |L^e|, we have H∩∂H =D₁⁰ +D₂⁰, whereD₁⁰ and D₂⁰ are effective nonzero (Cartier) divisors

in H. Since dim(H)≥ 3, there exist by the Lefschetz Theorem divisors D₁ and D₂ onA such that D₁ +D₂ ≡H and D_i|_H ≡D⁰_i. Since D_i⁰ is effective, the long exact sequence in cohomology associated with the exact sequence

0→OA(D_i−H)→OA(D_i)→OH(D⁰_i)→0

shows that, for each i∈ {1,2}, either H⁰(A, D_i)6= 0 or H¹(A, D_i−H)6= 0.

The case where bothH¹(A, D₁−H) andH¹(A, D₂−H) are zero is impossible, since we would then have a section of L^e with divisor H ∩∂H on H. The case where bothH¹(A, D₁−H) andH¹(A, D₂−H) are nonzero is impossible because we would have⁴

dim(A) =i−(−H) = i−(D₁−H+D₂−H)≤i−(D₁−H) +i−(D₂−H)≤2 So we may assume H¹(A, D2−H)6= 0 andH¹(A, D1−H) = 0, and takeD1

effective such that D₁∩H =D₁⁰. Furthermore,

i−(D₁) = 0 , i₊(D₁) = 4−i−(−D₁) = 4−i−(D₂−H) = 3

It follows that A contains an elliptic curve E such that, if B is the neutral component of the kernel of the composed morphism

A−→^φH Pic⁰(A)→Pic⁰(E)

the addition map π : E ×B → A is an isogeny, ∂ is tangent to E, and π^∗(D₁) = p^∗₁(D_E) for some effective divisorD_E onE. Pick a basis (t₁, . . . , t_d) for H⁰(B, L^e|_B) and a section s of L^e with divisor H, and write

π^∗s=

d

X

i=1

s_i⊗t_i

with s1, . . . , sd ∈H⁰(E, L^e|E), so that π⁻¹ H∩∂H

is defined by

d

X

i=1

s_i⊗t_i =

d

X

i=1

∂s_i⊗t_i = 0

4For any line bundle M onA, either all the cohomology groupsHⁱ(A, M) vanish, or they don’t vanish exactly for i∈[i₋(M), i+(M)], the group K =K(M)⁰ has dimension i+(M)−i₋(M), andM pulls back fromA/K.

Since D₁⁰ =H∩D₁ is contained in H∩∂H, for every point xof the support of D_E, we have

divX^d

i=1

s_i(x)t_i

⊂divX^d

i=1

∂s_i(x)t_i

⊂B

Since these two divisors belong to the same linear series |L^e|_B| on B, they must be equal and

rank

s₁(x) · · · s_d(x)

∂s₁(x) · · · ∂s_d(x)

≤1

Since H is irreducible, the sections s1, . . . , sd have no common zero and the morphism ψ_H :E →P^d−1 that they define is ramified atx.

The vector subspace ofH⁰(E, L^e|_E) generated by s₁, . . . , s_donly depends ons, not on the choice of the basis (t1, . . . , td). Ifb1, . . . , bdare general points of B, it is also generated bys(·+b₁), . . . , s(·+b_d) and

rank

s(x+b₁) · · · s(x+b_d)

∂s(x+b₁) · · · ∂s(x+b_d)

≤1

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_x of H⁰(X, L^e). If s is in

H_x⁰ =H_x∩ {t∈H⁰(X, L^e)|t(x+b₁) =t(x+b₂) = 0}

it also satisfies ∂s(x+b₁) = ∂s(x+b₂) = 0. Since H_x⁰ has codimension at most 3 in H⁰(X, L^e), this means that L^e is not 3-jet ample and contradicts a theorem of Bauer and Szemberg (or Pareschi and Popa): the lemma is

proved.

Using vanishing results of Sommese, we get information on the cohomol- ogy of symmetric tensors on a complete intersection in an abelian variety.

Corollary 7 LetX be the intersection ofc sufficiently ample general hyper- surfaces in an abelian variety A of dimension n. We have⁵

h^q(X,S^rΩ_X)







= 0 for q >max{n−2c,0} and r0

=h^q(A,S^rΩ_A) for q < n−2c and r≥0

≥h^q(A,S^rΩ_A) for q=n−2c and r≥0

5Recallh^q(A,S^rΩA) =h^q(A,OA) dimS^rΩA,0= ⁿ_q n+r−1 n−1

.

4 Subvarieties of the projective space

I would like to discuss the following conjecture (that maybe should be called a question).

Conjecture 8 The cotangent bundle of the intersection in Pⁿ of at least n/2 general hypersurfaces of sufficiently high degrees is ample.

This is much more difficult than the analogous theorem for complete intersections in an abelian variety, because the cotangent bundle has no a priori reason to be generated by global sections. We can reformulate the conjecture in a form that applies to (general) complete intersections of any dimension.

Conjecture 9 LetX be the intersection in Pⁿ of cgeneral hypersurfaces of sufficiently high degrees and let m be an integer. For r0, we have

H^q(X,S^rΩ_X(m)) = 0 (4)

except for q= max{n−2c,0}.

Forc≥n/2, this is equivalent to Conjecture 8. Whenq <2n−c, this fol- lows from results of Schneider. The difficult case is q >2n−c. Conjecture 9 holds for c≤1 by results of Bogomolov and Demailly on vanishing of covari- ant tensors on varieties of general type (both Bogomolov and Demailly need the existence on X of (some kind of) a K¨ahler–Einstein metric. Bogomolov uses the fact that the tangent bundle T_X is then stable. Demailly only needs an “approximate” K¨ahler–Einstein metric (but that still requires the theory of Monge–Amp`ere equations) and differential geometric calculations). The conjecture agrees with the sign of the polynomial P_m(r) = χ(X,S^rΩ_X(m)).

Unfortunately, cohomological calculations are very hard, so I am not sure that Conjecture 9 is the right approach. Another way to attack Conjecture 8, would be to try instead to determine the slope

t_amp= inf{t∈Q|Ω_Xhtiample}

for any (general) complete intersection of general type in Pⁿ and prove that it is negative when dim(X)≤n/2 and the degrees are large enough.

This is also a very difficult problem that is unsolved even for quartics in P³. Note that for a surface of general type, with Chern classes c₁ and c₂, we have

0 ≤ c₁(OP(ΩX)(1)ht_ampi)³

= (c²₁−c₂)(Ω_Xht_ampi)

= 3t²h²+ 3tc1 ·h+c²₁−c2 =PX(tamp)

In particular, t_amp ≥ρ, where ρis the largest root of the polynomial P_X. A famous trick of Bogomolov’s yields an upper bound on the other slope

tbig = inf{t ∈Q|ΩXhti big} ≤tamp

as follows. For any integers p and q >0 such that qK_X + 2pH is ample, H²(X,(S^qrΩ_X)(pr))^∗ ' H⁰(X,(S^qrΩ_X)(−(qr−1)K_X)(−pr))

⊂ H⁰(X,(S^qrΩ_X)(pr))

because H⁰(X,(qr−1)K_X + 2prH) is nonzero for r0. By the Riemann–

Roch theorem, we have h⁰(X,(S^qrΩ_X)(pr)) ≥ 1

2 h⁰(X,(S^qrΩ_X)(pr)) +h²(X,(S^qrΩ_X)(pr))

≥ 1

2χ(X,(S^qrΩ_X)(pr))

≥ 1

2χ(P(Ω_X),OP(ΩX)(qr)⊗π^∗O(pr))

= r³

12q³P_X(p

q) +O(r²)

for r 0. It follows that for p/q > ρthe twisted Q-bundle ΩXhp/qi is big.

So we get

t_big ≤ρ≤t_amp (5)

It is expected that for a general complete intersection surface in Pⁿ, n ≥4, both inequalities in (5) are equalities. Since

ρ≤ −1

5(e₁+· · ·+e_n−2−n−1)

when the e_i are large enough, this would certainly imply Conjecture 8 in this case.

Note also that it follows from a result of Miyaoka that given any ε >0, the vector bundle Ω_Xhεi is almost everywhere ample for a general complete intersection surface X in Pⁿ of sufficiently high multidegree.