On the irregularity of the image of the Iitaka fibration

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IITAKA FIBRATION

JUNGKAI A. CHEN AND CHRISTOPHER D. HACON Abstract. We characterize the irregularity of the image of the Iitaka fibration in terms of the dimension of certain cohomological support loci.

1. Introduction

Let L be a Cartier divisor on a projective variety X with κ(X, L) ≥ 0. The Iitaka fibration associated to L is a birational model of the map induced by the linear series |mL| for m sufficiently big and divisible. We will denote it by fL : X0 → YL, where X0 is an appropriate birational

model of X. It is a fundamental tool in the birational classification of higher dimensional varieties. Therefore, it is important to understand the geometry of the image of the Iitaka fibration.

In this note, we characterize the irregularity of YL in terms of

di-mension of certain cohomological support loci. To be more precise, let

Vm(L) = Vm := {P ∈ Pic0(X)|h0(X, L⊗m⊗P ) 6= 0},

and V = ∪m≥1Vm. We show that the maximal irreducible component

of V passing through the origin, denoted by G, is in fact the subgroup

f∗Pic0(Y

L) ⊂ Pic0(X). In particular, q(YL) = dimG.

In general, the locus V need not be a group, it could have infinitely many irreducible components and there may exists P, Q ∈ Pic0(X) such that fL⊗P and fL⊗Q are not birational. Nevertheless, we show

that when one considers the case of the canonical divisor L = KX,

then the set V (KX) is a subgroup of Pic0(X) such that its components

consist of finitely many torsion translates of Pic0(Y ). Moreover, we show that, for all P ∈ V , one has that the Iitaka fibration fKX⊗P is

birational to fKX.

Finally, we study the case when X is of maximal Albanese dimension. In [CH2], the authors have shown that the translates through the origin of the components of V1(KX) generate Pic0(X). In §4 we show that the

locous T given by the intersection of all (translates through the origin of the components of V1(KX)) plays a fundamental role in understanding

the geometry of X. In particular we show that: Let X be of general type

and maximal Albanese dimension. If P1(X) = 1 then q(X) = dim(X).

Moreover, we construct examples of such varieties. This answers a question of Koll´ar (cf. [Kol2], Conjecture (17.9.2)).

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Acknowledgement. The second author would like to thank R. Par-dini for valuable conversations.

Notation and conventions.

(1.1) Throughout this paper, we work over the field of complex num-bers C. X will always denote a projective variety.

(1.2) |D| will denote the linear series associated to the divisor D. We do not distinguish between line bundles, linear equivalence classes of divisors and invertible sheaves. Let |V | ⊂ |L| be a linear subsystem. A log resolution of |V | is a proper birational morphism µ : X0 −→ X

such that X0 is smooth, µ|V | = |W | + F , where |W | is base point free

and the union of support of F and the exceptional set of µ is a divisor with normal crossings support.

(1.3) For a real number a, let bac be the largest integer ≤ a and

dae be the smallest integer ≥ a. For a Q-divisor D = PaiDi, let

bDc =PbaicDi and dDe =

P

daieDi.

(1.4) We will denote by A(X) the Albanese variety of X, by albX :

X −→ A(X) the Albanese morphism. As usual Pic0(X) is the abelian variety dual to A(X) parameterizing all topologically trivial line bun-dles on X. Pic0(X)tors will denote the set of torsion elements in

Pic0(X). We will denote by m : A × A −→ A the group law of an

abelian variety A. Given subsets Z, W ⊂ A, we define Z + W :=

image (m|Z×W : Z × W −→ A).

(1.5) Let F be a coherent sheaf on X, then hi(X, F) denotes the

com-plex dimension of Hi(X, F). In particular, the plurigenera h0(X, ω⊗m X )

are denoted by Pm(X) and the irregularity h0(X, Ω1X) is denoted by

q(X).

(1.6) Let L be a Cartier divisor on X. If h0(X, L) > 0, then there is

a rational map φ|L| : X 99K P(H0(X, L)) defined by the sections of L.

The Iitaka dimension of a line bundle L is defined as

κ(L) := max{dim φ|mL|(X); m ∈ N}.

If |mL| = ∅ for all m > 0, we set κ(L) = −∞. A nonsingular rep-resentative of the Iitaka fibration of X is a morphism of smooth com-plex projective varieties f0

L = f0 : X0 −→ Y such that for all

suffi-ciently big and divisible integers m, f0 : X0 −→ Y is birational to

f|mL| : X 99K f|mL|(X). It is characterized up to birational equivalence

by the following properties:

i) f0 : X0 −→ Y is an algebraic fiber space (i.e. it is surjective with

connected fibers); ii) dim(Y ) = κ(L); iii) κ(X0

y, L|Xy0) = 0 (where Xy0 is a generic geometric fiber of f0).

There is a semi-group N(L) = N(X, L) := {m ∈ N; H0(X, L⊗m) 6=

0}. The exponent e(L) is the greatest common divisor of all elements of N(L). For all m À 0, one has that m ∈ N(L) iff e(L)|m.

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2. cohomological support loci

Lemma 2.1. Let X be a closed subvariety (reduced and irreducible) of

an abelian variety A passing through the origin. If X is closed under the group law, then X is an abelian subvariety.

Proof. Let g be the dimension of X. Assume that g > 0. Since X is

closed under multiplication, we have an induced morphism

m : X × X → X.

Let Fx be the fiber over x ∈ X. It is clear that dimFx ≥ g. On the

other hand, one notices that Fx ∩ ({a} × X) consists of at most one

point for all a ∈ X. This, together with the fact that dimFx ≥ g, shows

that Fx∩ ({a} × X) has exactly one point for all a ∈ X. Let x = 0 then

it follows that X is a subgroup of A hence an abelian subvariety. ¤ Let X be a smooth projective variety with q(X) > 0, L a Cartier divisor on X with κ(L) ≥ 0 and exponent e := e(L). We define

Vm = Vm(X, L) := {P ∈ Pic0(X)|h0(X, L⊗m⊗P ) 6= 0}.

By semi-continuity, each component of Vmis a closed subset of Pic0(X).

Let V = ∪∞

m=1Vm. Then V ⊂ Pic0(X) is a semi-group. We define Gm

to be the union of all irreducible components of Vm passing through the

origin and let G = ∪m>0Gm. Note that Gm is non-empty only when

m ∈ N(L). Recall that for all m À 0, we have m ∈ N(L) iff e(L)|m.

Lemma 2.2. There exists an integer m > 0 such that G = Gm is an

abelian subvariety of Pic0(X).

Proof. Pick any maximal irreducible component W ⊂ G, that is, if W0 is an irreducible component of G containing W then W = W0.

(This is possible since Pic0(X) is of course Noetherian). Assume that

W ⊂ Gm0. We claim that W = G and W is closed under multiplication.

Then we are done by Lemma 2.1. To see the claim, observe that if Z is any irreducible component of Gm, then Z + W is irreducible since it

is the image of m : Z × W → Pic0(X). Moreover,

Z + W ⊂ Gm0 + Gm ⊂ Gm0+m⊂ G.

Let W0 be an irreducible component of G

m0+m containing Z + W . It

follows that W ⊂ Z + W ⊂ W0. By the maximality of W , one has

W = W0. In particular, Z ⊂ W and hence W = G.

It then suffices to check that W is closed under multiplication. But

W ⊂ W + W ⊂ G + G = G = W and hence W + W = W . ¤

Corollary 2.3. There exists an integer t0 > 0 such that Gte = G for

all t ≥ t0.

Question: Are the loci V (resp. Vm) union of translates of subgroups

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Remark: It is easy to see that if k(L) = 0 and P ∈ V is a non torsion element, then P∨ is not in V . In particular, dimG = 0. The following

example, shows that dimVm could be positive even if κ(L) = 0.

Example 2.4. Consider A = E × E a product of elliptic cures and

let M = p∗

1H + p∗2P with H ample of degree 1 and P ∈ Pic0(E) −

Pic0(E)

tors. Let X = P(OA⊕ M), π : X −→ A and L := OX(1). Since

π : X −→ A is a projective bundle, it follows that π is the Albanese morphism of X. Since

π∗OX(m) = Sm(M ⊕ OA) = M⊗m⊕ M⊗m−1⊕ .... ⊕ M ⊕ OA,

one sees that

Vm = {OA} ∪ m [ i=1 (p∗ 1Pic0(E) + ip∗2P∨). We have (1) κ(L) = 0 and dimG(L) = 0. (2) For all m > 0, Q ∈ (p∗

1Pic0(E) + mp∗2P∨), κ(L⊗Q) = 1 and

dimVm = dimV = 1.

This gives an example in which κ(L) = 0 but κ(L⊗P ) > 0 for some P ∈ Pic0(X). Moreover, V = ∪V

m is a semi-group (with infinitely

many components) but not a group.

Consider the following diagram relating the Iitaka fibration fL and

the Albanese morphism albX

X albX −−−→ A(X) fL   y ψL   y YL albY −−−→ A(YL).

Notice that fL is not a morphism but simply a rational map. However

there exists a birational model µ : X0 −→ X such that the induced

map f0

L : X0 −→ YL is a morphism. Let πL : X0 −→ A(YL) be the

induced map.

Lemma 2.5. If P ∈ Vm then for all sufficiently big and divisible

in-tegers, s > 0 one has that sP + Pic0(Y

L⊗m⊗P) ⊂ V . In particular, if κ(L) ≥ 0, then Pic0(YL) ⊂ G(L)

Proof. Let N = L⊗m⊗P . Fix H an ample line bundle on A(Y

L⊗m⊗P).

For k À 0 sufficiently divisible, we have that h0(X0, µN⊗k⊗πH) >

0. Clearly, µ∗OX0 = OX and RiµOX0 = 0 for all i > 0. Since µ∗N⊗k⊗π

NH∨ = µ∗(N⊗k⊗alb∗Xψ∗NH∨), by the projection formula, we

have h0(X, N⊗k⊗alb

XψN∗H∨) > 0. It is easy to see that for any integer

j À 0, one has h0(Hj⊗Q) > 0 for any Q ∈ Pic0(Y

L⊗m⊗P). It follows

that h0(X, N⊗kj⊗Q⊗j) > 0, i.e. kjP + jQ ∈ V

mkj. The Lemma follows

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In particular, take P = OX ∈ Vm, then OX ∈ Pic0(YL) ⊂ Vmkj and

hence Pic0(YL) ⊂ G. ¤

It follows that for any m > 0 and P ∈ Pic0(X), we have q(YL⊗P) ≤

dimG(L⊗P ) ≤ dimV (L). Recall though that it is possible dimV (L) >

G(L).

It also follows that if L is big, then there exists an integer m0 > 0

such that Vm = Pic0(X) for all m ≥ m0.

Lemma 2.6. i) ψ : A(X) −→ A(Y ) is surjective and has connected

fibers.

ii) Let K := ker(ψ). Then there is an exact sequence

0 −→ Pic0(Y ) −→ Pic0(X) −→ Pic0(K) −→ 0.

iii) Fix a general fiber i : Xy −→ X. Let

H := ker¡i∗ : Pic0(X) −→ Pic0(X y)

¢

. Then, H ⊃ Pic0(Y ) and H/Pic0(Y ) is a finite group. Proof. i) See [HP] Proposition 2.1.b.

ii) This is an immediate consequence of i) obtained by dualizing the exact sequence of abelian varieties

0 −→ K −→ A(X) −→ A(Y ) −→ 0.

iii) Let Jy be the abelian subvariety generated by the translates of

albX(Xy) through origin. Since there are at most countably many

sub-abelian varieties of a fixed abelian variety, we have that J = Jy

does not depend on y ∈ Y for general y. Let B := A(X)/J. Then

Y −→ A(Y ) factors through B, and hence there is a map of abelian

varieties A(Y ) −→ B inducing the map Y −→ B. The image of X in

A(X) generates A(X), hence its image in B generates B as well. It

follows that image of A(Y ) in B generates B. And hence, one sees that

B = A(Y ) i.e. K = J.

We have the following commutative diagram Pic0(X) −−−→ Pic0(K) i∗   y =   y Pic0(X y) ←−−− Picα 0(J),

where i : Xy → X is the inclusion and α is induced from albX : Xy → J.

Note that, since A(Xy) → J is surjective, the homomorphism α has

finite kernel. The assertion now follows from ii). ¤ Theorem 2.7. G(L) = Pic0(YL).

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Proof. By Lemma 2.5, one has G(L) ⊃ Pic0(YL).

If κ(L) = 0, it suffices to prove that dim G = 0. This is clear as otherwise one may consider m > 0 such that G(L) ⊂ Vm and the

morphisms

|mL + P | × |mL − P | −→ |2mL|

for P ∈ G. It is easy to see that if dim G > 0 then dim |2mL| > 0 a contradiction.

Assume now that κ(L) > 0. If P ∈ G(L), then for some m À 0,

h0(L⊗m⊗P ) 6= 0 in particular h0(X

y, L⊗m⊗P |Xy) 6= 0. It follows that

i∗G(X, L) ⊂ V

m(Xy, L|Xy). And hence i∗G(X, L) ⊂ G(Xy, L|Xy).

However, G(Xy, L|Xy) = {OXy} as we have seen in the preceding

para-graph. Thus we have

i∗G(X, L) = G(X

y, L|Xy) = {OXy}.

By Lemma 2.6, G(X, L) ⊂ H. Since G(X, L) is connected and contains the origin, it follows that G(X, L) ⊂ Pic0(Y

L). ¤

3. Iitaka fibrations of canonical divisors ¿From now on we will assume L = KX and κ(X) ≥ 0.

Proposition 3.1. Fix an integer m ≥ 2. For all Q ∈ Pic0(X)

tors and

all P ∈ Pic0(Y

KX) one has h0⊗m

X ⊗P ⊗Q) = h0(ω⊗mX ⊗Q).

Proof. Just follow the proof of [HP] Proposition 2.12. ¤ Theorem 3.2. The loci Vm(KX) consist of a finite union of torsion

translates of abelian subvarieties of Pic0(X).

Proof. By a result of Simpson, the loci V1(KX) are torsion translates

of abelian subvarieties of Pic0(X). Fix m ≥ 2, and let P ∈ V

m(KX).

Let µ : X0 −→ X be a log resolution of the non empty linear series

|r((m−1)KX+P )|. Let D be a general member of µ∗|r((m−1)KX+P )|.

D has normal crossings support. Let N := OX0((m−1)KX0+P −bD

rc),

then the locus

V1(ωX0⊗N) := {Q ∈ Pic0(X)|h0X0⊗N⊗Q) > 0}

consist of a finite union of torsion translates of subtori of Pic0(X)

(cf. [Sim] and [ClH] §7). Comparing base loci, it is easy to see that

h0

X0⊗N) = h0X⊗m0 ⊗P ) and for all Q ∈ TP we have h0⊗m

X0 ⊗Q) = h0X0(bD

r c)⊗N⊗Q⊗P

) ≥ h0

X0⊗N⊗Q⊗P∨) > 0.

Hence for each P ∈ Vm(KX) there exists TP a torsion translate of a

subtorous such that P ∈ TP ⊂ Vm(KX) ⊂ Pic0(X). It follows that

Vm(KX) = ∪TP. Since Vm(KX) ⊂ Pic0(X) is closed, it follows that

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Lemma 3.3. If κ(X) = 0 then there exists an integer c(X) = c > 0 and

an element P ∈ Pic0(X)tors such that for all m > 0, the set Vm(KX)

is non-empty if and only if c divides m. Moreover, if c divides m, then Vm = {P⊗

m

c}. In particular V is a subgroup of Pic0(X).

Proof. It is easy to see that since κ(X) = 0, each Vm can have at most

1 torsion point. Therefore Vm(KX) consists of at most one element say

Pm ∈ Pic0(X)tors. Let

c := min{m > 0|Vm 6= ∅}.

It is clear that Vm 6= ∅ for all m divisible by c. Let d be any positive

integer such that Vd6= ∅. We claim that c|d. We may write d = cq + r

with q > 0 and 0 ≤ r ≤ c − 1. Let Dc (resp. Dd) be the unique

divisor in the linear series |cKX + Pc| (resp. |dKX + Pd|). Since Vcd

has only one point, we have dPc = cPd and moreover cDd = dDc as

h0(cdK

X + dPc) = 1.

Consider now the divisor Dd−qDc. It is effective since c(Dd−qDc) =

rDc is effective. It follows that h0(rKX + Pd − qPc) 6= 0 and hence

Vr 6= ∅. One sees that r = 0. Therefore, Vm(KX) is non-empty iff c

divides m. The Lemma now follows easily. ¤

Corollary 3.4. Let eH := i∗−1(V (X

y)). For all m > 0 one has Vm

e

H. Moreover, V ⊂ eH is a subgroup of finite index containing G. Proof. For any Q ∈ Vm, if h0X⊗m⊗Q) 6= 0, then h0X⊗my ⊗Q) 6= 0.

Therefore, Vm ⊂ eH for all m > 0 and hence, V ⊂ eH.

Since eH/G is finite and G ⊂ V ⊂ eH, it suffices to show that the

semigroup V is in fact a group.

To this end, pick any P ∈ V , we may assume that P ∈ Vm for m ≥ 2

since V1 ⊂ V1+t for all sufficiently big and divisible t. By Proposition

3.1 and Theorem 3.2, one sees that P is in an irreducible component of the type G + P0 with P0 torsion. Let k be the order of P0. One sees

that

−P ⊂ G − P0 = (k − 1)(G + P0) ⊂ V.

It follows that V is a group. ¤

Corollary 3.5.

q(X) − q(Y ) ≤ q(Xy).

Proof. Since κ(Xy) = 0, by [Kaw], the map albXy : Xy −→ A(Xy) is

surjective. The map Xy −→ J is factored through A(Xy). It follows

that A(Xy) −→ J is surjective by the definition of J. Thus q(Xy) =

dimA(Xy) ≥ dimJ. But J = K and dimK = dimA(X) − dimA(Y ).

The inequality follows. ¤

Corollary 3.6. X is of maximal Albanese dimension if and only if Y

is of maximal Albanese dimension and

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Proof. If X is of maximal Albanese dimension, then albX : X → A(X)

is generically finite. It follows that albX : Xy → A(X) is generically

finite. Therefore, Xy −→ J is an ´etale map of abelian varieties and

dimXy = dimJ. Moreover, since albX is generically finite, for a general

point z ∈ albY(Y ), the preimage in X consits of at most finite union

of fibers Xy. Thus albY is generically finite and hence Y is of maximal

Albanese dimension.

On the other hand, let Y → S → A(Y ) be the Stein factorization. And let T := S ×A(Y ) A(X). It suffices to check that X → T is

generically finite. Notice that κ(Xy) = 0 and q(Xy) ≥ dimXy. It

follows by [Kaw] that Xy is birational to an abelian variety and hence

that Xy → K is generically finite. Therefore Xy → T is generically

finite, and so is X → T . ¤

Corollary 3.7. For all P ∈ Pic0(Y ), the Iitaka fibration Y (KX + P )

is birational to the Iitaka fibration Y (KX).

Proof. Let P, Q ∈ Pic0(Y ) such that P = 2Q. From the morphism |mKX| × |m(KX + P )| −→ |2m(KX + Q)|,

one sees that Y (KX + Q) dominates Y (KX) and Y (KX + P ). By

Proposition 3.1,

dim Y (KX + Q) = dim Y (KX + P ) = dim Y (KX).

since the various Iitaka fibrations X −→ Y (...) have connected fibers, then Y (KX + Q) is birational to Y (KX) and to Y (KX + P ). ¤

Lemma 3.8. Let a : X → A(X) be the Albanese map. If q(Y ) = 0,

then a∗ωX is a homogeneous vector bundle. In particular, c1(a∗ωX) =

0.

Proof. Let a = albX. If q(Y ) = 0, then V1 is supported on finitely

many points. We recall that by definition

Vi(a

∗ωX) := {P ∈ Pic0(A(X))|hi(a∗ωX⊗P ) 6= 0}.

By [CH1], we have

V1 = V0 ⊇ V1 ⊇ ... ⊇ Vq,

where q = q(X). It follows that all the Vi are supported on finitely

many points.

We now follow [Laz] to show that a∗ωX is a homogeneous vector

bundle. Let pi be the projection of A(X) × Pic(X) = A × bA on to the

i-th factor and P be the normalized Poincare’ line bundle on A × bA.

Then RSa∗ωX = Rp2∗(p∗1a∗ωX⊗P) is the Fourier-Mukai transform of

a∗ωX. By [Muk], it is enough to show that RgSa∗ωX ∼= RSa∗ωX is a

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that for an open affine U = Spec(B) ⊂ ˆA, there is a complex of finitely

generated free B-modules, denoted by E•, such that

RiSa

∗ωX ∼= Hi(E•),

and

Hi(X

y, a∗ωX⊗Py) ∼= Hi(E•⊗k(y)).

It suffices to show that Hi(E) = 0 for all i < q and Hq(E) is

sup-ported on finite point. We may assume that B is local since localization is flat. Let

Wi(E•) := coker(Ei−1→ Ei).

It easy to see that Wi(E)⊗k(y) = Wi(E⊗k(y)) since ⊗ is right exact.

One has exact sequence

0 → Hi(E•) → Wi(E•) → Ei+1.

Our hypothesis on the cohomologies of a∗ωX implies that Hi(E•) is

Artinian for all i. Thus by Mumford’s acyclic lemma ([Mum], page

127), we are done. ¤

4. Varieties of Maximal Albanese dimension

Throughout this section, we assume that X is of maximal Albanese dimension i.e. dim (albX(X)) = dimX.

In (17.9.3) of [Kol2], Koll´ar Conjectured that If X is of general type

and Maximal Albanese dimension, then χ(X, OX) > 0. A consequence

of this conjecture would be: i) V1(X, KX) = Pic0(X) and

ii) (cf. (17.9.2) of [Kol2]) that P1(X) ≥ 2.

In [EL], Ein and Lazarsfeld show that χ(X, OX) ≥ 0, and they produce

a counterexample to the above conjecture of Koll´ar ((17.9.3) of [Kol2]). In [CH2], the authors show that:

Theorem 4.1. If X is of maximal Albanese dimension then (the

trans-lates through the origin of) the irreducible components of V1(X, KX)

generate Pic0(Y ), and for all m ≥ 2 one has G

m(X, KX) = Pic0(Y ).

In particular if X is of general type, they generate Pic0(X) (see i)

above).

Let X be of maximal Albanese dimension. If χ(OX) = 0, let Si0 be

the translates of the components of V1(X, KX) through the origin and

T := ∩S0 i.

Theorem 4.2. Let X be a variety of general type, maximal Albanese

dimension with χ(ωX) = 0. Let A(X) −→ T∨ be the corresponding

map of abelian varieties, and π : X −→ T∨ the induced morphism. For

general w in W := π(X), one has that χ(ωXw) = 0 and albX(Xw) is a translate of the abelian subvariety ker (A(X) −→ T∨). In particular,

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Proof. Recall that by [EL] Theorem 3, the Albanese image Z := albX(X)

⊂ Alb(X) is fibered by tori. In fact [EL] shows the following: Let S be any positive dimensional component of V0

X) = V1(X, KX).

Con-sider the quotient map π : A := Alb(X) −→ C dual to the inclu-sion S ,→ Pic0(A). Let g : X −→ C be the induced morphism, then dim g(X) = dim(X) − (dim(A) − dim(C)), i.e. Z −→ g(X) is fibered

by tori. By [CH2] Theorem 2.3, the translates through the origin of

the components of V1(ωX) generate Pic0(X). X is of general type and

hence not ruled by tori. Assume that the intersection of the translates through the origin of the components of V1(ωX) is just {OX}. Let Si

be any component of V1(ωX) and Bi be the kernel of the

correspond-ing projection of abelian varieties πi : A −→ Ci := Si∨. Then the

union of the Bi generates A. As observed above, by the result of [EL],

Z + Bi = Z and hence Z = A, i.e. q(X) = dimX.

Suppose now that dimT > 0 and consider the following dual exact sequences of abelian varieties

0 −→ T −→ Pic0(X)−→ Picφ 0(X)/T −→ 0,

0 −→ F −→ A(X) −→ T∨ −→ 0.

Let Si := Si/T ⊂ Pic0(F ) = Pic0(X)/T and π : X −→ T∨ the induced

morphism. For any P ∈ Pic0(X)

torssuch that P /∈ V1(X, KX), one has

that

φ−1(φ(P )) = P + T ∩ V

1(X, KX) = ∅.

So for all Q ∈ T one has h0(X, ω

X⊗P ⊗π∗Q) = 0 and therefore also

hi(X, ω

X⊗P ⊗π∗Q) = 0 for all i ≥ 0. By a result of Koll´ar (cf. [Kol1],

Corollary 3.3) one sees that for all Q ∈ T , the group Hi

∗(ωX⊗P )⊗Q)

is a direct summand of the group Hi(X, ω

X⊗P ⊗π∗Q) and hence also

vanishes. By a result of Mukai (cf. [Muk]) it follows that π∗(ωX⊗P ) =

0. So for general w ∈ W = π(X), one has h0(X

w, ωXw⊗P ) = 0.

Therefore, χ(ωXw) = 0 and the locous V1(Xw, KXw, Pic

0(F )) := {R ∈ Pic0(F )|h0

Xw⊗R) > 0}

(which is determined by its torsion points) is a subset of V1(X, KX)/T .

It follows that the intersection of the translates through the origin of the components of V1(Xw, KXw, Pic

0(F )) is just {O

F}. Therefore,

arguing as in the preceeding paragraph, one sees that albX(Xw) = F

is an abelian subvariety of A(X). ¤

Corollary 4.3. Let X be a 3-fold of general type and maximal Albanese

dimension. If χ(ωX) = 0 then q(X) = dimX.

Proof. We may assume that dimX > dimT > 0 and hence Xw is of

general type and has dimension 1 or 2. By the classification theory of curves and surfaces χ(ωXw) > 0. This is the required contradiction. ¤

Corollary 4.4. Let X be a variety of general type, maximal Albanese

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Proof. If P1(X) = 1 then χ(ωX) = 0. Let ˜W −→ W be an appropriate

desingularization (of W = π(X) as above). Since X is of maximal Al-banese dimension, one has P1( ˜W ) = 1 and hence since W is a subvariety

of the abelian variety T∨, it follows that W = T. Since alb

X(Xw) = F

one sees that albX(X) = A(X) and hence q(X) = dim(X). ¤

Corollary 4.5. Let X be a variety of general type, maximal Albanese

dimension with χ(ωX) = 0. If dimT = 0, then h1(X, P ) = 0 for all

P ∈ Pic0(X) − {OX}.

Proof. Assume that h1(X, P ) > 0 for some P ∈ Pic0(X) − {O X}.

By [Sim], we may assume that P is torsion. Consider the ´etale cover

Y −→ X corresponding to the subgroup < P >⊂ Pic0(X). Clearly Y

is of general type, maximal Albanese dimension and χ(ωY) = 0 so by

the Theorem 4.2

dim(Y ) = q(Y ) = h1(O Y) =

X

h1(X, P⊗i).

In particular h1(X, P⊗i) = 0 for i > 0. ¤

The following example shows that Koll´ar’s conjecture (17.9.2) of [Kol2] also fails:

Example. Let d : D −→ E be the Z2

2 cover of an elliptic curve defined

by d∗OD = OE⊕L∨⊕P ⊕L∨⊗P where deg(L) = 1 and P is a non-zero

2-torsion element of Pic0(E). Then g(D) = 3 and we will denote by l, p, lp the elements of Z2

2 whose eigensheves with eigenvalue 1 are OE

and L∨, P, L⊗P respectively. It follows that

δ : D × D × D −→ E × E × E

is a Z6

2 cover. Let X := D × D × D/G where G ∼= Z42 is the subgroup

of Z6

2 generated by {(id, p, l), (p, l, id), (l, id, p), (p, p, p)}. By a direct

check, one can see that

(δ∗OD×D×D)G =

OE£ OE£ OE⊕ L∨£ L∨⊗P £ P ⊕ P £ L∨£ L∨⊗P ⊕ L∨⊗P £ P £ L∨.

The singularities of X are of type 1

2(1, 1, 1) i.e. locally isomorphic to

C3/ < (−1, −1, −1) >. In particular all singularities are rational. Let

˜

X −→ X be a resolution of the singularities of X and ν : ˜X −→ E × E × E be the induced morphism. One can compute that

ν∗(ωX˜) = OE£OE£OE⊕L£L⊗P £P ⊕P £L£L⊗P ⊕L⊗P £P £L.

Therefore, ˜X is a threefold of maximal Albanese dimension with P1(X) = 1 and q(X) = 3.

We remark that this example easily generalizes to dim X ≥ 3. The product X × C with C a curve of genus 2, is a variety of general type and maximal Albanese dimension with χ(ωX×C) = 0 and 5 = q(X) >

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χ( ˜X) = 0. One can then construct (infinitely many) examples where P1( ˜X) = P1(X), q( ˜X) = q(X).

References

[CH1] J. A. Chen, C. D. Hacon, On algebraic fiber spaces over varieties of maximal

Albanese dimension, to appear in Duke Math. Jour.

[CH2] A. J. Chen, C. D. Hacon, Pluricanonical maps of varieties of maximal

Al-banese dimension, Math. Annalen 320 (2001) 2, 367-380.

[ClH] H. Clemens and C. D. Hacon, Deformations of the trivial line bundle and

vanishing theorems, Preprint math.AG/0011244. To appear in Amer. Jour.

Math.

[EL] L. Ein, R. Lazarsfeld, Singularities of theta divisors, and birational geometry

of irregular varieties, Jour. AMS 10, 1 (1997), 243–258.

[HP] C. D. Hacon and R. Pardini, On the birational geometry of varieties of

max-imal Albanese dimension To appear in Jour. f¨ur die Reine Angew.

[Kaw] Y. Kawamata, Characterization of abelian varieties, Comp. Math. 41, (1981).

[Kol1] J. Koll`ar, Higher direct images of dualizing sheaves II, Ann. Math. (1987) 124

[Kol2] J. Koll`ar, Shafarevich Maps and Automorphic Forms, Princeton University Press (1995).

[Laz] R. Lazarsfeld, Personal communication

[Muk] S. Mukai, Duality between D(X) and D( ˆX) with its application to Picard sheaves, Nagoya Math. Jour. (1981).

[Mum] D. Mumford, Abelian Varieties, Oxford University Press.

[Sim] Simpson, C., Subspaces of moduli spaces of rank one local systems., Ann. scient. ´Ecole Norm. Sup. 26 (1993), 361-401.

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