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 ∈ Pic*0*(X)|h*0*(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∗*_{Pic}0_{(Y}

*L) ⊂ Pic*0*(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 ∈ Pic*0*(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 V*1*(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 V*1*(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 P*1*(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)).

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 =* P*aiDi*, let

*bDc =*P*baicDi* *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 Pic*0*(X) is the abelian*
*variety dual to A(X) parameterizing all topologically trivial line *
*bun-dles on X. Pic*0*(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 h}*0

_{(X, ω}⊗m*X*)

*are denoted by Pm(X) and the irregularity h*0*(X, Ω*1*X*) is denoted by

*q(X).*

*(1.6) Let L be a Cartier divisor on X. If h*0_{(X, L) > 0, then there is}

*a rational map φ|L|* *: X 99K P(H*0*(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* _{: X}0_{−→ Y is birational to}

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

by the following properties:

*i) f0* _{: X}0_{−→ 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; H*0_{(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.*

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 ∈ Pic*0*(X)|h*0*(X, L⊗m⊗P ) 6= 0}.*

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

*Let V = ∪∞*

*m=1Vm. Then V ⊂ Pic*0*(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 Pic*0*(X).*

*Proof. Pick any maximal irreducible component W ⊂ G, that is, if*
*W0* _{is an irreducible component of G containing W then W = W}0_{.}

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

*W ⊂ Gm*0*. 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 → Pic*0*(X). Moreover,*

*Z + W ⊂ Gm*0 *+ Gm* *⊂ Gm*0*+m⊂ G.*

*Let W0* _{be an irreducible component of G}

*m*0*+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 t*0 *> 0 such that Gte* *= G for*

*all t ≥ t*0*.*

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

*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∗*

1*H + p∗*2*P with H ample of degree 1 and P ∈ Pic*0*(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∗*2*P∨).*
*We have*
*(1) κ(L) = 0 and dimG(L) = 0.*
*(2) For all m > 0, Q ∈ (p∗*

1Pic0*(E) + mp∗*2*P∨), 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 ∈ Pic*0_{(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 alb*X*

*X* alb*X*
*−−−→ A(X)*
*fL*
y *ψL*
y
*YL*
alb*Y*
*−−−→ 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 + Pic*0_{(Y}

*L⊗m _{⊗P}) ⊂ V . In particular, if*

*κ(L) ≥ 0, then Pic*0

*(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 h*0_{(X}0_{, µ}∗_{N}⊗k_{⊗π}∗_{H}∨_{) >}

*0. Clearly, µ∗OX0* *= O _{X}*

*and Riµ*

_{∗}O_{X}0*= 0 for all i > 0. Since*

*µ∗*

_{N}⊗k_{⊗π}∗*NH∨* *= µ∗(N⊗k⊗alb∗Xψ∗NH∨*), by the projection formula, we

*have h*0_{(X, N}⊗k_{⊗alb}∗

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

*j À 0, one has h*0* _{(H}j_{⊗Q) > 0 for any Q ∈ Pic}*0

_{(Y}*L⊗m _{⊗P}*). It follows

*that h*0_{(X, N}⊗kj_{⊗Q}⊗j_{) > 0, i.e. kjP + jQ ∈ V}

*mkj*. The Lemma follows

*In particular, take P = OX* *∈ Vm, then OX* *∈ Pic*0*(YL) ⊂ Vmkj* and

hence Pic0*(YL) ⊂ G.* ¤

*It follows that for any m > 0 and P ∈ Pic*0*(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 m*0 *> 0*

*such that Vm* = Pic0*(X) for all m ≥ m*0.

*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 −→ Pic*0*(Y ) −→ Pic*0*(X) −→ Pic*0*(K) −→ 0.*

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

*H := ker*¡*i∗* _{: Pic}0* _{(X) −→ Pic}*0

_{(X}*y*)

¢

*.*
*Then, H ⊃ Pic*0* _{(Y ) and H/Pic}*0

_{(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

alb*X(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) −−−→ Pic*0*(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) = Pic*0*(YL).*

*Proof. By Lemma 2.5, one has G(L) ⊃ Pic*0*(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,*

*h*0* _{(L}⊗m_{⊗P ) 6= 0 in particular h}*0

_{(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) ⊂ Pic*0_{(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 ∈ Pic*0_{(X)}

*tors* *and*

*all P ∈ Pic*0_{(Y}

*KX) one has*
*h*0_{(ω}⊗m

*X* *⊗P ⊗Q) = h*0*(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 Pic*0_{(X).}

*Proof. By a result of Simpson, the loci V*1*(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)K _{X}0+P −bD*

*rc),*

then the locus

*V*1*(ωX0⊗N) := {Q ∈ Pic*0*(X)|h*0*(ω _{X}0⊗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*

*h*0_{(ω}

*X0⊗N) = h*0*(ω _{X}⊗m0*

*⊗P ) and for all Q ∈ T*we have

_{P}*h*0

_{(ω}⊗m*X0* *⊗Q) = h*0*(ω _{X}0(bD*

*r* *c)⊗N⊗Q⊗P*

*∨ _{) ≥ h}*0

_{(ω}*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) ⊂ Pic*0*(X). It follows that*

*Vm(KX) = ∪TP. Since Vm(KX) ⊂ Pic*0*(X) is closed, it follows that*

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

*an element P ∈ Pic*0*(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 Pic*0*(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* *∈ Pic*0*(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

*h*0_{(cdK}

*X* *+ dPc*) = 1.

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

*rDc* *is effective. It follows that h*0*(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 h*0*(ω _{X}⊗m⊗Q) 6= 0, then h*0

*(ω*

_{X}⊗m_{y}*⊗Q) 6= 0.*

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

Since e*H/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 V*1 *⊂ 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 + P*0 *with P*0 *torsion. Let k be the order of P*0. One sees

that

*−P ⊂ G − P*0 *= (k − 1)(G + P*0*) ⊂ 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 alb*Xy* *: 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*

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

is generically finite. It follows that alb*X* *: 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 alb*Y* *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 ∈ Pic*0*(Y ), the Iitaka fibration Y (KX* *+ P )*

*is birational to the Iitaka fibration Y (KX).*

*Proof. Let P, Q ∈ Pic*0_{(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, c*1*(a∗ωX*) =

*0.*

*Proof. Let a = albX. If q(Y ) = 0, then V*1 is supported on finitely

many points. We recall that by definition

*Vi _{(a}*

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

By [CH1], we have

*V*1 *= V*0 *⊇ V*1 *⊇ ... ⊇ 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∗*1*a∗ωX⊗P) is the Fourier-Mukai transform of*

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

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

*generated free B-modules, denoted by E•*_{, such that}

*Ri _{Sa}*

*∗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 H}q_{(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) = W}i_{(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 (alb*X(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) V*1*(X, KX*) = Pic0*(X) and*

*ii) (cf. (17.9.2) of [Kol2]) that P*1*(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 V*1*(X, KX*)

*generate Pic*0_{(Y ), and for all m ≥ 2 one has G}

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

*In particular if X is of general type, they generate Pic*0_{(X) (see i)}

above).

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

*the translates of the components of V*1*(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,}*

*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 V*0_{(ω}

*X) = V*1*(X, KX). *

*Con-sider the quotient map π : A := Alb(X) −→ C dual to the *
*inclu-sion S ,→ Pic*0*(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 V*1*(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 V*1*(ωX) is just {OX}. Let Si*

*be any component of V*1*(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 −→ Pic*0* _{(X)}_{−→ Pic}φ* 0

_{(X)/T −→ 0,}*0 −→ F −→ A(X) −→ T∨* *−→ 0.*

*Let Si* *:= Si/T ⊂ Pic*0*(F ) = Pic*0*(X)/T and π : X −→ T∨* the induced

*morphism. For any P ∈ Pic*0_{(X)}

*torssuch that P /∈ V*1*(X, KX*), one has

that

*φ−1 _{(φ(P )) = P + T ∩ V}*

1*(X, KX) = ∅.*

*So for all Q ∈ T one has h*0_{(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 _{(π}*

*∗(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 h*0_{(X}

*w, ωXw⊗P ) = 0.*

*Therefore, χ(ωXw*) = 0 and the locous
*V*1*(Xw, KXw, Pic*

0* _{(F )) := {R ∈ Pic}*0

*0*

_{(F )|h}

_{(ω}*Xw⊗R) > 0}*

*(which is determined by its torsion points) is a subset of V*1*(X, KX)/T .*

It follows that the intersection of the translates through the origin
*of the components of V*1*(Xw, KXw, Pic*

0_{(F )) is just {O}

*F}. Therefore,*

arguing as in the preceeding paragraph, one sees that alb*X(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*

*Proof. If P*1*(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 P*1( ˜*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 alb*X(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 h*1*(X, P ) = 0 for all*

*P ∈ Pic*0*(X) − {OX}.*

*Proof. Assume that h*1* _{(X, P ) > 0 for some P ∈ Pic}*0

_{(X) − {O}*X}.*

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

*Y −→ X corresponding to the subgroup < P >⊂ Pic*0_{(X). Clearly Y}

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

the Theorem 4.2

*dim(Y ) = q(Y ) = h*1_{(O}*Y*) =

X

*h*1_{(X, P}⊗i_{).}

*In particular h*1_{(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 Z*2

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 Z*2

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∗(ω _{X}*˜

*) = 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*
*P*1*(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) >*

*χ( ˜X) = 0. One can then construct (infinitely many) examples where*
*P*1( ˜*X) = P*1*(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.