ON THE IRREGULARITY OF THE IMAGE OF THE 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 f_{L} : X′ → Y_{L}, where X′ 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 Y_{L} in terms of dimension of certain cohomological support loci. To be more precise, let

V_{m}(L) = V_{m} := {P ∈ Pic^{0}(X)|h^{0}(X, L^{⊗m}⊗P) ≠ 0},

and V = ∪_{m≥1}V_{m}. 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(Y_{L}) = 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 f_{L⊗P} and f_{L⊗Q} are not birational. Nevertheless, we show that when one considers the case of the canonical divisor L = K_{X}, then the set V (K_{X}) is a subgroup of Pic^{0}(X) such that its components consist of finitely many torsion translates of Pic^{0}(Y). Moreover, we show that, for all P ∈ V, one has that the Iitaka fibration f_{K_{X}⊗P} is birational to f_{K_{X}}.

Finally, we study the case when X is of maximal Albanese dimension. In [1], the authors have shown that the translates through the origin of the components of V_{1}(K_{X}) generate Pic^{0}(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}(K_{X})) 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 Kollár (cf. [2], Conjecture (17.9.2)).

Acknowledgement. We would like to thank R. Lazarsfeld and R. Pardini for valuable conversations. This work was completed during a visit of the first author to the University of Michigan, Ann Arbor. The first author would like to thank the University of Michigan for its hospitality. The first author was partially supported by National Science Council, Taiwan NSC-39151F.

Notation and conventions.

(1.1) Throughout this paper, we work over the field of complex numbers 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 : X′ ⟶ X such that X′ 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 ⌊a⌋ be the largest integer ≤ a and ⌈a⌉ be the smallest integer ≥ a. For a ℚ-divisor D = ∑ a_{i}D_{i}, let ⌊D⌋ = ∑⌊a_{i}⌋D_{i} and ⌈D⌉ = ∑⌈a_{i}⌉D_{i}.

(1.4) We will denote by A(X) the Albanese variety of X, by alb_{X} : X ⟶ A(X) the Albanese morphism. As usual Pic^{0}(X) is the abelian variety dual to A(X) parameterizing all topologically trivial line bundles on X. Pic^{0}(X)_{tors} will denote the set of torsion elements in Pic^{0}(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 h^{i}(X, F) denotes the complex dimension of H^{i}(X, F). In particular, the plurigenera h^{0}(X, ω_{X}^{⊗m}) are denoted by P_{m}(X) and the irregularity h^{0}(X, Ω_{X}^{1}) 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 −−→ 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 representative of the Iitaka fibration of X is a morphism of smooth complex projective varieties f′_{L} = f′ : X′ ⟶ Y such that for all sufficiently big and divisible integers m, f′ : X′ ⟶ Y is birational to f_{|mL|} : X −−→ f_{|mL|}(X). It is characterized up to birational equivalence by the following properties:

i) f′ : X′ ⟶ Y is an algebraic fiber space (i.e. it is surjective with connected fibers);

ii) dim(Y) = κ(L);

iii) κ(X′_{y}, L|X′_{y}) = 0 (where X′_{y} is a generic geometric fiber of f′).

There is a semi-group N(L) = N(X, L) := {m ∈ ℕ; H^{0}(X, L^{⊗m}) ≠ 0}. The exponent e(L) is the greatest common divisor of all elements of N(L). For all m ≫ 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 F_{x} be the fiber over x ∈ X. It is clear that dimF_{x} ≥ g. On the other hand, one notices that F_{x} ∩ ({a} × X) consists of at most one point for all a ∈ X. This, together with the fact that dimF_{x} ≥ g, shows that F_{x} ∩({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

V_{m} = V_{m}(X, L) := {P ∈ Pic^{0}(X)|h^{0}(X, L^{⊗m}⊗P) ≠ 0}.

By semi-continuity, each component of V_{m} is a closed subset of Pic^{0}(X). Let V = ∪_{m=1}^{∞}V_{m}. Then V ⊂ Pic^{0}(X) is a semi-group. We define G_{m} to be the union of all irreducible components of V_{m} passing through the origin and let G = ∪_{m>0}G_{m}. Note that G_{m} is non-empty only when m ∈ N(L). Recall that for all m ≫ 0, we have m ∈ N(L) iff e(L)|m. Lemma 2.2. There exists an integer m > 0 such that G = G_{m} is an abelian subvariety of Pic^{0}(X).

Proof. Pick any maximal irreducible component W ⊂ G, that is, if W′ is an irreducible component of G containing W then W = W′. (This is possible since Pic^{0}(X) is of course Noetherian). Assume that W ⊂ G_{m_{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 G_{m}, then Z + W is irreducible since it is the image of m : Z × W → Pic^{0}(X). Moreover,

Z + W ⊂ G_{m_{0}} + G_{m} ⊂ G_{m_{0}+m} ⊂ G.

Let W′ be an irreducible component of G_{m_{0}+m} containing Z + W. It follows that W ⊂ Z + W ⊂ W′. By the maximality of W, one has W = W′. 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 G_{te} = G for all t ≥ t_{0}.

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 dimV_{m} 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) − Pic^{0}(E)_{tors}. Let X = ℙ(O_{A} ⊕M), π : X ⟶ A and L := O_{X}(1). Since π : X ⟶ A is a projective bundle, it follows that π is the Albanese morphism of X. Since

π_{∗}O_{X}(m) = S^{m}(M ⊕ O_{A}) = M^{⊗m} ⊕ M^{⊗m−1} ⊕ .... ⊕ M ⊕ O_{A},

one sees that

V_{m} = {O_{A}} ∪ ⋃_{i=1}^{m}(p_{1}^{∗}Pic^{0}(E) + ip_{2}^{∗}P^{∨}).

We have (1) κ(L) = 0 and dimG(L) = 0.

(2) For all m > 0, Q ∈ (p_{1}^{∗}Pic^{0}(E) + mp_{2}^{∗}P^{∨}), κ(L⊗Q) = 1 and

dimV_{m} = 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 f_{L} and the Albanese morphism alb_{X}

X ⟶^{alb_{X}} A(X)
f_{L}↓  ^{ψ_{L}}↓
Y_{L} ⟶^{alb_{Y}} A(Y_{L}).

Notice that f_{L} is not a morphism but simply a rational map. However there exists a birational model μ : X′ ⟶ X such that the induced map f′_{L} : X′ ⟶ Y_{L} is a morphism. Let π_{L} : X′ ⟶ A(Y_{L}) be the induced map. Lemma 2.5. If P ∈ V_{m} then for all sufficiently big and divisible integers, s > 0 one has that sP + Pic^{0}(Y_{L^{⊗m}⊗P}) ⊂ V. In particular, if κ(L) ≥ 0, then Pic^{0}(Y_{L}) ⊂ G(L)

Proof. Let N = L^{⊗m}⊗P. Fix H an ample line bundle on A(Y_{L^{⊗m}⊗P}). For k ≫ 0 sufficiently divisible, we have that h^{0}(X′, μ^{∗}N^{⊗k}⊗π^{∗}H^{∨}) > 0. Clearly, μ_{∗}O_{X′} = O_{X} and R^{i}μ_{∗}O_{X′} = 0 for all i > 0. Since μ^{∗}N^{⊗k}⊗π_{N}^{∗}H^{∨} = μ^{∗}(N^{⊗k}⊗alb_{X}^{∗}ψ_{N}^{∗}H^{∨}), by the projection formula, we have h^{0}(X, N^{⊗k}⊗alb_{X}^{∗}ψ_{N}^{∗}H^{∨}) > 0. It is easy to see that for any integer j ≫ 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 by setting s = kj.

In particular, take P = O_{X} ∈ V_{m}, then O_{X} ∈ Pic^{0}(Y_{L}) ⊂ V_{mkj} and hence Pic^{0}(Y_{L}) ⊂ G. □

It follows that for any m > 0 and P ∈ Pic^{0}(X), we have q(Y_{L⊗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 V_{m} = Pic^{0}(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 : X_{y} ⟶ 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 [3] Proposition 2.1.b. fact

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 J_{y} be the abelian subvariety generated by the translates of alb_{X}(X_{y}) through origin. Since there are at most countably many sub-abelian varieties of a fixed abelian variety, we have that J = J_{y} 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

Pic^{0}(X) ⟶ Pic^{0}(K)
i^{∗}↓  ^{=}↓
Pic^{0}(X_{y}) ⟵^{α} Pic^{0}(J),

where i : X_{y} → X is the inclusion and α is induced from alb_{X} : X_{y} → J. Note that, since A(X_{y}) → J is surjective, the homomorphism α has finite kernel. The assertion now follows from ii). □

Theorem 2.7. G(L) = Pic^{0}(Y_{L}).

Proof. By Lemma 2.5, one has G(L) ⊃ Pic^{0}(Y_{L}).

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) ⊂ V_{m} 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 ≫ 0, h^{0}(L^{⊗m}⊗P) ≠ 0 in particular h^{0}(X_{y}, L^{⊗m}⊗P |X_{y}) ≠ 0. It follows that i^{∗}G(X, L) ⊂ V_{m}(X_{y}, L|X_{y}). And hence i^{∗}G(X, L) ⊂ G(X_{y}, L|X_{y}). However, G(X_{y}, L|X_{y}) = {O_{X_{y}}} as we have seen in the preceding paragraph. Thus we have

i^{∗}G(X, L) = G(X_{y}, L|X_{y}) = {O_{X_{y}}}.

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 = K_{X} 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_{K_{X}}) one has

h^{0}(ω_{X}^{⊗m}⊗P ⊗Q) = h^{0}(ω_{X}^{⊗m}⊗Q).

Proof. Just follow the proof of [3] Proposition 2.12. □

Theorem 3.2. The loci V_{m}(K_{X}) 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}(K_{X}) are torsion translates of abelian subvarieties of Pic^{0}(X). Fix m ≥ 2, and let P ∈ V_{m}(K_{X}). Let μ : X′ ⟶ X be a log resolution of the non empty linear series |r((m−1)K_{X}+P)|. Let D be a general member of μ^{∗}|r((m−1)K_{X}+P)|. D has normal crossings support. Let N := O_{X′}((m−1)K_{X′} +P −⌊\frac{D}{r}⌋), then the locus

V_{1}(ω_{X′}⊗N) := {Q ∈ Pic^{0}(X)|h^{0}(ω_{X′}⊗N⊗Q) > 0}

consist of a finite union of torsion translates of subtori of Pic^{0}(X) (cf.

[4] and [5] §7). Comparing base loci, it is easy to see that h^{0}(ω_{X′}⊗N) = h^{0}(ω_{X′}^{⊗m}⊗P) and for all Q ∈ T_{P} we have

h^{0}(ω_{X′}^{⊗m}⊗Q) = h^{0}(ω_{X′}(⌊\frac{D}{r}⌋)⊗N⊗Q⊗P^{∨}) ≥ h^{0}(ω_{X′}⊗N⊗Q⊗P^{∨}) > 0.

Hence for each P ∈ V_{m}(K_{X}) there exists T_{P} a torsion translate of a subtorous such that P ∈ T_{P} ⊂ V_{m}(K_{X}) ⊂ Pic^{0}(X). It follows that V_{m}(K_{X}) = ∪T_{P}. Since V_{m}(K_{X}) ⊂ Pic^{0}(X) is closed, it follows that V_{m}(K_{X}) consists of a finite union of the T_{P}. □

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 V_{m}(K_{X}) is non-empty if and only if c divides m. Moreover, if c divides m, then V_{m} = \{P^{⊗^{\frac{m}{c}}} \}. In particular V is a subgroup of Pic^{0}(X).

Proof. It is easy to see that since κ(X) = 0, each V_{m} can have at most 1 torsion point. Therefore V_{m}(K_{X}) consists of at most one element say P_{m} ∈ Pic^{0}(X)_{tors}. Let

c := min{m > 0|V_{m} ≠ ∅}.

It is clear that V_{m} ≠ ∅ for all m divisible by c. Let d be any positive integer such that V_{d} ≠ ∅. We claim that c|d. We may write d = cq + r with q > 0 and 0 ≤ r ≤ c − 1. Let D_{c} (resp. D_{d}) be the unique divisor in the linear series |cK_{X} + P_{c}| (resp. |dK_{X} + P_{d}|). Since V_{cd} has only one point, we have dP_{c} = cP_{d} and moreover cD_{d} = dD_{c} as h^{0}(cdK_{X} + dP_{c}) = 1.

Consider now the divisor D_{d}−qD_{c}. It is effective since c(D_{d}−qD_{c}) = rD_{c} is effective. It follows that h^{0}(rK_{X} + P_{d} − qP_{c}) ≠ 0 and hence V_{r} ≠ ∅. One sees that r = 0. Therefore, V_{m}(K_{X}) is non-empty iff c divides m. The Lemma now follows easily. □

Corollary 3.4. Let H̃ := i^{∗}^{−1}(V (X_{y})). For all m > 0 one has V_{m} ⊂ H̃. Moreover, V ⊂ H̃ is a subgroup of finite index containing G.

Proof. For any Q ∈ V_{m}, if h^{0}(ω_{X}^{⊗m}⊗Q) ≠ 0, then h^{0}(ω_{X_{y}}^{⊗m}⊗Q) ≠ 0. Therefore, V_{m} ⊂ H̃ for all m > 0 and hence, V ⊂ H̃.

Since H̃/G is finite and G ⊂ V ⊂ H̃, 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 ∈ V_{m} for m ≥ 2 since V_{1} ⊂ V_{1+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(X_{y}).

Proof. Since κ(X_{y}) = 0, by [6], the map alb_{X_{y}} : X_{y} ⟶ A(X_{y}) is surjective. The map X_{y} ⟶ J is factored through A(X_{y}). It follows that A(X_{y}) ⟶ J is surjective by the definition of J. Thus q(X_{y}) = dimA(X_{y}) ≥ 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

q(X) − q(Y) = dimX − dimY.

Proof. If X is of maximal Albanese dimension, then alb_{X} : X → A(X) is generically finite. It follows that alb_{X} : X_{y} → A(X) is generically finite. Therefore, X_{y} ⟶ J is an étale map of abelian varieties and dimX_{y} = dimJ. Moreover, since alb_{X} is generically finite, for a general point z ∈ alb_{Y} (Y), the preimage in X consits of at most finite union of fibers X_{y}. 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(X_{y}) = 0 and q(X_{y}) ≥ dimX_{y}. It follows by [6] that X_{y} is birational to an abelian variety and hence that X_{y} → K is generically finite. Therefore X_{y} → T is generically finite, and so is X → T. □

Corollary 3.7. For all P ∈ Pic^{0}(Y), the Iitaka fibration Y (K_{X} + P) is birational to the Iitaka fibration Y (K_{X}).

Proof. Let P, Q ∈ Pic^{0}(Y) such that P = 2Q. From the morphism

|mK_{X}| × |m(K_{X} + P)| ⟶ |2m(K_{X} + Q)|,

one sees that Y (K_{X} + Q) dominates Y (K_{X}) and Y (K_{X} + P). By Proposition 3.1,

dim Y (K_{X} + Q) = dim Y (K_{X} + P) = dim Y (K_{X}).

since the various Iitaka fibrations X ⟶ Y (...) have connected fibers, then Y (K_{X} + Q) is birational to Y (K_{X}) and to Y (K_{X} + P). □

Lemma 3.8. Let a : X → A(X) be the Albanese map. If q(Y) = 0, then a_{∗}ω_{X} is a homogeneous vector bundle. In particular, c_{1}(a_{∗}ω_{X}) = 0.

Proof. Let a = alb_{X}. If q(Y) = 0, then V_{1} is supported on finitely many points. We recall that by definition

V^{i}(a_{∗}ω_{X}) := {P ∈ Pic^{0}(A(X))|h^{i}(a_{∗}ω_{X}⊗P) ≠ 0}.

By [7], we have

V_{1} = V^{0} ⊇ V^{1} ⊇ ... ⊇ V^{q},

where q = q(X). It follows that all the V^{i} are supported on finitely many points.

We now follow [8] to show that a_{∗}ω_{X} is a homogeneous vector bundle. Let p_{i} be the projection of A(X) × Pic(X) = A × Â on to the i-th factor and P be the normalized Poincare’ line bundle on A × Â. Then RSa_{∗}ω_{X} = Rp_{2}_{∗}(p_{1}^{∗}a_{∗}ω_{X}⊗P) is the Fourier-Mukai transform of a_{∗}ω_{X}. By [9], it is enough to show that R^{g}Sa_{∗}ω_{X} ≅ RSa_{∗}ω_{X} is a coherent sheaf supported on finitely many points. To see this, recall that for an open affine U = Spec(B) ⊂ Â, there is a complex of finitely generated free B-modules, denoted by E^{•}, such that

R^{i}Sa_{∗}ω_{X} ≅ H^{i}(E^{•}),

and

H^{i}(X_{y}, a_{∗}ω_{X}⊗P_{y}) ≅ H^{i}(E^{•}⊗k(y)).

It suffices to show that H^{i}(E^{•}) = 0 for all i < q and H^{q}(E^{•}) is supported on finite point. We may assume that B is local since localization is flat. Let

W^{i}(E^{•}) := coker(E^{i−1} → E^{i}).

It easy to see that W^{i}(E^{•})⊗k(y) = W^{i}(E^{•}⊗k(y)) since ⊗ is right exact. One has exact sequence

0 → H^{i}(E^{•}) → W^{i}(E^{•}) → E^{i+1}.

Our hypothesis on the cohomologies of a_{∗}ω_{X} implies that H^{i}(E^{•}) is Artinian for all i. Thus by Mumford’s acyclic lemma ([10], 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 [2], Kollár Conjectured that If X is of general type and Maximal Albanese dimension, then χ(X, O_{X}) > 0. A consequence of this conjecture would be:

i) V_{1}(X, K_{X}) = Pic^{0}(X) and

ii) (cf. (17.9.2) of [2]) that P_{1}(X) ≥ 2. In [11], Ein and Lazarsfeld show that χ(X, O_{X}) ≥ 0, and they produce a counterexample to the above conjecture of Kollár ((17.9.3) of [2]). In [1], the authors show that: Theorem 4.1. If X is of maximal Albanese dimension then (the translates through the origin of) the irreducible components of V_{1}(X, K_{X}) generate Pic^{0}(Y), and for all m ≥ 2 one has G_{m}(X, K_{X}) = Pic^{0}(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(O_{X}) = 0, let S′_{i} be the translates of the components of V_{1}(X, K_{X}) through the origin and T := ∩S′_{i}. Theorem 4.2. Let X be a variety of general type, maximal Albanese dimension with χ(ω_{X}) = 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(ω_{X_{w}}) = 0 and alb_{X}(X_{w}) is a translate of the abelian subvariety ker (A(X) ⟶ T^{∨}). In particular, dimT ≠ 1 and if dimT = 0 then q(X) = dim(X).

Proof. Recall that by [11] Theorem 3, the Albanese image Z := alb_{X}(X) ⊂ Alb(X) is fibered by tori. In fact [11] shows the following: Let S be any positive dimensional component of V^{0}(ω_{X}) = V_{1}(X, K_{X}). Consider the quotient map π : A := Alb(X) ⟶ C dual to the inclusion 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 [1] Theorem 2.3, the translates through the origin of the components of V_{1}(ω_{X}) generate Pic^{0}(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}(ω_{X}) is just {O_{X}}. Let S_{i} be any component of V_{1}(ω_{X}) and B_{i} be the kernel of the corresponding projection of abelian varieties π_{i} : A ⟶ C_{i} := S_{i}^{∨}. Then the union of the B_{i} generates A. As observed above, by the result of [11], Z+B_{i} = 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) ⟶^{ϕ} Pic^{0}(X)/T ⟶ 0,

0 ⟶ F ⟶ A(X) ⟶ T^{∨} ⟶ 0.

Let \overline{S}_{i} := S_{i}/T ⊂ Pic^{0}(F) = Pic^{0}(X)/T and π : X ⟶ T^{∨} the induced morphism. For any P ∈ Pic^{0}(X)_{tors} such that P ∉ V_{1}(X, K_{X}), one has that

ϕ^{−1}(ϕ(P)) = P + T ∩ V_{1}(X, K_{X}) = ∅.

So for all Q ∈ T one has h^{0}(X, ω_{X}⊗P ⊗π^{∗}Q) = 0 and therefore also h^{i}(X, ω_{X}⊗P ⊗π^{∗}Q) = 0 for all i ≥ 0. By a result of Kollár (cf. [12], Corollary 3.3) one sees that for all Q ∈ T, the group H^{i}(π_{∗}(ω_{X}⊗P)⊗Q) is a direct summand of the group H^{i}(X, ω_{X}⊗P ⊗π^{∗}Q) and hence also vanishes. By a result of Mukai (cf. [9]) it follows that π_{∗}(ω_{X}⊗P) = 0. So for general w ∈ W = π(X), one has h^{0}(X_{w}, ω_{X_{w}}⊗P) = 0. Therefore, χ(ω_{X_{w}}) = 0 and the locous

V_{1}(X_{w}, K_{X_{w}}, Pic^{0}(F)) := {R ∈ Pic^{0}(F)|h^{0}(ω_{X_{w}}⊗R) > 0}

(which is determined by its torsion points) is a subset of V_{1}(X, K_{X})/T. It follows that the intersection of the translates through the origin of the components of V_{1}(X_{w}, K_{X_{w}}, Pic^{0}(F)) is just {O_{F} }. Therefore, arguing as in the preceeding paragraph, one sees that alb_{X}(X_{w}) = 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(ω_{X}) = 0 then q(X) = dimX.

Proof. We may assume that dimX > dimT > 0 and hence X_{w} is of general type and has dimension 1 or 2. By the classification theory of curves and surfaces χ(ω_{X_{w}}) > 0. This is the required contradiction. □

Corollary 4.4. Let X be a variety of general type, maximal Albanese dimension. If P_{1}(X) = 1 then q(X) = dimX.

Proof. If P_{1}(X) = 1 then χ(ω_{X}) = 0. Let W̃ ⟶ W be an appropriate desingularization (of W = π(X) as above). Since X is of maximal Albanese dimension, one has P_{1}(W̃) = 1 and hence since W is a subvariety of the abelian variety T^{∨}, it follows that W = T^{∨}. Since alb_{X}(X_{w}) = 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(ω_{X}) = 0. If dimT = 0, then h^{1}(X, P) = 0 for all P ∈ Pic^{0}(X) − {O_{X}}.

Proof. Assume that h^{1}(X, P) > 0 for some P ∈ Pic^{0}(X) − {O_{X}}. By [4], we may assume that P is torsion. Consider the étale cover Y ⟶ X corresponding to the subgroup < P >⊂ Pic^{0}(X). Clearly Y is of general type, maximal Albanese dimension and χ(ω_{Y}) = 0 so by the Theorem 4.2

dim(Y) = q(Y) = h^{1}(O_{Y}) = ∑ h^{1}(X, P^{⊗i}).

In particular h^{1}(X, P^{⊗i}) = 0 for i > 0. □

The following example shows that Kollár’s conjecture (17.9.2) of [2] also fails:

Example. Let d : D ⟶ E be the ℤ_{2}^{2} cover of an elliptic curve defined by d_{∗}O_{D} = O_{E} ⊕L^{∨}⊕P ⊕L^{∨}⊗P where deg(L) = 1 and P is a non-zero 2-torsion element of Pic^{0}(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 O_{E} and L^{∨}, P, L^{∨}⊗P respectively. It follows that

δ : D × D × D ⟶ E × E × E

is a ℤ_{2}^{6} cover. Let X := D × D × D/G where G ≅ ℤ_{2}^{4} is the subgroup of ℤ_{2}^{6} generated by {(id, p, l), (p, l, id), (l, id, p), (p, p, p)}. By a direct check, one can see that

(δ_{∗}O_{D×D×D})^{G} ≅

O_{E} ⊠ O_{E} ⊠ O_{E} ⊕ L^{∨} ⊠ L^{∨}⊗P ⊠ P ⊕ P ⊠ L^{∨} ⊠ L^{∨}⊗P ⊕ L^{∨}⊗P ⊠ P ⊠ L^{∨}. The singularities of X are of type \frac{1}{2}(1, 1, 1) i.e. locally isomorphic to ℂ^{3}/ < (−1, −1, −1) >. In particular all singularities are rational. Let X̃ ⟶ X be a resolution of the singularities of X and ν : X̃ ⟶ E × E × E be the induced morphism. One can compute that

ν_{∗}(ωX̃) = O_{E} ⊠O_{E} ⊠O_{E} ⊕L⊠L⊗P ⊠P ⊕P ⊠L⊠L⊗P ⊕L⊗P ⊠P ⊠L.

Therefore, X̃ is a threefold of maximal Albanese dimension with P_{1}(X) = 1 and q(X) = 3. type.

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(ω_{X×C}) = 0 and 5 = q(X) > dimX = 4. Moreover, for any étale cover X̃ ⟶ X one also has χ(X̃) = 0. One can then construct (infinitely many) examples where P_{1}(X̃) = P_{1}(X), q(X̃) = q(X).

References

[1] Chen, J.A.; Hacon, C.D. Pluricanonical maps of varieties of maximal Albanese dimension. Math. Ann. 2001,320 (2), 367-380.

[2] Kollàr, J. Shafarevich Maps and Automorphic Forms, Princeton University Press: Princeton, NJ, 1995.

[3] Hacon, C.D.; Pardini, R. On the birational geometry of varieties of maximal Albanese dimension. J. Reine Angew. Math. 2002,546, 177-199.

[4] Simpson, C. Subspaces of moduli spaces of rank one local systems. Ann. scient. École Norm. Sup. 1993 ,26, 361-401.

[5] Clemens, H; Hacon, C.D. Deformations of the trivial line bundle and vanishing theorems. Amer. J. Math. 2002, 124 (4), 769-815.

[6] Kawamata, Y. Characterization of abelian varieties. Comp. Math., 1981, 43 (2), 253-276.

[7] Chen, J.A.; Hacon, C.D. On algebraic fiber spaces over varieties of maximal Albanese dimension. Duke Math. J. 2002, 111 (1), 159-175.

[8] Lazarsfeld, R. Personal communication.

[9] Mukai, S. Duality between D(X) and D(X̂) with its application to Picard sheaves. Nagoya Math. J. 1981, 81, 153–175

[10] Mumford, D. Abelian Varieties; Tata Institute of Fundamental Research Studies in Mathematics, No. 5; Oxford University Press:London 1970.

[11] Ein, L; Lazarsfeld, R. Singularities of theta divisors, and birational geometry of irregular varieties, Jour. AMS 1997, 10 (1), 243-258.

[12] Kollàr, J. Higher direct images of dualizing sheaves II, Ann. Math. 1986, 124 (1), 171–202.

Jungkai Alfred Chen, Department of Mathematics, National Chung Cheng University, Ming Hsiung, Chia Yi, 621, Taiwan

  E-mail address: jkchen@math.ccu.edu.tw

Christopher Derek Hacon, Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA

  E-mail address: hacon@math.utah.edu