Math. Nachr. xxx (2003), 0 – 0

Varieties with P_{3} = 3 and q = dim(X)

By Christopher D. Hacon of Salt Lake City

(Received ,; revised version , 2003; accepted ,)

Abstract. We prove that if X is a complex projective variety with P_{3}(X) = 3 and q(X) = dim(X), then X is birational to a bidouble cover of A(X).

1. Introduction

It is an interesting problem in algebraic geometry to try and classify complex projective varieties modulo birational isomorphism in terms of their invariants. In the case of curves (varieties of dimension 1), there is a very satisfactory answer depending on the genus g := h^{0}(Ω_{C}^{1}). In the case of surfaces (varieties of dimension 2), the problem is essentially unsolvable and only a much coarser description is available. In dimension higher than two, the problem becomes very complex. It is therefore surprising that for certain irregular varieties it is possible to give a complete answer independently of their dimension. One example is the case of abelian varieties:

Theorem 1.1. (Kawamata, [Ka]) If X is a smooth complex projective variety with κ(X) = 0 and q(X) = dim(X), then X is birational to an abelian variety.

The statement is somewhat unsatisfactory as κ(X) is not a naive invariant (it depends on P_{m}(X) for all sufficiently big integers m). An effective version was given by Kollár in [Ko4], where it is shown that one may replace the condition κ(X) = 0 by the a priori weaker condition P_{m}(X) = 1 for some m ≥ 4. This result was further refined in [CH1], [CH4]:

Theorem 1.2. (Chen-Hacon) If X is a smooth complex projective variety with P_{m}(X) = 1 for some m ≥ 2 and q(X) = dim(X), then X is birational to an abelian variety.

It is natural to ask if one can obtain similar results for other varieties with large irregularity. The first result in this direction was obtained in [HP]:

1991 Mathematics Subject Classification. Primary: 14J10

Keywords and phrases. Birational classification, irregular varieties.

\begin{matrix}c \\ ◯ \end{matrix}WILEY-VCH Verlag Berlin GmbH, 13086 Berlin, 2003 0025-584X/03/xxx00-0000 $ 17.50+.50/0

Theorem 1.3. (Hacon-Pardini) If X is a smooth complex projective variety with P_{3}(X) = 2 and q(X) = dim(X), then X is birational to a smooth double cover of the abelian variety A(X) with branch locus given by the union of two distinct fibers of a surjective morphism A(X) ⟶ E where E is an elliptic curve.

The purpose of this paper is to classify the case when P_{3}(X) = 3 and q(X) = dim(X). Consider the following:

Example. Let q : A ⟶ E be a surjective morphism with connected fibers from an abelian variety of dimension n ≥ 2 to an elliptic curve. Let F be a general fiber of A ⟶ E and fix P, Q, R ∈ Pic^{0}(A) such that {O_{F} , P |F, Q|F, R|F } is a subgroup of Pic^{0}(F) which is isomorphic to ℤ_{2} ⊕ ℤ_{2}. In particular 2P, 2Q, 2R, P + Q + R ∈ Pic^{0}(E). Let L be a line bundle of degree 1 on E. Define M_{P} := q^{∗}L⊗P, M_{Q} := q^{∗}L⊗Q, M_{R} := q^{∗}L⊗R and E := M_{P}^{∨} ⊕ M_{Q}^{∨} ⊕ M_{R}^{∨}. Pick non-zero sections c_{1,4} ∈ H^{0}(M_{P} ⊗M_{Q}⊗M_{R}^{∨}), c_{1,6} ∈ H^{0}(M_{P} ⊗M_{Q}^{∨}⊗M_{R}) and c_{4,6} ∈ H^{0}(M_{P}^{∨}⊗M_{Q}⊗M_{R}). We will assume that these sections correspond to distinct divisors or equivalently that 2P, 2Q, 2R are distinct points of Pic^{0}(E). Following [HM] §7 and [Ca], there is a 4 − 1 cover a : X ⟶ A defined by a_{∗}O_{X} = O_{A} ⊕ E and by coefficients c_{i,j} chosen as above if (i, j) ∈ {(1, 4), (1, 6), (4, 6)} and c_{i,j} = 0 otherwise. In particular the multiplication map ϕ : S^{2}E ⟶ O_{A} ⊕ E has the form ϕ(x^{2}) = −c_{1,4}c_{1,6}, ϕ(xy) = c_{1,4}z, ϕ(xz) = −c_{1,6}y, ϕ(y^{2}) = c_{1,4}c_{4,6}, ϕ(yz) = c_{4,6}x, ϕ(z^{2}) = −c_{1,6}c_{4,6} where x, y, z are local generators of M_{P} , M_{Q}, M_{R} respectively. We remark that there is a ℤ_{2}⊕ℤ_{2} action on a_{∗}O_{X} induced by

(1, 0) · (x, y, z) = (x, −y, −z), (0, 1) · (x, y, z) = (−x, y, −z).

One sees that (a_{∗}O_{X})_{(1,0)} = O_{A} ⊕ M_{P}^{∨} is a O_{A} module of rank 2. The multiplication map S^{2}(a_{∗}O_{X})_{(1,0)} ⟶ (a_{∗}O_{X})_{(1,0)} can be defined (as an O_{A} linear map) by x^{2} ⟶ −c_{1,4}c_{1,6}. This corresponds to a degree 2 (smooth and projective) cover X_{P} ⟶ A. One sees that (a_{∗}O_{X})_{(1,0)} is a sheaf of subrings of a_{∗}O_{X}. Therefore the morphism X ⟶ A factors through a degree 2 morphism X ⟶ X_{P}. Proceeding analogously with (0, 1), (1, 1) one obtains degree 2 covers X ⟶ X_{Q} and X ⟶ X_{R}. Let D_{P} , D_{Q}, D_{R} be the divisors defined by c_{4,6}, c_{1,6}, c_{1,4}. Then the branch loci of the morphisms from X, X_{P} , X_{Q}, X_{R} to A are given by D_{P} + D_{Q} + D_{R}, D_{Q} + D_{R}, D_{P} + D_{R}, D_{P} + D_{Q} respectively.

One sees that

a_{∗}(ω_{X}) = O_{A} ⊕ M_{P} ⊕ M_{Q} ⊕ M_{R}.

and ω_{X}^{⊗2} = a^{∗}O_{A}(D_{P} + D_{Q} + D_{R}). By the projection formula, we have P_{2}(X) = P_{3}(X) = 3.

Theorem 1.4. Let X be a smooth complex projective variety with P_{3}(X) = 3 and q(X) = dim(X), then dim(X) ≥ 2 and X is a bidouble cover of its Albanese variety A(X) as described above.

Hacon, Varieties with P_{3} = 3 and q = dim(X)  5

The proof of this result relies on the generic vanishing theorems of Green and Lazarsfeld (and their generalizations cf. [ClH]), on the theory of Mukai transforms and on results of Kollár on higher direct images of dualizing sheaves. We also make use of Theorems 1.1, 1.2 and 1.3.

Remark 1.5. By [CH4] and [HP] Theorem 5.1, if P_{2}(X) = 1 or if 0 < P_{3}(X) ≤ 3, then the Albanese map a_{X} : X ⟶ A(X) is surjective. It follows that in the above theorems, one may replace q(X) = dim(X) by q(X) ≥ dim(X).

Acknowledgments. The author would like to thank A. J. Chen and R. Pardini for useful conversations.

Notation and conventions. We work over the field of complex numbers. We identify Cartier divisors and line bundles on a smooth variety, and we use the additive and multiplicative notation interchangeably. If X is a smooth projective variety, we let K_{X} be a canonical divisor, so that ω_{X} = O_{X}(K_{X}), and we denote by κ(X) the Kodaira dimension, by q(X) := h^{1}(O_{X}) the irregularity and by P_{m}(X) := h^{0}(ω_{X}^{⊗m}) the m−th plurigenus. We denote by a_{X}: X → A(X) the Albanese map and by Pic^{0}(X) the dual abelian variety to A(X) which parameterizes all topologically trivial line bundles on X. For a ℚ−divisor D we let ⌊D⌋ be the integral part and {D} the fractional part. Numerical equivalence is denoted by ≡ and we write D ≺ E if E − D is an effective divisor. If f: X → Y is a morphism, we write K_{X/Y} := K_{X} − f^{∗}K_{Y} and we often denote by F_{X/Y} the general fiber of f. A ℚ-Cartier divisor L on a projective variety X is nef if for all curves C ⊂ X, one has L.C ≥ 0. For a surjective morphism of projective varieties f : X ⟶ Y, we will say that a Cartier divisor L on X is Y -big if for an ample line bundle H on Y, there exists a positive integer m > 0 such that h^{0}(L^{⊗m}⊗f^{∗}H^{∨}) > 0. The rest of the notation is standard in algebraic geometry.

2. Preliminaries

Here we recall several results from the literature and we prove some technical statements that will be needed later.

2.1. Direct images of dualizing sheaves

The following Theorem summarizes some important facts on direct images of adjoint bundles. We refer to [Ko2], (10.1.5) for the definition of “klt” (Kawamata log terminal), and to [Ko2] (10.1.5), [Ko3] Theorem 2.1 and [HP] Lemma 2.7 for the proof.

Theorem 2.1. Let f: X → Y be a surjective map of projective varieties, X smooth and Y normal. Let M be a line bundle on X such that M ≡ f^{∗}L + Δ, where L is a ℚ−divisor on Y and (X, Δ) is klt. Then

a) R^{j}f_{∗}(ω_{X}⊗M) is torsion free for j ≥ 0;

b) R^{i}f_{∗}ω_{X} is zero for i > dim(X) − dim(Y);

c) If L is nef and big, then H^{i}(Y, R^{j}f_{∗}(ω_{X}⊗M)) = 0 for i > 0, j ≥ 0; d) If L is nef and big and g: Y → Z is any morphism with Z projective, then

H^{i}(Y, R^{j}f_{∗}(K_{X} + M)) = H^{i}(Z, g_{∗}(R^{j}f_{∗}(K_{X} + M))) i, j ≥ 0.

We also need to understand the behavior of direct images of dualizing sheaves with respect to the Stein factorization.

Let Y be a smooth variety. For a coherent sheaf F on Y, we denote by F^{∨} the dual sheaf Hom_{O_{Y}} (F, O_{Y}). A sheaf F is said to be reflexive if the natural map F → F^{∨∨} is an isomorphism. Following [OSS], we say that F is normal iff for every open set U ⊂ Y and every closed subset C ⊂ Y of codimension > 1 the restriction map F(U) → F(U \ C) is an isomorphism.

Proposition 2.2. Let X, Y be smooth projective varieties, f: X → Y a generically finite morphism, and let X →^{h} Z →^{g} Y be the Stein factorization of f. If f_{∗}ω_{X} is locally free on Y, then g is a flat morphism and g_{∗}O_{Z} = (f_{∗}ω_{X})^{∨} ⊗ ω_{Y}.

Proof. By the definition of Stein factorization, h is birational, g is finite and g_{∗}O_{Z} = f_{∗}O_{X}. The variety Z is normal, since X is normal, hence g_{∗}O_{Z} is a normal sheaf, as defined above. Denote by ω_{Z} the dualizing sheaf of Z. By [Re], Corollary (8) p. 283, if j: Z_{0} → Z is the inclusion of the smooth locus Z_{0} of Z, one has ω_{Z} = j_{∗}ω_{Z_{0}}. If we write X_{0} = h^{−1}(Z_{0}) and denote by h_{0} : X_{0} → Z_{0} the restriction of h to X_{0}, then we have (h_{0})_{∗}ω_{X_{0}} = ω_{Z_{0}} since h_{0} is a birational morphism of smooth varieties. This identification induces a sheaf map ψ: f_{∗}ω_{X} → g_{∗}ω_{Z} that is an isomorphism outside the image Σ of the singular locus of Z, which is a closed subset of Y of codimension > 1. The sheaf f_{∗}ω_{X} is normal, since it is locally free and Y is smooth, while g_{∗}ω_{Z} is normal by the definition of ω_{Z} and by the fact that g is finite. It follows that ψ is an isomorphism.

By [Hart], Chapter III, ex. 6.10b, there is an isomorphism g_{∗}(g^{!}ω_{Y}) ≅ (g_{∗}O_{Z})^{∨}⊗ω_{Y}, where, for a coherent sheaf F of Y and a finite morphism g: Z → Y, one defines g^{!}F as the coherent sheaf on Z corresponding to the g_{∗}O_{Z}−module Hom_{O_{Y}} (g_{∗}O_{Z}, F). Now g^{!}ω_{Y} = ω_{Z}, by [Hart], Chapter III, ex. 7.2, hence g_{∗}ω_{Z} = (g_{∗}O_{Z})^{∨} ⊗ ω_{Y}. Taking duals, one has (g_{∗}O_{Z})^{∨∨} ≅ (g_{∗}ω_{Z})^{∨} ⊗ ω_{Y} ≅ (f_{∗}ω_{X})^{∨} ⊗ ω_{Y}. The sheaf g_{∗}O_{Z} is normal and torsion free, since Z is normal, and thus it is reflexive by Lemma 1.1.12 of [OSS]. Thus we have g_{∗}O_{Z} ≅ (f_{∗}ω_{X})^{∨} ⊗ ω_{Y}. In particular, g_{∗}O_{Z} is locally free, namely g is flat. □

2.2. The Albanese map and the Iitaka fibration

Let X be a smooth projective variety. If κ(X) > 0, then the Iitaka fibration of X is a morphism of projective varieties f: X′ → Y, with X′ birational to X and Y of dimension κ(X), such that the general fiber of f is smooth, irreducible, of zero Kodaira dimension. The Iitaka fibration is determined only up to birational equivalence. Since we are interested in questions of a birational nature, we usually assume that X = X′ and that Y is smooth.

X has maximal Albanese dimension if dim(a_{X}(X)) = dim(X). We will need the following facts (cf. [HP] Propositions 2.1, 2.3, 2.12 and Lemma 2.14 respectively).

Proposition 2.3. Let X be a smooth projective variety of maximal Albanese dimension, and let f: X → Y be the Iitaka fibration (assume Y smooth). Denote by f_{∗}: A(X) → A(Y) the homomorphism induced by f and consider the commutative diagram:

X ⟶^{a_{X}} A(X)
f↓  ^{f_{∗}}↓
Y ⟶^{a_{Y}} A(Y).

Then:

a) Y has maximal Albanese dimension;

b) f_{∗} is surjective and ker f_{∗} is connected of dimension dim(X) − κ(X);

c) There exists an abelian variety P isogenous to ker f_{∗} such that the general fiber of f is birational to P.

Let K := ker f_{∗} and F = F_{X/Y}. Define

G := ker (Pic^{0}(X) → Pic^{0}(F)) .

Then

Lemma 2.4. G is the union of finitely many translates of Pic^{0}(Y) corresponding to the finite group

\overline{G} := G/Pic^{0}(Y) ≅ ker (Pic^{0}(K) → Pic^{0}(F)) .

2.3. Sheaves on abelian varieties

Recall the following easy corollary of the Theory of Fourier-Mukai transforms cf. [M]:

Proposition 2.5. Let ψ: F ↪ G be an inclusion of coherent sheaves on A inducing isomorphisms H^{i}(A, F⊗P) → H^{i}(A, G⊗P) for all i ≥ 0 and all P ∈ Pic^{0}(A). Then ψ is an isomorphism of sheaves.

Following [M], we will say that a coherent sheaf F on an abelian variety A is I.T. 0 if h^{i}(A, F⊗P) = 0 for all i > 0. We will say that an inclusion of coherent sheaves on A, ψ: F ↪ G is an I.T. 0 isomorphism if F, G are I.T. 0 and h^{0}(G) = h^{0}(F). From the above proposition, it follows that every I.T. 0 isomorphism F ↪ G is an isomorphism.

Lemma 2.6. Let f : X ⟶ E be a morphism from a smooth projective variety to an elliptic curve, such that K_{X} is E-big. Then, for all m ≥ 2, f_{∗}(ω_{X}^{⊗m}) is I.T. 0.

Proof. Let H be an ample divisor on E. Since K_{X} is E-big, for some r ≫ 0, the linear series |r(m − 1)K_{X} − f^{∗}H| is non-empty. Let

B ∈ |r(m − 1)K_{X} − f^{∗}H|

be a general member. Replacing X by an appropriate birational model, we may assume that B has simple normal crossings. We may write

L := O_{X} ((m − 1)K_{X} − ⌊B/r⌋) ≡ (1/r)f^{∗}H + {B/r}.

By Theorem 2.1, h^{i}(E, f_{∗}(ω_{X}⊗L)⊗P) = 0 for all i > 0 and P ∈ Pic^{0}(E). Proceeding as in the proof of Proposition 10.2 of [Ko4], one sees that the inclusion of torsion free coherent sheaves f_{∗}(ω_{X}⊗L) ⟶ f_{∗}(ω_{X}^{⊗m}) is an isomorphism at the generic point of E, and therefore we also have h^{i}(E, f_{∗}(ω_{X}^{⊗m})⊗P) = 0 for all i > 0 and P ∈ Pic^{0}(E). □

Corollary 2.7. Let X be a smooth projective variety of maximal Albanese dimension and f : X ⟶ Y the corresponding Iitaka fibration. Fix an integer m ≥ 2. Let P_{m}(Y) = P_{m}(X) and assume that the Iitaka fibration of Y is birational to g : Y ⟶ E where E is an elliptic curve. Then κ(X) = 1 and Y = E.

Proof. We have an inclusion of coherent sheaves on E

g_{∗}(ω_{Y}^{⊗m}) ⟶ g_{∗}f_{∗}(ω_{X}^{⊗m}).

By the preceding Lemma, both sheaves are I.T. 0. Since P_{m}(X) = P_{m}(Y), one has h^{0}(f_{∗}(ω_{Y}^{⊗m})) = h^{0}(g_{∗}f_{∗}(ω_{X}^{⊗m})), and hence the above inclusion is an I.T. 0 isomorphism. It follows that P_{m}(F_{X/E}) = P_{m}(F_{Y/E}) = 1 and by Theorem 1.2, F_{X/E} is birational to an abelian variety. By easy addition one has

1 = κ(Y) ≤ κ(X) ≤ κ(F_{X/E}) + dim(E) = 1

and so κ(X) = dim(E) = 1. Since f, g have connected fibers, g ∘ f : X ⟶ E also has connected fibers (of Kodaira dimension zero). So by definition of the Iitaka fibration of X, this implies that Y is birational (and hence equal) to E. □

2.4. Cohomological support loci

Let π : X ⟶ A be a morphism from a smooth projective variety to an abelian variety, T ⊂ Pic^{0}(A) be the translate of a subtorous and F be a coherent sheaf on X. One can define the cohomological support loci of F as follows:

V^{i}(X, T, F) := {P ∈ T |h^{i}(X, F⊗π^{∗}P) > 0}.

If T = Pic^{0}(X) we write V^{i}(F) or V^{i}(X, F) instead of V^{i}(X, Pic^{0}(X), F). When F = ω_{X}, the geometry of the loci V^{i}(ω_{X}) is governed by the following result of Green and Lazarsfeld (cf. [GL], [EL]):

Theorem 2.8.(Generic Vanishing Theorem) Let X be a smooth projective variety. Then:

a) V^{i}(ω_{X}) has codimension ≥ i − (dim(X) − dim(a_{X}(X)));

b) Every irreducible component of V^{i}(X, ω_{X}) is a translate of a sub-torus of Pic^{0}(X) by a torsion point; c) Let T be an irreducible component of V^{i}(ω_{X}), let P ∈ T be a point such that

V^{i}(ω_{X}) is smooth at P, and let v ∈ H^{1}(X, O_{X}) ≅ T_{P} Pic^{0}(X). If v is not tangent to T, then the sequence

H^{i−1}(X, ω_{X} ⊗ P) ⟶^{∪v} H^{i}(X, ω_{X} ⊗ P) ⟶^{∪v} H^{i+1}(X, ω_{X} ⊗ P)

is exact. Moreover, if P is a general point of T and v is tangent to T then both maps vanish;

d) If X has maximal Albanese dimension, then there are inclusions:

V^{0}(ω_{X}) ⊇ V^{1}(ω_{X}) ⊇ ⋯ ⊇ V^{n}(ω_{X}) = {O_{X}}.

When X is of maximal Albanese dimension, its geometry is very closely connected to the properties of the loci V^{i}(ω_{X}). We recall the following two results from [CH2]:

Theorem 2.9. Let X be a variety of maximal Albanese dimension. The translates through the origin of the irreducible components of V^{0}(ω_{X}) generate a subvariety of Pic^{0}(X) of dimension κ(X) − dim(X) + q(X). In particular, if X is of general type then V^{0}(X, ω_{X}) generates Pic^{0}(X).

Proposition 2.10. Let X be a variety of maximal Albanese dimension and G, Y defined as in Proposition 2.3. Then

a) V^{0}(X, Pic^{0}(X), ω_{X}) ⊂ G;

b) For every P ∈ G, the loci V^{0}(X, Pic^{0}(X), ω_{X}) ∩ (P + Pic^{0}(Y)) are non-empty;

c) If P is an isolated point of V^{0}(X, Pic^{0}(X), ω_{X}), then P = O_{X}.

The following result governs the geometry of V^{0}(ω_{X}^{⊗m}) for all m ≥ 2:

Proposition 2.11. Let X be a smooth projective variety of maximal Albanese dimension, f: X → Y the Iitaka fibration (assume Y smooth) and G defined as in Proposition 2.3. If m ≥ 2, then V^{0}(ω_{X}^{⊗m}) = G Moreover, for any fixed Q ∈ V^{0}(ω_{X}^{⊗m}), and all P ∈ Pic^{0}(Y) one has h^{0}(mK_{X} + Q + P) = h^{0}(mK_{X} + Q).

Proof. See [HP] Proposition 2.12 and [CH3] Theorem 3.2. □

We will also need the following technical result that follows from Theorem 2.8:

Proposition 2.12. Let X be a smooth projective variety of maximal Albanese dimension such that χ(ω_{X}) = 0. Let T be a component of V^{0}(ω_{X}) and let W ⊂ H^{1}(O_{X}) be a linear subspace complementary to the tangent space to T (recall that T is a translate of an abelian subvariety of Pic^{0}(X)). Let P ∈ T be a point such that h^{i}(ω_{X}⊗P) = 0 for i > dim(W) and such that P is a smooth point of V^{0}(ω_{X}). Then the natural map induced by cup product:

H^{0}(ω_{X}⊗P)⊗ ∧^{j} W → H^{j}(ω_{X}⊗P)

is an isomorphism for every j ≥ 0.

Before proving the Proposition, we wish to point out that the assumptions on P are satisfied if P ∈ T is general (cf. [EL], proof of Theorem 3).

Proof. Notice that by Theorem 2.8, a), the assumptions on X imply that V^{0}(ω_{X}) is a proper subset of Pic^{0}(X). Write n := dim(X) and w := dim(W). As explained in the proof of Theorem 3 of [EL], T is a component of V^{w}(ω_{X}) and T ⊄ V^{i}(ω_{X}) for i > w. By Theorem 2.8, c), the assumptions on P imply that the complex D(v):

… H^{j−1}(ω_{X}⊗P) ⟶^{∪v} H^{j}(ω_{X}⊗P) ⟶^{∪v} H^{j+1}(ω_{X}⊗P) …

is exact for all v ∈ W ⊂ H^{1}(X, O_{X}). The complexes D(v) fit together to give an exact complex K^{•} of vector bundles on ℙ := ℙ(W):

0 → H^{0}(ω_{X}⊗P)⊗O_{ℙ}(−w) → H^{1}(ω_{X}⊗P)⊗O_{ℙ}(−w + 1) → …
⋯ → H^{n}(ω_{X}⊗P)⊗O_{ℙ}(−w + n) → 0.

Similarly, there is an exact sequence K_{0}^{•} of vector bundles on ℙ:

⋯ → H^{0}(ω_{X}⊗P)⊗ ∧^{j−1} W ⊗O_{ℙ}(−w + j − 1) → H^{0}(ω_{X}⊗P)⊗ ∧^{j} W ⊗O_{ℙ}(−w + j) →

→ H^{0}(ω_{X}⊗P)⊗ ∧^{j+1} W ⊗O_{ℙ}(−w + j + 1) → …

There is a map of complexes K_{0}^{•} → K^{•} induced by

H^{0}(ω_{X}⊗P)⊗ ∧^{j} W ⊗O_{ℙ}(−w + j) ↪ H^{0}(ω_{X}⊗P)⊗ ∧^{j} H^{1}(X, O_{X})⊗O_{ℙ}(−w + j)

→ H^{j}(ω_{X}⊗P)⊗O_{ℙ}(−w + j)

Clearly for j = 0 there is an isomorphism

H^{0}(ω_{X}⊗P) ≅ H^{0}(ω_{X}⊗P)⊗ ∧^{0} W.

Proceeding by induction, assume 1 ≤ j ≤ dim(X) and

H^{l}(ω_{X}⊗P) ≅ H^{0}(ω_{X}⊗P)⊗ ∧^{l} W

for all l < j. Tensoring by O_{ℙ}(−j) and taking cohomology, one gets:

…  …
↓  ↓
H^{0}(ω_{X}⊗P)⊗∧^{j−1} W⊗H^{w−1}(O_{ℙ}(−w − 1)) ⟶^{∼} H^{j−1}(ω_{X}⊗P)⊗H^{w−1}(O_{ℙ}(−w − 1))
↓  ↓
H^{0}(ω_{X}⊗P)⊗ ∧^{j} W ⊗H^{w−1}(O_{ℙ}(−w)) ⟶ H^{j}(ω_{X}⊗P)⊗H^{w−1}(O_{ℙ}(−w))
↓  ↓
0  0

If the vertical rows are exact, then the required isomorphism follows from the five lemma. Consider the spectral sequence associated to the complex K^{•}⊗O_{ℙ}(−j). We have ′E_{1}^{s,t} = 0 for any s, t, thus ′E_{∞}^{s,t} = 0. On the other hand for 1 ≤ j ≤ dim(X) one has ′′E_{1}^{s,t} = 0 if s ≠ w −1 and s ≠ 0. If s = 0 then ′′E_{1}^{0,t} = H^{t}(ω_{X}⊗P)⊗H^{0}(O_{ℙ}(−w + t − j)). For 0 ≤ t ≤ w this group is 0 as −w + t − j < 0, while for t > w it is 0 since h^{t}(ω_{X}⊗P) = 0 by assumption. Comparing spectral sequences one sees that the sequence

... ⟶ H^{j−1}(ω_{X}⊗P)⊗H^{w−1}(O_{ℙ}(−w − 1)) ⟶ H^{j}(ω_{X}⊗P)⊗H^{w−1}(O_{ℙ}(−w)) ⟶ ...

is exact. The exactness of the first vertical row can be shown in the same way. □

We will also need the following lemma proved in [CH2] §3.

Lemma 2.13. Let X be a smooth projective variety and E an effective a_{X}- exceptional divisor on X. If O_{X}(E)⊗P is effective for some P ∈ Pic^{0}(X), then P = O_{X}.

Let L be a line bundle on X and denote by f_{L} : X ⟶ Y_{L} (a birational model of) the Iitaka fibration of X induced by the linear series |mL| for m sufficiently big and divisible. Let V = ∪_{m≥1}V^{0}(L^{⊗m}).

Lemma 2.14. Let T be an abelian subvariety of Pic^{0}(X) such that for some m > 0, an appropriate torsion translate of T is contained in V^{0}(L^{⊗m}). Let S be the dual abelian variety to T, then L is S-big.

Proof. By [CH3] Theorem 2.7, T ⊂ Pic^{0}(Y_{L}) and so the natural morphism X ⟶ S factors through Y_{L}. By definition of Y_{L}, L is Y_{L}-big and hence also S-big. □

We will need the following result of Ein and Lazarsfeld (see [HP] Lemma 2.13):

Lemma 2.15. Let X be a variety such that χ(ω_{X}) = 0 and such that a_{X} : X ⟶ A(X) is surjective and generically finite. Let T be an irreducible component of V^{0}(ω_{X}), and let π_{E} : X ⟶ E := Pic^{0}(T) be the morphism induced by the map A(X) ⟶ Pic^{0}(Pic^{0}(X)) ⟶ E corresponding to the inclusion T ↪ Pic^{0}(X).

Then there exists a divisor D_{T} ≺ R := Ram(a_{X}) = K_{X}, vertical with respect to π_{E} (i.e. π_{E}(D_{T}) ≠ E), such that for general P ∈ T, G_{T} := R − D_{T} is a fixed divisor of each of the linear series |K_{X} + P |.

We have the following useful Corollary (cf. [HP] Step 8 of the proof of Theorem 6.1):

Corollary 2.16. In the notation of Lemma 2.15, if dim T = 1, then for general P ∈ T, there exists a line bundle of degree 1 on E such that π_{E}^{∗}L_{P} ≺ K_{X} + P.

Finally we recall the following result:

Lemma 2.17.Let X be a smooth projective variety, let L and M be line bundles on X, and let T ⊂ Pic^{0}(X) be an irreducible subvariety of dimension t. If for all P ∈ T, dim |L + P | ≥ a and dim |M − P | ≥ b, then dim |L + M| ≥ a + b + t.

Proof. This follows by considering the morphism

|L + P | × |L − P | ⟶ |L + M|

for all P ∈ T, and the fact that any effective divisor has finitely many decompositions as the sum of two effective divisors (see [Ko2] p. 179, [HP] Lemma 2.14). □

3. Main result

In this section we will prove Theorem 1.4. Assume from now on that P_{3}(X) = 3 and q(X) = dim(X). By Theorem 5.1 of [HP], the Albanese map is surjective and since q(X) = dim(X), it is generically finite. Recall that f : X ⟶ Y denotes the Iitaka fibration of X, and Y is also of maximal Albanese dimension with q(Y) = dim(Y).

The proof of Theorem 1.4 is divided into several steps. We proceed by induction on dim(X). Since there is no curve with P_{3} = 3 and q(X) = 1, the case when dim(X) = 1 is trivially true.

Step 1.

(i) For all P ∈ Pic^{0}(X)−Pic^{0}(Y) and any irreducible component T of (P^{∨} + Pic^{0}(Y))∩ V^{0}(ω_{X}), one has h^{0}(ω_{X}^{⊗2}⊗P) + dim T ≤ 3. In particular, h^{0}(ω_{X}^{⊗2}⊗P) ≤ 2.

(ii) If P^{⊗2} ∉ Pic^{0}(Y), then h^{0}(ω_{X}⊗P) ≤ 1.

(iii) If P ∈ Pic^{0}(Y), then h^{0}(ω_{X}^{⊗2}⊗P) ≤ 3.

(iv) If T ⊂ Pic^{0}(Y) ∩ V^{0}(ω_{X}), then dim T ≤ 1.

(i) By Proposition 2.10, there exists a component T of (P^{∨} + Pic^{0}(Y)) ∩ V^{0}(ω_{X}) with dim T ≥ 1, moreover T ⊂ G. By Proposition 2.11 and Lemma 2.17, one sees that P_{3}(X) ≥ h^{0}(ω_{X}^{⊗2}⊗P) + dim T and hence that h^{0}(ω_{X}^{⊗2}⊗P) ≤ 2 and if h^{0}(ω_{X}^{⊗2}⊗P) = 2, then dim T = 1.

(ii) Suppose now that P^{⊗2} ∉ Pic^{0}(Y) and that h^{0}(ω_{X}⊗P) ≥ 2, then proceeding as above h^{0}(ω_{X}^{⊗2}⊗P^{⊗2}) ≥ 3 which is impossible.

Assertions (iii) and (iv) are similar.

Step 2. Y is birational to an abelian variety.

By Theorem 1.2, we have that P_{2}(X) ≥ 2. Since K_{X} is effective, we also have P_{2}(X) ≤ 3. Assume that P_{2}(X) = 3. Proceeding as in Step 1, one sees that V^{0}(X, Pic^{0}(Y), ω_{X}) = {O_{X}}. Since K_{X/Y} is effective, then V^{0}(ω_{Y}) = {O_{Y} } and by a result of Ein and Lazarsfeld (cf. Theorem 2.9), it follows that Y is birational to an abelian variety.

Assume now that P_{2}(X) = 2. If X is not of general type, then dim(X) > dim(Y) ≥ 1. Since K_{X/Y} is effective, also P_{2}(Y) ≤ 2 and P_{3}(Y) ≤ 3. We remark that Y is also of maximal Albanese dimension, so if P_{3}(Y) = 1 then we are done by Theorem 1.2. If P_{3}(Y) = 3 then by the induction hypothesis Y is a bidouble cover of A(Y) with P_{2}(Y) = 3. This a contradiction. If P_{3}(Y) = 2, then by Theorem 1.3, dim(Y) ≥ 2, κ(Y) = 1, P_{2}(Y) = P_{2}(X) = 2 and the Iitaka fibration ϕ_{Y} : Y ⟶ E is a surjection on to an elliptic curve. By Corollary 2.7, this is impossible.

Therefore, we may assume that X is of general type (i.e. X = Y), P_{2}(X) = 2, P_{3}(X) = 3. As we have seen above

V^{0}(X, Pic^{0}(X), ω_{X}) ≠ {O_{X}}.

It follows by Step 1 that there is a 1-dimensional component T of V^{0}(ω_{X}). Let E be the elliptic curve dual to T and π : X ⟶ E be the induced morphism. By Corollary 2.16, for general P ∈ T there exists a line bundle L of degree 1 on E and an inclusion π^{∗}L ↪ ω_{X}⊗P. The corresponding inclusion L^{⊗2} ⟶ π_{∗}(ω_{X}^{⊗2}⊗P^{⊗2}) of I.T. 0 sheaves (cf. Lemma 2.6 and Lemma 2.14) is an I.T. 0 isomorphism, since by Proposition 2.11

h^{0}(L^{⊗2}) = 2 = P_{2}(X) = h^{0}(ω_{X}^{⊗2}⊗P^{⊗2}).

It follows that the rank of the vector bundle π_{∗}(ω_{X}^{⊗2}⊗P^{⊗2}) is 1. Let F be a general fiber of π : X ⟶ E, then F is of general type. By Proposition 2.11

1 = h^{0}(ω_{F}^{⊗2}⊗P^{⊗2}) = P_{2}(F).

By Theorem 1.2, this implies that κ(F) = 0, which is a contradiction.

Step 3. Let T be any irreducible component of V^{0}(ω_{X}) then dim T ≤ 1.

Assume that dim T ≥ 2, then κ(X) ≥ 2 and h^{0}(ω_{X}^{⊗2}⊗P^{⊗2}) ≥ dim T + 1 for all P ∈ T. By Step 1, we see that dim T = 2 and 2T ⊂ Pic^{0}(Y). Let W be the abelian variety dual to T and π : X ⟶ W the induced morphism. Since K_{X} is effective, one has h^{0}(2K_{X} + P) ≥ h^{0}(K_{X} + P) ≥ 1 for all P ∈ T. By Step 1, one sees that for general (and hence by Proposition 2.11, for all) P ∈ T, h^{0}(2K_{X} + P) = 1. It follows that h^{0}(K_{X} + P) = 1 for all P ∈ T.

Assume that κ(X) = 2. By Step 2, Y ⟶ A(Y) is a birational morphism. Let π : X ⟶ A(Y) be the induced morphism. Let F be the general fiber of π, since P ∈ G, one has that P |F = O_{F} and hence h^{0}(ω_{X}⊗P |F) = h^{0}(ω_{F}) = 1. In particular, π_{∗}(ω_{X}⊗P) has generic rank 1. Proceeding as in [HP] Step 1 of the proof of Proposition 4.2, one sees that all higher cohomologies π_{∗}(ω_{X}⊗P)⊗Q also vanish. (cf. [ClH] for the corresponding generic vanishing theorems for the sheaf π_{∗}(ω_{X}⊗P).) By [Hac], it follows that π_{∗}(ω_{X}⊗P) = L is an ample line bundle with h^{0}(A(Y), L) = 1, i.e. a principal polarization. One sees that there is an inclusion π^{∗}L^{⊗2} ↪ (ω_{X}⊗P)^{⊗2}, and hence

P_{2}(X) = h^{0}(ω_{X}^{⊗2}⊗P^{⊗2}) ≥ h^{0}(A(Y), L^{⊗2}) = 4

which is impossible. Therefore, we may assume that κ(X) ≥ 3.

Since κ(X) ≥ 3, by Theorem 2.9, there exists a component S of V^{0}(K_{X}) such that S is not parallel to T. Let C be the abelian variety dual to S and p : X ⟶ C the induced morphism. By Lemma 2.15, there exists divisors D_{T} , D_{S} ≺ K_{X} vertical with respect to π, q such that for general Q ∈ T, Q′ ∈ S, the divisors G_{T} := K_{X} − D_{T} , G_{S} := K_{X} −D_{S} are fixed divisors for each of the linear series |K_{X} +Q|, |K_{X} +Q′|. Since h^{0}(D_{T} + Q), h^{0}(D_{S} + Q′) ≥ 1 for general Q ∈ T, Q′ ∈ S, this is true for any Q ∈ T, Q′ ∈ S. Moreover, for all Q ∈ T, one has h^{0}(D_{T} + Q) = h^{0}(2K_{X} + Q) = 1. Therefore,

|2K_{X} + Q| = K_{X} + G_{T} + |D_{T} + Q|, for all Q ∈ T.

Since T contains a 2-torsion element, it follows that for all Q ∈ T, one has −Q ∈ T. Following Lemma 2.17, one sees that the morphism

|K_{X} + Q| × |2K_{X} − Q| ⟶ |3K_{X}| Q ∈ T

is surjective. In particular, one may write |3K_{X}| = K_{X} + 2G_{T} + |2D_{T} |, and any divisor in |2D_{T} | is π vertical. From the morphism |K_{X} + Q′| × |2K_{X} − Q′| ⟶ |3K_{X}| with Q′ ∈ S, one sees that for Q′ ∈ S, there exist divisors Δ ∈ |D_{S} + Q′| and Γ ∈ |G_{S} + 2K_{X} − Q′| such that

Δ + Γ ∈ |3K_{X}| = K_{X} + 2G_{T} + |2D_{T} |.

Since Δ is p vertical and hence not π vertical, it follows that Δ ≺ K_{X} + 2G_{T}. But Δ varies with Q′ ∈ S and so this is a contradiction.

Step 4. If κ(X) ≥ 2, then \overline{G} ≅ (ℤ_{2})^{s}.

Assume that there is a torsion element P ∈ V^{0}(ω_{X}) such that the order of \overline{P} := P + Pic^{0}(Y) ∈ \overline{G} is greater than 2. Let T be an irreducible component of V^{0}(ω_{X}) containing P. By Proposition 2.10, dim T > 0. Following Step 1, one sees that dim T = 1. Let E be the elliptic curve dual to T and π : X ⟶ E the corresponding morphism. We may assume that P is general in T. Then, by Corollary 2.16, there is a line bundle L on E of degree 1 and an inclusion π^{∗}L ⟶ ω_{X}⊗P. It follows that there is an inclusion L^{⊗2} ⟶ π_{∗}(ω_{X}^{⊗2}⊗P^{⊗2}) which is easily seen to be an I.T. 0 isomorphism (see Lemma 2.6 and Step 1). Let F be a general fiber of X ⟶ E. It follows that h^{0}(ω_{F}^{⊗2}⊗P^{⊗2}) = 1. As above, let |K_{X} + P | = G_{T} + |D_{T} + P |. Notice that the linear series |D_{T} +P | is effective and D_{T} is X ⟶ E vertical. Therefore, (D_{T} +P)|F = P |F is also effective and hence P |F = O_{F}. So, P_{2}(F) = 1 and by Theorem 1.2, F is birational to an abelian variety. In particular, κ(X) ≤ 1, which is a contradiction.

Step 5. κ(X) = 1.

Assume to the contrary that κ(X) ≥ 2. Let T be a positive dimensional component of V^{0}(ω_{X}). By Step 3, T is 1-dimensional. Let E the dual elliptic curve and π : X ⟶ E the induced morphism. By Corollary 2.16, for general P ∈ T, there is a line bundle L on E of degree 1 and an inclusion π^{∗}L ↪ ω_{X}⊗P. Let F be a general fiber of π. Since κ(X) ≥ 2, we have that κ(F) ≥ 1 and by Theorem 1.2, P_{2}(F) ≥ 2. Therefore, the rank of the vector bundle π_{∗}(ω_{X}^{⊗2}⊗P) is at least 2. The inclusion of I.T. 0 sheaves L ⟶ π_{∗}(ω_{X}^{⊗2}⊗P) is not an isomorphism and hence h^{0}(π_{∗}(ω_{X}^{⊗2}⊗P)) > h^{0}(L) = 1. By an easy Riemann Roch computation, it follows that

h^{0}(ω_{X}^{⊗2}⊗P ⊗π^{∗}L) = h^{0}(π_{∗}(ω_{X}^{⊗2}⊗P)⊗L) ≥ rankπ_{∗}(ω_{X}^{⊗2}⊗P) + deg π_{∗}(ω_{X}^{⊗2}⊗P) ≥ 4.

However, h^{0}(ω_{X}⊗P ⊗π^{∗}L^{∨}) > 0, and by Step 4, P^{⊗2} ∈ Pic^{0}(X). By Proposition 2.11, one sees that

P_{3}(X) = h^{0}(ω_{X}^{⊗3}⊗P^{⊗2}) ≥ h^{0}(ω_{X}^{⊗2}⊗P ⊗π^{∗}L) ≥ 4.

This is the required contradiction.

Step 6. The degree of a_{X} is 4 and \overline{G} ≅ ℤ_{2} ⊕ ℤ_{2}.

Since, κ(X) = 1, the Iitaka fibration of X is birational to a morphism π : X ⟶ E where (by Step 2) E is an elliptic curve. We remark that K_{X} is X ⟶ E vertical and by Proposition 2.10,

V^{0}(ω_{X}) = \{O_{X}\} + ∑_{\overline{O}_{X}≠\overline{P} ∈\overline{G}} P + Pic^{0}(E).

We observe that the generic degree of a_{X} (and hence also the order of \overline{G}) is not two. If in fact the degree were 2, then as in [HP] Step 12 of the proof of Theorem 6.1, one can see that a_{X,∗}O_{X} = O_{A} ⊕ M^{∨}⊗P where M is the pull back of L via A(X) ⟶ E and P ∈ Pic^{0}(A) is 2 torsion. In this case P_{3}(X) ≠ 3.

By Corollary 2.16, for any general P ∈ V^{0}(ω_{X}) − Pic^{0}(Y), we have

|K_{X} + P | ⊃ |π^{∗}L_{P} | + G_{P}

where L_{P} is a line bundle of degree 1 on E and G_{P} is effective. Let \overline{P} , \overline{Q} ∈ \overline{G} be two distinct non-zero elements. Assume that the non a_{X}-exceptional components of G_{P} , G_{Q} coincide. Let E_{P} , E_{Q} be the corresponding a_{X}-exceptional components. One sees that E_{P} , E_{Q} are effective divisors on X such that O_{X}(E_{P} −E_{Q}) = O_{X}(G_{P} −G_{Q}) ≠ O_{X} ∈ Pic^{0}(X). This contradicts Lemma 2.13. Therefore, we may assume that there are distinct irreducible reduced non a_{X} exceptional components R_{P} ⊂ G_{P} , R_{Q} ⊂ G_{Q}. After replacing X by an appropriate birational model, we may assume that R_{P} , R_{Q} are smooth non a_{X} exceptional, X ⟶ E vertical divisors. For η ∈ Pic^{0}(E), consider the exact sequence

0 ⟶ ω_{X}⊗π^{∗}η ⟶ ω_{X}(R_{P} + R_{Q})⊗π^{∗}η ⟶ ω_{R_{P}} ⊕ ω_{R_{Q}} ⟶ 0.

By Theorem 2.1 R^{1}π_{∗}(ω_{X}⊗π^{∗}η) is torsion free and hence the homomorphism

π_{∗}(ω_{R_{P}} ⊕ ω_{R_{Q}}) ⟶ R^{1}π_{∗}(ω_{X}⊗π^{∗}η)

is zero. So, pushing forward via π, one obtains a short exact sequence of sheaves on E

0 ⟶ π_{∗} (ω_{X}) ⊗η ⟶ π_{∗} (ω_{X}(R_{P} + R_{Q})) ⊗η ⟶ π_{∗} (ω_{R_{P}} ⊕ ω_{R_{Q}}) ⟶ 0.

By [ClH], for general η ∈ Pic^{0}(E) H^{1}(π_{∗}ω_{X}⊗η) = 0 and so for general η ∈ Pic^{0}(E) this sequence is exact on global sections. Therefore,

h^{0}(ω_{X}(R_{P} + R_{Q})⊗π^{∗}η) ≥ h^{0}(ω_{R_{P}} ⊕ ω_{R_{Q}}) ≥ 2.

Assume that one can choose P ∈ G such that \overline{P} ≠ −\overline{P} in \overline{G}. Then

|K_{X} + R_{P} + R_{P^{∨}} + η| × π^{∗}|L_{P} + L_{P^{∨}} − η| + (G_{P} − R_{P} + G_{P^{∨}} − R_{P^{∨}}) ⟶ |3K_{X}|.

Since for all η ∈ Pic^{0}(E), dim |K_{X} + R_{P} + R_{P^{∨}} + η| ≥ 1 and dim |L_{P} + L_{P^{∨}} − η| = 1, it follows by Lemma 2.17 that h^{0}(3K_{X}) ≥ 4 a contradiction. So for all P ∈ G, 2P ∈ Pic^{0}(Y). In particular \overline{G} ≅ (ℤ_{2})^{s}.

Similarly, one sees that for any P ∈ V^{0}(ω_{X}) − Pic^{0}(Y), the divisor G_{P} contains at most one irreducible non a_{X} exceptional component R_{P}. Suppose in fact that R_{1}, R_{2} are two distinct non a_{X} exceptional components of R_{P}, then proceeding as above, one sees that for all η ∈ Pic^{0}(E), h^{0}(ω_{X}(R_{1} + R_{2})⊗η) ≥ 2. The divisor 2R_{P} − R_{1} − R_{2} is effective and so for all η ∈ Pic^{0}(E) there is an inclusion

π^{∗}|L_{P}^{⊗2}⊗η^{∨}| × |ω_{X}(R_{1} + R_{2})⊗η| ⟶ |3K_{X}|.

By Lemma 2.17, dim |3K_{X}| ≥ 3 which is impossible.

Since |\overline{G}| > 2, for any \overline{P} ∈ \overline{G}, there exist \overline{Q} ∈ \overline{G} such that \overline{P} , \overline{Q}, \overline{P} + \overline{Q} are distinct non-zero elements. The inclusion

π^{∗}|L_{P} + L_{Q} + L_{P +Q}| + G_{P} + G_{Q} + G_{P +Q} ⟶ |3K_{X}|

is an isomorphism, and hence (G_{P})_{red} ≺ K_{X}. \overline{P} , \overline{Q} ∈ \overline{G} are distinct non-zero elements of \overline{G}, so R_{P} ≠ R_{Q} as observed above. Since R_{P} ≺ K_{X}, one sees that

3 = h^{0}(3K_{X}) ≥ h^{0}(K_{X} + ∑_{O_{X}≠P ∈\overline{G}} R_{P}) ≥ |\overline{G}| − 1

and so |\overline{G}| ≤ 4 and hence |\overline{G}| is 4.

Step 7. a_{X,∗}(ω_{X}) is locally free coherent sheaf of rank 4.

Assume now that \overline{G} ≅ ℤ_{2} ⊕ ℤ_{2}. We have that

V^{0}(ω_{X}) = {O_{X}} ∪ P + Pic^{0}(E) ∪ +Q + Pic^{0}(E) ∪ (P + Q) + Pic^{0}(E)

where π : X ⟶ E is birational to the Iitaka fibration and \overline{P} , \overline{Q} generate \overline{G}. The sheaf π_{∗}ω_{X} is a line bundle with h^{0}(π_{∗}ω_{X}) = 1 and h^{0}(π_{∗}ω_{X}⊗R) = 0 for all O_{E} ≠ R ∈ Pic^{0}(E). It follows that π_{∗}ω_{X} = O_{E}. Similarly, the sheaf π_{∗}(ω_{X}⊗P) is a line bundle with h^{0}(π_{∗}(ω_{X}⊗P)⊗R) = 1 for general R ∈ Pic^{0}(E), and therefore L_{P} := π_{∗}(ω_{X}⊗P) is a line bundle of degree 1. It follows by the projection formula, that for all R ∈ Pic^{0}(E)

h^{0}(ω_{X}⊗P ⊗π^{∗}R) = h^{0}(L_{P} ⊗R) = 1.

So we have a homomorphism of coherent sheaves of rank 4

ψ : O_{A(X)} ⊕ q^{∗}L_{P} ⊗P^{∨} ⊕ q^{∗}L_{Q}⊗Q^{∨} ⊕ q^{∗}L_{P +Q}⊗P^{∨}⊗Q^{∨} ⟶ a_{X,∗}ω_{X}

where q : A(X) ⟶ E is such that π = q ∘ a_{X}. By restricting to a general fiber of q, one sees that ψ is an inclusion. Let

F := O_{A(X)} ⊕ q^{∗}L_{P} ⊗P^{∨} ⊕ q^{∗}L_{Q}⊗Q^{∨} ⊕ q^{∗}L_{P +Q}⊗P^{∨}⊗Q^{∨}.

We wish to show that ψ induces isomorphisms of all cohomology groups

H_{R}^{i}(ψ) : H^{i}(F⊗R) ⟶ H^{i}(a_{X,∗}ω_{X}⊗R)

for all i ≥ 0 and all R ∈ Pic^{0}(A(X)) and hence by Proposition 2.5, ψ is an isomorphism of sheaves.

If R ∉ V^{0}(ω_{X}), then H^{i}(F⊗R) = H^{i}(a_{X,∗}ω_{X}⊗R) = 0 (cf. Theorem 2.8, d)) and there is nothing to prove. If R ∈ V^{0}(ω_{X}), then by dimension count, H_{R}^{0}(ψ) is an isomorphism. We will prove that for i > 0, H_{R}^{i}(ψ) is an isomorphism for all R ∈ P + Pic^{0}(E). (The proof for R ∈ Q + Pic^{0}(E), R ∈ P + Q + Pic^{0}(E) is identical and for R = O_{X} it proceeds analogously). For such a choice of R one has:

H^{i} (F ⊗ R) = H^{i} (q^{∗}L_{P} ⊗P^{∨}⊗R) .

Let W ⊂ H^{1}(O_{X}) be a subspace complementary to the tangent space to V^{0}(ω_{X}) in Pic^{0}(X). The assumptions of Proposition 2.12 are satisfied for all R ∈ P + Pic^{0}(E), hence for all 0 ≤ j ≤ dim(X) and R ∈ P + Pic^{0}(E) there are isomorphisms

H^{j}(ω_{X}⊗R) ≅ H^{0}(ω_{X}⊗R)⊗ ∧^{j} W.

induced by cup product. The required isomorphism is given by the following commutative diagram

H^{0}(L_{P} ⊗P^{∨}⊗R)⊗ ∧^{i} W ⟶^{≅} H^{0}(a_{X,∗}ω_{X}⊗R)⊗ ∧^{i} W
≅↓  ↓≅
H^{i}(L_{P} ⊗P^{∨}⊗R)  ⟶  H^{i}(a_{X,∗}ω_{X}⊗R)

Step 8. Conclusion.

Let X ⟶ Z ⟶^{g} A(X) be the Stein factorization of the Albanese morphism of X. Then Z is normal and g : Z ⟶ A(X) is finite of degree 4. By Proposition 2.2, one has that g_{∗}O_{Z} = (a_{X,∗}ω_{X})^{∨}.

Let E := q^{∗}L_{P}^{∨}⊗P ⊕q^{∗}L_{Q}^{∨}⊗Q⊕q^{∗}L_{P +Q}^{∨}⊗P ⊗Q. Following [HM] §7, one sees that the 4−1 cover Z ⟶ A is defined by a section of Λ^{2}S^{2}E^{∨}⊗Λ^{3}E. The only possible non zero coefficients are c_{1,4} ∈ H^{0}(A, q^{∗}(L_{P} ⊗L_{Q}⊗L_{P +Q}^{∨})), c_{1,6} ∈ H^{0}(A, q^{∗}(L_{P} ⊗L_{Q}^{∨}⊗L_{P +Q})⊗(P^{∨})^{⊗2}), c_{4,6} ∈ H^{0}(A, q^{∗}(L_{P}^{∨}⊗L_{Q}⊗L_{P +Q})⊗(Q^{∨})^{⊗2}) and therefore this is a bidouble cover as described in the introduction.

References

[Ca] F. Catanese, On the moduli of surfaces of general type, J. Diff. Geom. 19 (1984) 483-515.

[ClH] H. Clemens, C. D. Hacon, Deformations of flat line bundles and their metrics, Amer. Journal of Math. 124.4 August 2002, 769-815.

[CH1] A. J. Chen, C. D. Hacon, Characterization of Abelian Varieties, Inv. Math. 143 (2001) 2, 435-447.

[CH2] A. J. Chen, C. D. Hacon, Pluricanonical maps of varieties of maximal Albanese dimension, Math. Annalen 320 (2001) 2, 367-380.

[CH3] A. J. Chen, C. D. Hacon, On the irregularity of the image of the Iitaka fibration, Preprint 2002.

[CH4] A. J. Chen, C. D. Hacon, On Algebraic fiber spaces over varieties of maximal Albanese dimension, Duke Math. Jour. 111 issue 1, 159-175.

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

[GL] M. Green, R. Lazarsfeld, Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville, Invent. Math. 90 (1987 ), 389–407.

[Hac] C. D. Hacon Fourier transforms, generic vanishing theorems and polarizations of abelian varieties, Math. Zeischrift 235 (2000) 717-726.

[Hart] R. Hartshorne, Algebraic Geometry, Graduate texts in Math 52, Springer Verlag, (1977).

[HP] C. D. Hacon, R. Pardini On the birational geometry of varieties of maximal albanese dimension, Jour. Reine Angew. Math. 546 (2002).

[HM] D. W. Hahn, R. Miranda, Quadruple covers of algebraic varieties, J. Algebraic Geom. 8 (1999) 1-30

[Ka] Y. Kawamata, Characterization of Abelian Varieties, Comp. Math. 43 (1981), 253-276.

[Ko1] J. Kollár Subadditivity of the Kodaira Dimension: Fibers of General Type, Adv. Stud. in Pure Math., Algebraic Geometry, Sendai, 1985, pp. 361-398

[Ko2] J. Kollàr, Shafarevich Maps and Automorphic Forms, Princeton University Press (1995).

[Ko3] J. Kollàr, Higher direct images of dualizing sheaves I, Ann. Math. 123 (1986), 11–42.

[Ko4] J. Kollàr, Shafarevich maps and plurigenera of algebraic varieties, Invent. Math. 113 (1993), 177–215.

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

[OSS] C. Okonek, M. Schneider, H. Spindler, Vector bundles on complex projective spaces, Progress in Mathematics, 3, Birkhäuser, Boston, Mass., (1980).

[Re] M. Reid, Canonical 3-folds, Journées de Géometrie Algébrique d’Angers, Juillet 1979, A. Beauville ed., Sijthoff & Noordhoff, Alphen aan den Rijn (1980), 273– 310.

Christopher D. Hacon Department of Mathematics, University of Utah 155 South 1400 East, JWB 233 Salt Lake City, UT 84112-0090 - USA email: hacon@math.utah.edu fax: 801-581-4148