EFFECTIVE GENERATION OF ADJOINT LINEAR SERIES OF IRREGULAR VARIETIES

JUNGKAI A. CHEN AND CHRISTOPHER D. HACON

Abstract. We give a nonvanishing theorem for coherent sheaves on abelian varieties. As application, we study adjoint linear series on irregular varieties. In particular we prove that if X is a variety of maximal Albanese dimension, then |6K_{X}| defines the stable canonical map.

1. Introduction

Let A be an abelian variety over ℂ. Lefschetz’s theorem asserts that for an ample line bundle L, L^{2} is free (and L^{3} is very ample). One way to see this is as follows: for any fixed point x ∈ A, for a general Q ∈ Pic^{0}(A), there is a section of L ⊗ Q not passing through x. Hence x is not a base point of L^{2} = (L⊗Q)⊗(L⊗Q^{∨}).

In §2, using the theory of Fourier-Mukai transforms, we show that a similar argument works for coherent sheaves F on an abelian variety A satisfying the I.T. of index 0 (i.e. such that h^{i}(A, F⊗P) = 0 for all i > 0 and P ∈ Pic^{0}(A)).

As an application of this, in §3, we study adjoint linear series of irregular varieties. For any variety X with q(X) > 0, there is a natural morphism to an abelian variety (the Albanese map) which we denote by a : X → Alb(X). We study adjoint line bundles ω_{X}⊗L on X by considering the pushforward sheaf a_{∗}(ω_{X}⊗L). We are able to prove that the linear series |2(K_{X} + mL)| is base point free on smooth fibers for any ample line bundle L and any integer m ≥ \frac{e^{2}+e+2}{2}, where e denotes the dimension of a general fiber.

In §4, when X is of general type, we also obtain an effective bound for the pluricanonical map being birational (depending on the pluricanonical maps of the fibers of the Albanese map). In particular if X is of general type and dim(alb_{X}(X)) ≥ dim(X) − 1, then |6K_{X}| induces a birational map.

Acknowledgments. The first author was partially supported by National Science Council, Taiwan (NSC-89-2115-M-194-016). The first author is grateful to Y. Kawamata and I-Chau Huang for helpful conversations.

Notation and conventions.

(1.1) Throughout this paper, we work over the field of complex numbers ℂ. (1.2) For D_{1}, D_{2} ℚ-divisors on a variety X, we write D_{1} ≺ D_{2} if D_{2} − D_{1} is

effective, and D_{1} ≡ D_{2} if D_{1} and D_{2} are numerically equivalent.

(1.3) |D| will denote the linear series associated to the divisor D, and Bs|D| denotes the base locus of |D|.

(1.4) 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}.

(1.5) We will denote by Alb(X) the Albanese variety of X, by a = alb_{X} : X ⟶ Alb(X) the Albanese morphism. As usual Pic^{0}(X) is the abelian variety dual to Alb(X) parameterizing all topologically trivial line bundles on X.

1991 Mathematics Subject Classification. 14J40 (14E05).

(1.6) We will say that a ℚ-divisor Δ is klt (Kawamata log terminal) if Δ has normal crossings support and ⌊Δ⌋ = 0. We refer to [Ko2] 10.1.5 for the general definition of klt divisor.

(1.7) We will say that a Cartier divisor L on X is nef if for every curve C ⊂ X, one has C.L ≥ 0. For a morphism f : X ⟶ Y, we will say that L is Y -big if for any ample Cartier divisor H on Y, there exists a positive integer m > 0 such that h^{0}(X, L^{m}⊗f^{∗}H^{∨}) > 0.

(1.8) 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.

2. Fourier Mukai Transforms and Essential base points

Let A be an abelian variety, and denote the corresponding dual abelian variety by Â. Let P be the normalized Poincaré bundle on A × Â. For any point y ∈ Â let P_{y} = P_{y} denote the associated topological trivial line bundle. Define the functor Ŝ of O_{A}-modules into the category of OÂ-modules by

Ŝ(M) = π_{Â,∗}(P⊗π_{A}^{∗}M).

The derived functor RŜ of Ŝ then induces an equivalence of categories between the two derived categories D(A) and D(Â). In fact, by [M]: There are isomorphisms of functors:

RS ∘ RŜ ≅ (−1_{A})^{∗}[−g]

and

RŜ ∘ RS ≅ (−1Â)^{∗}[−g],

where [−g] denotes ”shift the complex g places to the right”.

The index theorem (I.T.) is said to hold for a coherent sheaf F on A if there exists an integer i(F) such that for all j ≠ i(F), H^{j}(A, F⊗P) = 0 for all P ∈ Pic^{0}(A). It is easily seen that if F satisfies the I.T., then R^{j}Ŝ(F) = 0 for all j ≠ i(F). We will denote the coherent sheaf R^{i(F)}Ŝ(F) on Â by F̂.

Definition 2.1. Let F be a coherent sheaf on an irregular variety X. F is said to have an essential base point at y if there is a surjective map F → O_{y} such that for all P ∈ Pic^{0}(X), the induced map H^{0}(X, F⊗P) → H^{0}(X, O_{y}) is zero.

Proposition 2.2. If F is a non-zero sheaf on an abelian variety satisfying I.T. of index 0, then F has no essential base points.

Proof. Suppose that F has an essential base point at y. Let ϕ ≠ 0 be a surjective map as above, F_{y} := ker(F ⟶^{ϕ≠0} O_{y}). Then the exact sequence

0 → F_{y}⊗P → F⊗P → O_{y} → 0

gives

H^{0}(A, F_{y}⊗P) ⟶^{≅} H^{0}(A, F⊗P),

H^{0}(A, O_{y}) ⟶^{≅} H^{1}(A, F_{y}⊗P),

for all P ∈ Pic^{0}(A). Computing the Fourier-Mukai transform, one has

0 → R^{0}Ŝ(F_{y}) → R^{0}Ŝ(F) → P_{y} → R^{1}Ŝ(F_{y}) → 0.

By Grauert’s theorem, V := R^{0}Ŝ(F) and R^{0}Ŝ(F_{y}) are vector bundles of the same rank, and R^{1}Ŝ(F_{y}) is a line bundle. If follows that P_{y} → R^{1}Ŝ(F_{y}) is an isomorphism and hence V → P_{y} is the zero map. Therefore, one has a triangle

R^{•}Ŝ(F_{y}) ⟶ R^{•}Ŝ(F) ⟶^{0} R^{•}Ŝ(O_{y})

in D(Â). There is a second triangle in D(Â) given by (cf. [Ha] TR2)

R^{•}Ŝ(F) ⟶^{0} R^{•}Ŝ(O_{y}) ⟶ R^{•}Ŝ(F_{y})[1]

which is easily seen to be isomorphic to (cf. [Ha] TR3)

V ≅ R^{•}Ŝ(F) ⟶^{0} P_{y} ≅ R^{•}Ŝ(O_{y}) → P_{y} ⊕ V [1].

Taking the Fourier-Mukai transform again, one has F_{y} ≅ F ⊕ O_{y}[1] in D(A). This is the required contradiction. □

Definition 2.3. Let F be a coherent sheaf on a projective variety X, we will say that F is spanned if for any point x ∈ X, the homomorphism H^{0}(X, F) ⟶ H^{0}(X, F⊗O_{x}) is surjective. We will say that F is k-spanned if for any 0-dimensional subscheme Z ⊂ X of length k, the homomorphism H^{0}(X, F) ⟶ H^{0}(F⊗O_{Z}) is surjective. F is weakly k-generated if for every coherent sheaf Q with 0-dimensional support and length k, and for every surjective homomorphism F ⟶ Q with kernel K then h^{0}(X, K) = h^{0}(X, F) − k.

We remark that if h^{i}(X, F) = 0 for all i > 0, then F is weakly k-generated iff h^{i}(X, K) = 0 for all i > 0. If F is a vector bundle of rank r which is rk-weakly generated, then F is k-spanned.

Proposition 2.4. Let A be an abelian variety, L an ample line bundle on A, F be a non-zero coherent sheaf. And let t be a positive integer such that for all integers s, t ≥ s ≥ 0, the sheaf F⊗L^{s} satisfies the I.T. of index 0, then F⊗L^{t}⊗P is weakly t-generated for all P ∈ Pic^{0}(A).

Proof. Let Q be a coherent sheaf of length t supported at finitely many points x_{i} ∈ A. And let F ⟶ Q be a surjective homomorphism and K be the corresponding kernel. Let 0 = Q_{t} ⊂ Q_{t−1} ⊂ ... ⊂ Q_{0} = Q be a filtration of Q such that Q_{j}/Q_{j+1} ≅ ℂ(x_{i}) and let K_{i} := ker (F ⟶ Q/Q_{i}). Clearly, F = K_{0} ⊃ K_{1} ⊃ ... ⊃ K_{t} = K.

Since F has no essential base points, one sees that for general P_{1} ∈ Pic^{0}(A), h^{0}(A, K_{1}⊗P_{1}) = h^{0}(A, F⊗P_{1}) − 1. Since |L| also has no essential base points, one sees that for any given x_{i} ∈ A, general P_{2} ∈ Pic^{0}(A), x_{i} ∉ Bs|L+P_{2}|. In particular, for all P ∈ Pic^{0}(A), h^{0}(A, K_{1}⊗L⊗P) = h^{0}(A, F⊗L⊗P)−1 and h^{i}(A, K_{1}⊗L⊗P) = h^{i}(A, F⊗L⊗P) = 0 for all i > 0.

By induction, let’s assume that K_{j}⊗L^{j} satisfies I.T. of index 0. Similarly, one sees that K_{j+1}⊗L^{j+1} satisfies I.T. of index 0 and h^{i}(A, K_{j+1}⊗L^{j+1}⊗P) = 0 for all P ∈ Pic^{0}(A). Thus we are done. □

So, for example, we recover the following well known result: Let L be an ample line bundle on an abelian variety A. Then |2L| is base point free |3L| is very ample and |(k + 1)L| is k-spanned. More generally, since h^{j}(A, R^{i}a_{∗}ω_{X}⊗L⊗P) = 0 for all j > 0 and all P ∈ Pic^{0}(A) (cf. [Ko1] Theorem 2.1), one sees that for any s > 0, R^{i}a_{∗}ω_{X}⊗L^{s} satisfies the I.T. of index 0 and hence we have:

Corollary 2.5. Let a : X ⟶ A be a morphism from a smooth projective variety to an abelian variety, L an ample Cartier divisor on A. Then for i ≥ 0, t ≥ 1 and all P ∈ Pic^{0}(A), the coherent sheaf R^{i}a_{∗}ω_{X}⊗L^{t}⊗P is (t − 1)-weakly generated.

For example if z ∈ a(X) is a general point and t > h^{i}(X_{z}, K_{X_{z}}) then R^{i}a_{∗}ω_{X}⊗L^{t} is generated at z. (compare [Ka2] Conjecture 1.4).

3. Adjoint linear series of irregular varieties

In order to apply the criterion for essential base point freeness to linear series of irregular varieties, we will need the following result:

Theorem 3.1. Let X be a smooth projective varieties with q(X) > 0. Let a : X → A be a non-constant morphism to an abelian variety A. Let L be a line bundle on X and fix a point x ∈ X. Suppose that

i) h^{i}(A, a_{∗}L⊗P) = 0 for all i > 0 and P ∈ Pic^{0}(A).

ii) a_{∗}(L⊗I_{x}) ≠ a_{∗}(L).

Then x is not an essential base point of |L|. In particular, x is not a base point of |2L|.

Proof. For a fixed point x in X, we start by considering the sequence

0 → L⊗I_{x} → L → O_{x} → 0,

and its push-forward via a : X → A

0 → a_{∗}(L⊗I_{x}) → a_{∗}(L) → a_{∗}(O_{x}) → R^{1}a_{∗}(L⊗I_{x}) → ….

By assumption a_{∗}(L⊗I_{x}) → a_{∗}(L) is not an isomorphism. It is clear that a_{∗}(O_{x}) = O_{y} where y = a(x). Let Q be the kernel of O_{y} → R^{1}a_{∗}((L)⊗I_{x}). It follows that Q ≠ 0. However, O_{y} has no non-zero sub-sheaf other than itself. Hence Q = O_{y}. We therefore have an exact sequence:

0 → a_{∗}(L⊗I_{x}) → a_{∗}(L) → O_{y} → 0.

a_{∗}(L) satisfies the I.T. of index 0. By Proposition 2.2, y is not an essential base point of a_{∗}(L). Therefore, x is not a base point of L⊗a^{∗}Q for some Q ∈ Pic^{0}(A). By the semi-continuity of h^{0}(X, L⊗I_{x}⊗a^{∗}Q) as Q varies in Pic^{0}(A), it follows that x is not a base point of |L + a^{∗}Q| for general Q ∈ Pic^{0}(X), and hence the theorem. □

We now illustrate some applications of this theorem.

Corollary 3.2. Let X be a smooth projective variety, f : X ⟶ Z be a surjective morphism to a normal variety Z which is birational to its Albanese image. Let L be a Cartier divisor on X, M and Δ are ℚ-Cartier ℚ-divisors on X such that (X, Δ) is klt, and M is nef and either big on the general fiber or numerically trivial on X, H a nef and big ℚ-Cartier ℚ-divisor on Z such that L ≡ f^{∗}H + M + Δ. Let z ∈ Z be a general point, and y ∈ X_{z} such that y ∉ Bs|(K_{X} + L)|_{X_{z}}|. Then y in not an essential base base point of |K_{X} + L| and y is not a base point of |2K_{X} + 2L|.

Proof. Let ν : Z ⟶ A be the Albanese map of Z and let a := ν ∘ f. For all i > 0 and P ∈ Pic^{0}(A), one has h^{i}(A, a_{∗}(ω_{X}⊗L)⊗P) = h^{i}(f_{∗}(ω_{X}⊗L)⊗ν^{∗}P) = 0 (cf. [Ko2] Corollary 10.15 and [HP] Lemma 2.7) and hence a_{∗}(ω_{X}⊗L) satisfies the I.T. of index 0. Let z ∈ Z be a general point such that a_{∗}(ω_{X}⊗L) is free near ν(z). Consider the surjective homomorphism

a_{∗}(ω_{X}⊗L)⊗ℂ(ν(z)) ≅ f_{∗}(ω_{X}⊗L)⊗ℂ(z) ⟶^{≅} H^{0}(X_{z}, ω_{X}⊗L|_{X_{z}}) ⟶ ℂ(y).

It follows that a_{∗}(ω_{X}⊗L⊗I_{y}) ≠ a_{∗}(ω_{X}⊗L) and the proof follows by Theorem 3.1. □

Corollary 3.3. Let X be a variety of maximal Albanese dimension, and L a nef and big line bundle on X. Then Bs|2K_{X} +2L| is contained in the exceptional locus of a : X ⟶ Alb(X).

Proof. Let a : X → A be the Albanese map. We first note that a_{∗}(ω_{X}⊗L) is a sheaf satisfying I.T. of index 0 (cf. [Ko2] Corollary 10.15 and [HP] Lemma 2.7). Clearly, it is non-zero since a is generically finite onto its image.

Let x ∈ X be a point which is not in the exceptional divisor. There is an open neighborhood U of x such that a : U → a(U) is a finite morphism. Hence condition ii) is satisfied and we are done. □

One can generalize this result as follows:

Theorem 3.4. Let a : X ⟶ Z := alb_{X}(X) ⊂ A be the Albanese morphism of a nonsingular projective variety X to a subvariety Z contained in an abelian variety A. Let L be an ample line bundle on X, e := dim(X) − dim(Z) and m := \frac{e^{2}+e+2}{2}. Let U be an open subset of Z and V := a^{−1}(U) such that a|_{V} : V ⟶ U is a smooth morphism. Then |K_{X} + mL| has no essential base points contained in U. In particular Bs|2K_{X} + 2mL| ⊂ X − V.

Proof. By [AS] one has for all z ∈ U and x ∈ X_{z} that H^{i}(X_{z}, ω_{X_{z}}⊗L^{m}⊗I_{x}) = 0 for all i > 0. Claim. R^{i}(a_{∗}(ω_{X}⊗L^{m}⊗I_{x}))|_{U} = 0 for all i > 0.

It follows that for z ∈ U, we have an exact sequence of sheaves on Z

0 ⟶ a_{∗}(ω_{X}⊗L^{m}⊗I_{x}) ⟶ a_{∗}(ω_{X}⊗L^{m}) ⟶ O_{z} ⟶ 0.

By [Ko2] Corollary 10.15 and [HP] Lemma 2.7, a_{∗}(ω_{X}⊗L^{m}) Satisfies the I.T. with index 0. Therefore, by Proposition 2.2, for some (and hence general) P ∈ Pic^{0}(A)

h^{0}(ω_{X}⊗L^{m}⊗I_{x}⊗a^{∗}P) = h^{0}(a_{∗}(ω_{X}⊗L^{m}⊗I_{x})⊗P) = h^{0}(a_{∗}(ω_{X}⊗L^{m})⊗P) − 1.

It suffices therefore to prove the claim. If dim(Z) = 0 there is nothing to prove. We proceed by induction. For any m > 0 there is an exact sequence

O_{z} ⟶ R^{1}a_{∗}(ω_{X}⊗L^{m}⊗I_{x}) ⟶ R^{1}a_{∗}(ω_{X}⊗L^{m}) = 0

and R^{i}a_{∗}(ω_{X}⊗L^{m}⊗I_{x}) = 0 for all i ≥ 2. Therefore, we may assume that if R^{1}a_{∗}(ω_{X}⊗L^{m}⊗I_{x}) ≠ 0, then O_{z} ≅ R^{1}a_{∗}(ω_{X}⊗L^{m}⊗I_{x}). Let H be a general smooth very ample divisor on Z containing z = a(x), and let H̃ = a^{−1}(H). There is an exact sequence

0 ⟶ ω_{X}⊗L^{m}⊗I_{x} ⟶ ω_{X}⊗L^{m}⊗O_{X}(H̃)⊗I_{x} ⟶ ωH̃⊗L^{m}⊗I_{x} ⟶ 0.

Pushing forward via a, using the inductive hypothesis and the projection formula, one obtains the exact sequence

R^{1}a_{∗}(ω_{X}⊗L^{m}⊗I_{x}) ⟶^{0} R^{1}a_{∗}(ω_{X}⊗L^{m}⊗I_{x})⊗O_{Z}(H) ⟶ R^{1}a_{∗}(ωH̃⊗L^{m}⊗I_{x}) = 0.

(the first map is 0 since H is general containing z and R^{1}a_{∗}(ω_{X}⊗L^{m}⊗I_{x}) = O_{z}). It follows that R^{1}a_{∗}(ω_{X}⊗L^{m}⊗I_{x}) = 0. □

Of course if e ≤ 4 it suffices to assume m = e + 1 (cf. [Ka1]). Similarly one can show:

Example 3.5. Let X be a ℙ^{m}-bundle over a subvariety of an abelian variety, and L be an ample line bundle on X. Then 2(K_{X} + (m + 1)L) is base point free.

4. Pluricanonical maps of irregular varieties

Let X be a smooth projective variety and Z = alb_{X}(X) ⊂ Alb(X) its Albanese image.

Lemma 4.1. Let g : X ⟶ S be a surjective morphism of smooth projective varieties, L a nef and big Cartier divisor on S. Let X_{s} be a general fiber of X ⟶ S. For any integer m ≥ 2, assume that P_{m}(X_{s}) > 0 and write |mK_{X_{s}}| = |H^{m}| + F^{m} where |H^{m}| has no fixed divisors. Assume furthermore that K_{X} is S-big and κ(S) ≥ 0. Then, after replacing X by an appropriate birational model, there exists an integer b > 0 (any sufficiently big and divisible integer b > 0 will do) such that for any sufficiently big and divisible integers a > 0, there is a divisor B ∈ |ab(m − 1)K_{X} − bg^{∗}L|, B has normal crossings support and ⌊\frac{B}{ab}⌋|_{X_{s}} ≺ ⌊\frac{B}{ab}|_{X_{s}}⌋ ≺ F^{m}.

Proof. (m − 1)K_{X} is S-big, so for all sufficiently big and divisible integers r_{0}, the linear series |r_{0}(m − 1)K_{X} − 2g^{∗}L| is non-empty. Let D be a general member of this linear series.

Replacying X by an appropriate birational model, we may assume that |mK_{X_{s}}| = |H^{m}| + F^{m} where |H^{m}| is base point free and F^{m} has normal crossings support. Moreover we may assume that

|(r_{0}(m − 1)K_{X} − 2g^{∗}L)|_{X_{s}}| = |N| + G

where |N| is base point free, G+F^{m} has normal crossings support. Let D = ∑ d_{i}D_{i} with D_{i} reduced irreducible divisors, and let d := ∑ d_{i}. Since s ∈ S is a general point, also D_{i}|_{X_{s}} is reduced and so for G = ∑ g_{i}G_{i} we also have g_{i} ≤ d. Let t > \frac{md}{r_{0}}, then

⌊\frac{m}{m − 1}F^{m} + \frac{G}{tr_{0}} ⌋ ≺ F^{m}.

Since κ(S) ≥ 0, for r sufficiently big and divisible, the linear series g^{∗}|(t−1)r(m− 1)K_{S}| is non-empty.

By [V1] [V2] (see also [Ko3], §10), for all sufficiently big and divisible integers r > 0, there is a surjection

H^{0}(X, ω_{X/S}^{(t−1)r(m−1)}⊗g^{∗}L^{r/r_{0}})⊗ℂ(X) ⟶ H^{0}(X_{s}, ω_{X_{s}}^{(t−1)r(m−1)}).

Assume that r_{0}m divides r. Let B_{1}, B_{2}, B_{3} be general divisors in the linear series |ω_{X/S}^{(t−1)r(m−1)}⊗g^{∗}L^{r/r_{0}}|, |r_{0}(m − 1)K_{X} − 2g^{∗}L|, g^{∗}|(t − 1)r(m − 1)K_{S}| respectively. It is easy to see that

B_{1}|_{X_{s}} = H̄^{(t−1)r(m−1)} + F^{(t−1)r(m−1)} ≺ H̄^{(t−1)r(m−1)} + \frac{(t − 1)r(m − 1)}{m}F^{m},

where H̄^{(t−1)r(m−1)} is a general member of |H^{(t−1)r(m−1)}|.

Consider now the map of linear series

|(t−1)r(m−1)K_{X/S}+\frac{r}{r_{0}} g^{∗}L|×|r_{0}(m−1)K_{X}−2g^{∗}L|^{×r/r_{0}}×g^{∗}|(t−1)r(m−1)K_{S}| ⟶

⟶ |tr(m − 1)K_{X} − \frac{r}{r_{0}} g^{∗}L|.

We remark that B_{3}|_{X_{s}} = 0 and B_{2}|_{X_{s}} = N̄ + G where N̄ is a general member of the base point free linear series |N|. It is easy to see that

⌊\frac{B_{1} + \frac{r}{r_{0}} B_{2} + B_{3}}{tr} |X_{s}⌋ ≺ ⌊\frac{H̄^{(t−1)r(m−1)}}{tr} + \frac{(t − 1)r(m − 1)}{trm}F^{m} + \frac{N̄}{tr_{0}} + \frac{G}{tr_{0}} ⌋

≺ ⌊\frac{H̄^{(t−1)r(m−1)}}{tr} + \frac{N̄}{tr_{0}} + \frac{m − 1}{m}F^{m} + \frac{G}{tr_{0}} ⌋ ≺ F^{m}.

Define B := B_{1} + \frac{r}{r_{0}} B_{2} + B_{3}. Let ν : X′ ⟶ X be a birational morphism such that |tr(m − 1)K_{X′} − \frac{r}{r_{0}} ν^{∗}g^{∗}L| = Φ + |M| where Φ + K_{X′/X} is a divisor with normal crossings support and |M| is base point free. We have ν^{∗}B + tr(m − 1)K_{X′/X} ∈ Φ + |M|. X is smooth and \{\frac{B}{tr} |_{X_{s}}\} is klt and hence

⌊ν^{∗}\{\frac{B}{tr} |_{X_{s}}\}⌋ ≺ K_{X′_{s}/X_{s}}.

Since

|mK_{X′_{s}}| = |mK_{X_{s}}| + mK_{X′_{s}/X_{s}} = f^{∗}|H^{m}| + F^{m} + mK_{X′_{s}/X_{s}},

f^{∗}|H^{m}| is free and F^{′m} = f^{∗}F^{m} + mK_{X′_{s}/X_{s}} is the fixed divisor of |mK_{X′_{s}}|.

We remark that \frac{B}{tr} |_{X_{s}} = \{\frac{B}{tr} |_{X_{s}}\} + ⌊\frac{B}{tr} |_{X_{s}}⌋, and for general s ∈ S, ν^{∗}(B|_{X_{s}}) = (ν^{∗}B)|X′_{s}. For B′ a general member of Φ + |M|, one has

⌊\frac{B′}{tr} |_{X′_{s}}⌋ = ⌊\frac{Φ}{tr} |_{X′_{s}}⌋ ≺ ⌊\frac{ν^{∗}B}{tr} |_{X′_{s}} + (m − 1)K_{X′_{s}/X_{s}}⌋ =

= ⌊ν^{∗}\{\frac{B}{tr} |_{X_{s}}\} − K_{X′_{s}/X_{s}}⌋ + ν^{∗}⌊\frac{B}{tr} |_{X_{s}}⌋ + mK_{X′_{s}/X_{s}} ≺ ν^{∗}F^{m} + mK_{X′_{s}/X_{s}}.

So for a = tr_{0}, b = \frac{r}{r_{0}} , and for a general divisor B′ ∈ |ab(m − 1)K_{X′} − bν^{∗}g^{∗}L|, one sees that B′ has normal crossings support and ⌊\frac{B′}{ab} |_{X′_{s}}⌋ ≺ F^{′m}. Replacyng B with B′ and X with X′ the lemma follows. □

Theorem 4.2. Let X be a smooth projective variety of general type and alb_{X}(X) ⊂ Alb(X) its Albanese image. Let F := X_{z} be the fiber over a general z ∈ alb_{X}(X). Let m_{1} ≥ m_{2} ≥ 2 be any integers such that |m_{i}K_{F} | is non empty and |(m_{1}+m_{2})K_{F} | defines a birational map. Then |(m_{1} + 2m_{2})K_{X} + P | induces a birational map for any P ∈ Pic^{0}(X).

Proof. Let a := alb_{X} : X ⟶ A be the Albanese map, ν : Z ⟶ alb_{X}(X) an appropriate desingularization. Fix H an ample line bundle on A. Let m = m_{1}+m_{2}. For r ≫ 0 sufficiently divisible, pick a divisor

B ∈ |r(m_{1} − 1)(m_{2} − 1)(m − 1)K_{X} − a^{∗}H|.

Replacing X by an appropriate birational model, one may assume that alb_{X} factors through a morphism f : X ⟶ Z and that B has normal crossings support and by Lemma 4.1 that for general z ∈ Z (after replacying H by an appropriate multiple) ⌊\frac{B}{r(m−1)(m_{2}−1)}⌋|_{X_{z}} ≺ F^{m_{1}}, ⌊\frac{B}{r(m−1)(m_{1}−1)}⌋|_{X_{z}} ≺ F^{m_{2}} and ⌊\frac{B}{r(m_{1}−1)(m_{2}−1)}⌋|_{X_{z}} ≺ F^{m}.

We define

L_{1} := (m_{1}−1)K_{X}−⌊\frac{B}{r(m − 1)(m_{2} − 1)}⌋ ≡ \frac{a^{∗}H}{r(m − 1)(m_{2} − 1)}+\{\frac{B}{r(m − 1)(m_{2} − 1)}\},

L_{2} := (m_{2}−1)K_{X}−⌊\frac{B}{r(m − 1)(m_{1} − 1)}⌋ ≡ \frac{a^{∗}H}{r(m − 1)(m_{1} − 1)}+\{\frac{B}{r(m − 1)(m_{1} − 1)}\},

M := (m−1)K_{X}−⌊\frac{B}{r(m_{1} − 1)(m_{2} − 1)}⌋ ≡ \frac{a^{∗}H}{r(m_{1} − 1)(m_{2} − 1)}+\{\frac{B}{r(m_{1} − 1)(m_{2} − 1)}\}.

Therefore, since a^{∗}H = f^{∗}(ν^{∗}H), L_{1}, L_{2} and M are numerically equivalent to the pull back of a nef and big ℚ-divisor on Z plus a klt divisor. In particular

h^{i}(Z, f_{∗}(ω_{X}⊗L_{j})⊗ν^{∗}P) = h^{i}(A, a_{∗}(ω_{X}⊗L_{j})⊗P) = 0

for all i > 0, j = 1, 2, and all P ∈ Pic^{0}(A). Step 1. Let z ∈ Z be a general point, then for every P ∈ Pic^{0}(A), and y ∈ X_{z} such that y ∉ Bs|m_{1}K_{X_{z}}| ∪ Bs|m_{2}K_{X_{z}}|, one has y ∉ Bs|K_{X} + M + P |.

Since ⌊\frac{B}{r(m−1)(m_{1}−1)}⌋|_{X_{z}} ≺ F^{m_{2}}, we have

|(K_{X} + L_{2})|_{X_{z}}| + ⌊\frac{B}{r(m − 1)(m_{1} − 1)}⌋|_{X_{z}} = |m_{2}K_{X_{z}}|

and hence y ∉ Bs|(K_{X} + L_{2})|_{X_{z}}|. Similarly one sees that y ∉ Bs|(K_{X} + L_{1})|_{X_{z}}|. By Corollary 3.2, for some (and hence for general) Q ∈ Pic^{0}(A), one has that y is not contained in Bs|K_{X} + L_{j} + Q| for j = 1, 2. It follows that for all P ∈ Pic^{0}(A), y ∉ Bs|2K_{X} + L_{1} + L_{2} + P |.

Next consider the divisor

β := ⌊\frac{B}{r(m_{1} − 1)(m_{2} − 1)}⌋ − ⌊\frac{B}{r(m − 1)(m_{1} − 1)}⌋ − ⌊\frac{B}{r(m − 1)(m_{2} − 1)}⌋.

Write B = ∑ b_{i}B_{i} where the B_{i} are the distinct reduced irreducible components of B. Then from the inequalities

⌊\frac{b_{i}}{r(m_{1} − 1)(m_{2} − 1)}⌋ ≥ ⌊\frac{b_{i}}{r(m − 1)(m_{1} − 1)} + \frac{b_{i}}{r(m − 1)(m_{2} − 1)}⌋ ≥

≥ ⌊\frac{b_{i}}{r(m − 1)(m_{1} − 1)}⌋ + ⌊\frac{b_{i}}{r(m − 1)(m_{2} − 1)}⌋

it follows that β is effective. Therefore, we have a map of linear series

|K_{X} + M| ⟶^{+β} |2K_{X} + L_{1} + L_{2}|.

It is possible to arrange that ⌊\frac{B}{r(m_{1}−1)(m_{2}−1)}⌋ ≺ Bs|mK_{X}|. Comparing base loci, one sees that

h^{0}(X, O_{X}(K_{X} + M)) = h^{0}(X, O_{X}(2K_{X} + 2L)) = h^{0}(X, O_{X}(mK_{X})).

Moreover, it is easy to see that h^{0}(X, O_{X}(K_{X} +M)⊗P) = h^{0}(X, O_{X}(K_{X} +M)) for all P ∈ Pic^{0}(A). By [HP] Proposition 2.15, one also sees that h^{0}(X, O_{X}(mK_{X})) = h^{0}(X, O_{X}(mK_{X})⊗P) for all P ∈ Pic^{0}(A). In particular, for all P ∈ Pic^{0}(A) one has an isomorphism

|K_{X} + M + P | ⟶^{+β} |2K_{X} + L_{1} + L_{2} + P |

and therefore if y ∉ Bs|m_{1}K_{X_{z}}| ∪ |m_{2}K_{X_{z}}|, one has that y ∉ Bs|K_{X} + M + P |. Step 2. Let y_{1}, y_{2} ∈ X be general points. Then for general P ∈ Pic^{0}(A) one has φ_{|K_{X}+M+P |}(y_{1}) ≠ φ_{|K_{X}+M+P |}(y_{2}).

Let z_{i} = a(y_{i}). Assume that z = z_{1} = z_{2} (if z_{1} ≠ z_{2} the assertion will follow in a similar but easier fashion). Since y_{i} ∈ X are general, we may assume that z ∈ Z is also general. Let F = X_{z}, since φ_{|mK_{F} |} = φ_{|(K_{X}+M)|_{F} |} is birational, we may assume that the homomorphism H^{0}(F, (ω_{X}⊗M)|_{F}) ⟶ ℂ(y_{1})⊕ℂ(y_{2}) is surjective.

Since z ∈ Z is general, we have

a_{∗}(ω_{X}⊗M)⊗ℂ(ν(z)) ≅ f_{∗}(ω_{X}⊗M)⊗ℂ(z) ≅ H^{0}(F, (ω_{X}⊗M)|_{F}).

One sees that

a_{∗}(ω_{X}⊗M⊗I_{y_{1}})⊗ℂ(ν(z)) =
= ker (a_{∗}(ω_{X}⊗M)⊗ℂ(ν(z))⟶H^{0}(F, (ω_{X}⊗M)|_{F}) ⟶ ℂ(y_{1})) ,
a_{∗}(ω_{X}⊗M⊗I_{y_{1},y_{2}})⊗ℂ(ν(z)) =
= ker (a_{∗}(ω_{X}⊗M)⊗ℂ(ν(z)) ⟶ H^{0}(F, (ω_{X}⊗M)|_{F}) ⟶ ℂ(y_{1}) ⊕ ℂ(y_{2})) .

It follows that

a_{∗}(ω_{X}⊗M⊗I_{y_{1},y_{2}}) ≠ a_{∗}(ω_{X}⊗M⊗I_{y_{1}}).

Consider the exact sequence

0 ⟶ a_{∗}(ω_{X}⊗M⊗I_{y_{1},y_{2}}) ⟶ a_{∗}(ω_{X}⊗M⊗I_{y_{1}}) ⟶ Oz ⟶ 0.

By Step 1, h^{0}(a_{∗}(ω_{X}⊗M⊗I_{y_{1}}⊗P)) = h^{0}(a_{∗}(ω_{X}⊗M⊗P)) − 1 for any P ∈ Pic^{0}(A). Since h^{i}(a_{∗}(ω_{X}⊗M⊗P)) = 0 for all i > 0, it follows that a_{∗}(ω_{X}⊗M⊗I_{y_{1}}) satisfies the I.T. and hence by Proposition 2.2

h^{0}(a_{∗}(ω_{X}⊗M⊗I_{y_{1},y_{2}})⊗Q) = h^{0}(a_{∗}(ω_{X}⊗M⊗I_{y_{1}})⊗Q) − 1

for general Q ∈ Pic^{0}(A). Therefore for general Q ∈ Pic^{0}(A)

h^{0}(ω_{X}⊗M⊗I_{y_{1},y_{2}}⊗Q) = h^{0}(ω_{X}⊗M⊗I_{y_{1}}⊗Q) − 1 = h^{0}(ω_{X}⊗M⊗Q) − 2

and Step 2 follows. Step 3. Let y_{1}, y_{2} ∈ X be general points. Then for any P ∈ Pic^{0}(A) one has φ_{|2K_{X}+M+L_{2}+P |}(y_{1}) ≠ φ_{|2K_{X}+M+L_{2}+P |}(y_{2}).

Let y_{1}, y_{2} ∈ X be general points, it suffices to show that there exists a section s of 2K_{X} + M + L_{2} + P vanishing on y_{2} but not on y_{1}. By Steps 1 and 2, for general Q ∈ Pic^{0}(A) there is a section s_{1} ∈ H^{0}(X, ω_{X}⊗L⊗Q⊗P) not vanishing on y_{1} and a section s_{2} ∈ H^{0}(X, ω_{X}⊗M⊗Q^{∨}) vanishing on y_{2} but not vanishing on y_{1}. Let s := s_{1}⊗s_{2}.

It follows that the linear series |(m_{1} + 2m_{2})K_{X} + P | also induces birational morphisms. □

Corollary 4.3. Let X be of general type, then i) if dim(X) − dim(alb_{X}(X)) ≤ 1 then |6K_{X}| induces a birational morphism, ii) dim(X) − dim(alb_{X}(X)) = 2 then |7K_{X}| induces a birational morphism.

When X is of maximal Albanese dimension, it is possible to improve the above result.

Theorem 4.4. Let X be of general type and maximal Albanese dimension. If χ(ω_{X}) > 0, then |3K_{X}| is birational.

Proof. By [GL], since χ(ω_{X}) > 0, it follows that for all P ∈ Pic^{0}(X), h^{0}(X, ω_{X}⊗P) > 0. There exists an open subset U ⊂ X such that for any x ∈ U, and general Q ∈ Pic^{0}(X) (depending on x), there is a section of H^{0}(X, ω_{X}⊗Q) not vanishing at x. Therefore, for any x ∈ U and for any P ∈ Pic^{0}(X), there is a section of H^{0}(X, ω_{X}^{2}⊗P) not vanishing at x.

Let x_{1}, x_{2} be two general points in U. Proceeding as in the proof of the previous theorem, we see that for general Q ∈ Pic^{0}(X), there is a section of ω_{X}^{2}⊗Q vanishing on x_{1} but not on x_{2}. For a fixed P ∈ Pic^{0}(X), we choose general Q ∈ Pic^{0}(X) and then consider the map of linear series

|K_{X} − Q + P | × |2K_{X} + Q| ⟶ |3K_{X} + P |.

We conclude that for any fixed P ∈ Pic^{0}(X), there is a section of ω_{X}^{3}⊗P vanishing on x_{1} but not on x_{2}. □

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Jungkai Alfred Chen Department of Mathematics National Chung Cheng University Ming Hsiung, Chia Yi, 621, Taiwan jkchen@math.ccu.edu.tw Christopher D. Hacon Department of Mathematics Sproul Hall 2208 University of California Riverside, CA 92521-013 USA hacon@math.ucr.edu