PLURICANONICAL SYSTEMS ON IRREGULAR 3-FOLDS OF GENERAL TYPE

3-FOLDS OF GENERAL TYPE

JUNGKAI A. CHEN AND CHRISTOPHER D. HACON

Abstract. In this paper we prove that if X is an irregular 3-fold with χ(ω_X) > 0, then |mK_X| is birational for all m ≥ 5.

1. Introduction

Given a non-singular variety of general type, by definition, the pluri- canonical system |mKX| defines a birational map ϕm to a projective space, for all sufficiently big integers m. It is a natural question to try and find a reasonable effective bound for such integers m. When dimX = 1, it is classically known that ϕ_m is an embedding for m ≥ 3.

For surfaces, by a result of [Bo], ϕ_m is birational for m ≥ 5. Recently, following ideas of Tsuji (cf. [Ts1], [HM], [Ta]), it has been shown that for any positive integer n > 0, there exists a positive integer r_n > 0 such that ϕ_m is birational for m ≥ r_n for all varieties of general type and of dimension n. Tsuji has also shown [Ts2], that r₃ ≤ 18(2⁹· 3⁷)!.

Recently Todorov [To], has shown that for all but finitely many families of 3-folds of general type, the 5-th canonical map is birational.

The purpose of this note is to study pluricanonical systems on irreg- ular varieties, i.e. varieties with q(X) := h⁰(X, ΩX) > 0. In view of recent results on irregular varieties, one might hope that their geome- try behaves similarly to that of surfaces. In this article we are in fact able to prove the following:

Theorem 1.1. Let X be an irregular smooth projective threefold of general type with χ(ωX) > 0. Then |mKX+ P | is birational for m ≥ 5 and for all P ∈ Pic⁰(X).

It is easy to see that this result is optimal. Consider in fact a surface S of general type such that |4K_S| is not birational and a curve C of general type. For all P ∈ Pic⁰(C × S) ∼= Pic⁰(C) one sees that

|4K_C×S+ P | is not birational.

The main technical ingredient is the technique developed in [CH1], where the Fourier-Mukai transform and vanishing theorems are used to study adjoint linear series on irregular varieties. For the reader’s convenience, in Section 2 we recall the main result from [CH1]. Its

The first author was partially supported by NSC and NCTS of Taiwan.

The second author was partially supported by NSF research grant no: 0456363.

1

application to pluricanonical system is summarized in Theorem 2.8.

One finds a very sharp bound in Proposition 2.9 when the Albanese fiber dimension is small. Starting in Section 3, we will work on irregular varieties over curves. The case when X maps to a curve with genus g ≥ 2 is handled in Section 3 by a relatively straightforward application of the weak positivity of the relative dualizing sheaves. The case when X maps to an elliptic curve is more subtle. We are unable to recover a general result in arbitrary dimension. However, by a careful study of the positivity of vector bundles over elliptic curves in Section 4, we are able to prove our main theorem for threefolds in Section 5.

Acknowledgments. We are indebted to Meng Chen, De-Qi Zhang, Yujiro Kawamata, I-Hsun Tsai, H´el`ene Esnault, Eckart Viehweg and Chin-Lung Wang for useful conversations and comments on this sub- ject.

1.1. Notation. We work over the field of complex numbers C. A Q- Cartier divisor D on a normal variety X is nef if D · C ≥ 0 for any curve C ⊂ X. We say that two Q-divisors D₁, D₂ are Q-linearly equivalent (D₁ ∼_Q D₂) if there exists an integer m > 0 such that mD_i are linearly equivalent. Numerical equivalence is denoted ≡. And we write D₁ ≥ D₂ if D₁ − D₂ is effective. For any projective variety X, Pic⁰(X) denotes the abelian variety parametrizing the topologically trivial line bundles on X. If A is an abelian variety, then we can identify A with Picˆ ⁰(A). We will denote by P, the Poincar´e line bundle on A × ˆA.

2. Linear series on irregular varieties

We begin by recalling several important consequences of the theory of Fourier-Mukai transforms that will be needed throughout the paper.

Let A be an abelian variety. Recall that there is a functor ˆS from the category of O_A-modules to the category of OAˆ-modules defined by

S(M) = (pˆ Aˆ)_∗(P ⊗ p^∗_AM).

Here p_A and pAˆ denote the projections of A × ˆA to A and ˆA. Similarly we define S(N) = (p_A)_∗(P ⊗ p^∗_AˆN). Let RS (resp. R ˆS) be the derived functor of S (resp. ˆS) between the corresponding derived categories.

By a result of Mukai (cf. [Mu] Theorem 2.2), we have:

Theorem 2.1. There exist isomorphisms of functors

R ˆS ◦ RS ∼= (−1Aˆ)^∗[−g] and RS ◦ R ˆS ∼= (−1A)^∗[−g].

A coherent sheaf F on an abelian variety A is said to be IT⁰ if Hⁱ(A, F⊗P ) = 0 for all i > 0 and all P ∈ Pic⁰(A).

Lemma 2.2. Let F be a coherent IT⁰ sheaf on an abelian variety A.

If F 6= 0, then H⁰(A, F⊗P ) 6= 0 for all P ∈ Pic⁰(A).

Proof. Please see [CH1] Lemma 2.1. ¤ Lemma 2.3. Let F 6= 0 be an IT⁰ coherent sheaf on an abelian variety A. Suppose that there is a non-zero map F → C(z). Then the induced map H⁰(F⊗P ) → H⁰(C(z)) is non-zero for general P ∈ Pic⁰(A).

Proof. Please see [CH1] Proposition 2.3. ¤

Here are some useful consequences:

Corollary 2.4. Let X be a smooth projective variety and a : X → Z = a(X) ⊂ A a morphism to an abelian variety A. Let F be a general fiber and x a point on F . Let L be a divisor on X such that L ≡ a^∗H + ∆ for a nef and big Q-divisor H and a Q-divisor ∆ ≥ 0 so that ∆ has simple normal crossings support and b∆c = 0. Suppose furthermore that a_∗O_X(K_X + L) 6= 0. Then we have:

(1) If x 6∈ Bs|(K_X+L)|_F|, then x 6∈ Bs|K_X+L+a^∗P | for a general P ∈ Pic⁰(A).

(2) Let x₁, x₂ be two points on different general fibers F₁, F₂ respec- tively. If x_i 6∈ Bs|(K_X + L)|_Fi| for i = 1, 2, and one of the points x_i, say x₁, does not belong to Bs|K_X + L + a^∗P | for all P ∈ Pic⁰(A), then |K_X+ L + a^∗P | separates x₁, x₂ for a general P ∈ Pic⁰(A).

(3) Let x₁, x₂ be two distinct points on a general fiber F . If x_i 6∈

Bs|(K_X + L)|_F| for i = 1, 2, |(K_X+ L)|_F| separates x₁, x₂, and one of xi, say x1, 6∈ Bs|KX + L + a^∗P | for all P ∈ Pic⁰(A), then |K_X+ L + a^∗P | separates x₁, x₂ for a general P ∈ Pic⁰(A).

Proof. Let z ∈ Z be a general point and F the fiber over z. By z being general, we mean that

a_∗O_X(K_X + L) ⊗ C(z) ∼= H⁰(F, O_F(K_X + L)).

We first look at the short exact sequence, obtained by evaluated at x:

(∗) 0 → O_X(K_X + L)⊗I_x → O_X(K_X + L) → C(x) → 0.

Pushing forward to Z, we get

0 → a_∗(O_X(K_X + L)⊗I_x) → a_∗O_X(K_X + L) → C(z) → ...

Since x 6∈ Bs|(K_X + L)|_F| and

a∗OX(KX + L)⊗C(z) ∼= H⁰(F, OF(KX + L)), one sees that the induced map

a_∗O_X(K_X + L) → a_∗O_X(K_X + L) ⊗ C(z) → C(z) is non-zero, whence surjective.

By Corollary 10.15 of [Ko3], a_∗O_X(K_X+L) is IT⁰. Since a_∗O_X(K_X+ L) 6= 0 by assumption, applying Lemma 2.3 to the sheaf a_∗O_X(K_X+L), we get (1).

Before we move forward to the proof of the next two statements, we remark that the assumption that x1 6∈ Bs|KX + L + a^∗P | for all P implies that a_∗(O_X(K_X + L)⊗I_x1) is IT⁰ (by tensoring (∗) with a^∗P and replacing x by x₁).

To see (3), we begin by looking at the exact sequence

0 → a_∗(O_X(K_X + L)⊗I_x1,x2) → a_∗(O_X(K_X + L)⊗I_x1) → C(z) → ...

The last map factors as

a_∗(O_X(K_X + L)⊗I_x1) → a_∗(O_X(K_X + L)⊗I_x1)⊗C(z)

→ a∗OX(KX + L)⊗C(z) → C(z).

It is obtained by evaluating at x₂. The assumption that |(K_X + L)|_F| separates x₁, x₂ shows that this is surjective. Again, by applying Lemma 2.3 to the sheaf a_∗(O_X(K_X + L)⊗I_x1), we are done.

Finally we consider (2). Again we have (assuming now that x2 lies over z):

0 → a∗(OX(KX + L)⊗Ix1,x2) → a∗(OX(KX + L)⊗Ix1) → C(z) → ...

But now we look at the map

a_∗(O_X(K_X + L)⊗I_x1) → a_∗(O_X(K_X + L)⊗I_x1)⊗C(z)

→ a_∗O_X(K_X + L)⊗C(z) → C(z)

which is obtained by evaluating at x₂. Since a(x₁) 6= a(x₂), it follows that a_∗(O_X(K_X + L)⊗I_x1)⊗C(z) ∼= a∗O_X(K_X + L)⊗C(z) and hence the above map is surjective. By Lemma 2.3 we are done. ¤ We now fix some notation and conventions. For any P ∈ Pic⁰(X), we define B_m,P to be the fixed component of |mK_X + a^∗P | and B_m (resp. B_m,F) to be the fixed component of |mK_X| (resp. |mK_F|). After replacing X by an appropriate birational model, we may and we usually do assume that B_m,P, B_m,F have simple normal crossing support.

We need the following:

Lemma 2.5. Let g : X → W be a morphism with a general fiber F . Suppose that κ(W ) ≥ 0 and KX is W -big, i.e. sKX ≥ g^∗L for some ample divisor L and some integer s À 0 . Suppose further that Pm(F ) > 0 for some m ≥ 2. Then after replacing X by an appropriate birational model if necessary, there exist positive integers a, b and there is a normal crossing divisor B ∈ |ab(m − 1)K_X − g^∗bH| such that

bB

abc|_F ≤ B_m,F, bB

abc ≤ B_m,P,

for all P ∈ Pic⁰(W ). Here H is a given nef and big divisor on W and a, b are sufficiently large integers depending on H and KX.

Proof. Please see [CH1], Lemma 5.1. ¤

Definition 2.6. Let a : X → A be a morphism to an abelian variety and D a Cartier divisor on X. Let

V⁰(D) := {P ∈ Pic⁰(A)|h⁰(X, O_X(D)⊗a^∗P ) 6= 0}.

We say that D is full if V⁰(D) = Pic⁰(A).

Note that if a_∗O_X(D) is IT⁰ and non-zero, then by Lemma 2.2, D is full.

2.7. General setting. We can now work on a more general setting.

Let a : X → Z = a(X) ⊂ A be a morphism to an Abelian variety A. Usually, we take the Albanese map of X. Let ν : W → Z be a desingularization of the Stein factorization over Z. Replacing X by an appropriate birational model, we may assume that there is a morphism g : X → W with connected fibres. We assume that X is of general type. It follows that K_X is W -big. Let F be the general fiber of g.

In the following discussion, we will assume that P_m(F ) > 0 for some m ≥ 2. We then take Bm = _ab^B with B ∈ |ab(m − 1)KX − g^∗bH| as in Lemma 2.5 and

L_m := (m − 1)K_X − bB_mc ≡ 1

ag^∗H + {B_m}

where Bm = bBmc + {Bm}. By [Ko3] Theorem 10.15, since H is the pull back of a nef and big divisor on Z, one sees that

Hⁱ(A, a∗OX(KX + Lm)⊗P ) ∼= Hⁱ(W, g∗OX(KX + Lm)⊗ν^∗P ) = 0 for all P ∈ Pic⁰(A) and all i > 0. In particular, a_∗O_X(K_X + L_m) is IT⁰. Recall that by Lemma 2.5, one has bB_mc|_F ≤ B_m,F. Thus, H⁰(F, O_F((K_X+L_m)|_F)) ∼= H⁰(F, O_F(mK_F)). In particular, a_∗O_X(K_X+ L_m) 6= 0. By Lemma 2.2 K_X + L_m is full and hence so is mK_X.

Assume that for some integers m, n ≥ 1, mK_X and nK_X are full and consider the morphism

|mK_X + a^∗P₁| + |nK_X + a^∗P₂| → |(m + n)K_X + a^∗(P₁+ P₂)|

∼= |KX + L_m+n+ a^∗(P₁+ P₂)|,

for all P₁, P₂. If x ∈ X is a general point, then for general P ∈ Pic⁰(A), x is neither a base point of mK_X+ a^∗P nor a base point of nK_X+ a^∗P . Therefore x is not a base point of |(m+n)K_X+a^∗P | for all P ∈ Pic⁰(A).

And it follows that x is not a base point of |K_X + L_m+n+ a^∗P | since x is general.

By Corollary 2.4, we can now deduce the main result of this section.

Theorem 2.8. Let X be a smooth projective variety of general type, a : X → Z = a(X) ⊂ A a non-trivial morphism to an abelian variety, F a general fiber of a and m, n, t positive integers. If mK_X, nK_X, tK_X are full, then:

(1) |(m + n)K_X+ a^∗P | separates two general points on two distinct general fibers for general P ∈ Pic⁰(A).

(2) |(m + n + t)KX + a^∗P | separates two general points on two distinct general fibers for all P ∈ Pic⁰(A).

(3) If (m + n)KF is birational, then |(m + n)KX + a^∗P | separates two general points on a general fiber for a general P ∈ Pic⁰(A).

(4) If (m+n)K_F is birational, then |(m+n+t)K_X+a^∗P | separates two general points on a general fiber for all P ∈ Pic⁰(A).

Proof. (1) follows from Corollary 2.4 (2) since for two general points x1 and x2, we have seen that xi 6∈ Bs|KX + Lm+n + a^∗P | for any P ∈ Pic⁰(A) and so x_i 6∈ Bs|(K_X + L_m+n)|_F|.

(3) follows from Corollary 2.4 (3) since we assumed that |(m+n)K_F| is birational and so, since by Lemma 2.5 we have (Bm+n)|F ≤ Bm+n,F, then |(KX + Lm+n)|F| is also birational.

(2) and (4) now follow by considering the morphism

|(m + n)K_X+ a^∗P₁| + |tK_X + a^∗P₂| → |(m + n + t)K_X + a^∗(P₁+ P₂)|.

Since |(m + n)K_X+ a^∗P₁| separates x₁, x₂ for general P₁ ∈ Pic⁰(A) and

|tK_X + a^∗P₂| does not vanish along x₁, x₂ for general P₂ ∈ Pic⁰(A), then |(m + n + t)KX + a^∗P | separates x1, x2 for all P ∈ Pic⁰(A). ¤ Let F be an irreducible component of a general fibre of the Albanese map a = alb : X → a(X). Turning to a more detailed discussion, we now distinguish varieties according to their Albanese fiber dimension, i.e. f := dimX − dima(X). We would like to remark that an analogous result holds for arbitrary non-trivial morphism to an abelian variety.

Proposition 2.9. Let X be a smooth projective variety of general type with q(X) > 0.

(1) If f = 0, then

(a) |mK_X + P | is birational for a general (resp. all) P ∈ Pic⁰(X) for m ≥ 4 (resp. m ≥ 6).

(b) If moreover K_X is full, then |mK_X+ P | is birational for a general (resp. all) P ∈ Pic⁰(X) for m ≥ 2 (resp. m ≥ 3).

(2) If f = 1, then

(a) |mK_X + P | is birational for a general (resp. all) P ∈ Pic⁰(X) and for m ≥ 4 (resp. m ≥ 6).

(b) If moreover K_X is full, then |mK_X + P | is birational for a general (resp. all) P ∈ Pic⁰(X) and for m ≥ 3 (resp.

m ≥ 4).

(3) If f = 2, then

(a) |mK_X + P | is birational for a general (resp. all) P ∈ Pic⁰(X) and for m ≥ 5 (resp. m ≥ 7).

(b) If moreover KX is full, then |mKX + P | is birational for a general (resp. all) P ∈ Pic⁰(X) and for m ≥ 5 (resp.

m ≥ 6).

Proof. If f = 0, then X is generically finite over its Albanese variety.

By Lemma 2.2 KX + L2 is full and hence so is 2KX. Clearly |mKF| separates points on fibers for m ≥ 1, so (a) and (b) follow from Theorem 2.8, letting m = n = t = 2. If we assume that K_X is full, then we let m = n = t = 1.

If f ≥ 1, one sees that for m ≥ 2, mK_X is full if and only if P_m(F ) 6=

0. Since F is of general type, for f ≤ 2 it is well known that P_m(F ) > 0 for all m ≥ 2. Therefore mK_X is full for m ≥ 2.

If f = 1, mK_F is birational for m ≥ 3. By Theorem 2.8, we have (a) and (b), by taking m = n = t = 2. If we assume that K_X is full, then let m = 2, n = t = 1.

Finally, if f = 2, then mK_F is birational for m ≥ 5. By Theorem 2.8, we have (a) and (b) by letting m = 3, n = t = 2. If we assume that KX is full, then we let m = 4, n = t = 1. ¤

3. Varieties over curves of genus at least 2

In this section, we study the case in which the Albanese image is a curve of g ≥ 2.

Proposition 3.1. Let a : X → C be a morphism from a smooth projective variety of general type to a curve of genus g(C) ≥ 2 with a (connected) general fiber F . Suppose that for some m ≥ 4, |mKF| is birational. Then |mK_X + a^∗P | is birational for all P ∈ Pic⁰(C).

Proof. Fix an integer k À 0 such that P_2k(F ) > 0. We write 2kK_X − 3kF = 2kK_X/C+ 2ka^∗K_C− 3kF = 2kK_X/C+ a^∗H, for some ample line bundle H of degree (4g(C) − 7)k. By the weak positivity of a_∗(ω^2k_X/C) (cf. [V1]), one has that the restriction map

H⁰(X, O_X(2nK_X − 3nF )) → H⁰(F, O_F(2nK_F)),

is surjective for all n À k sufficiently large and divisible by k and by m.

Therefore, a_∗(ω²ⁿ_X(−3nF )) is a vector bundle of rank P_2n(F ) and degree at least P_2n(F )(4g(C) − 7)n. Since F is of general type, by Riemann- Roch, one sees that 2K_X − 3F is big. We write |2nK_X − 3nF | =

|N_n| + B_n where B_n is the fixed part. We have 2nK_X ∼ M_n+ B_n = (Nn + 3nF ) + Bn. After replacing X by an appropriate birational model, we may assume that |Nn|, |Mn| are free and that Bnis a divisor with normal crossing support. Note that Mn is nef and big. We write

|2nK_F| = |(M_n)|_F| + B_2n,F. By the above surjection, we have that B_2n,F = (B_n)|_F.

We have (m − 3)K_X ≡ ^(m−3)M_2n ⁿ+^(m−3)B_2n ⁿ and 2K_X− 3F ≡ ^N_nⁿ+^B_nⁿ. Let

L_m := (m − 1)K_X − b(m − 1)Bn

2n c.

Since m divides n, we have that Bn≤ _mⁿBm. Therefore, K_X + L_m ≥ mK_X − bm − 1

2n n

mB_mc ≥ mK_X − b1 2B_mc.

Similarly,

K_F + L_m|_F ≥ mK_F − bm − 1

2n B_2n,Fc ≥ mK_F − b1

2B_2m,Fc, and |K_F + L_m|_F| ∼= |mKF| 6= ∅. Note that

L_m ≡ (m − 3)M_n 2n +N_n

n + {(m − 1)B_n

2n } + 3F.

By Kawamata-Viehweg vanishing, one has that

H¹(X, O_X(K_X + L_m+ a^∗P − sF )) = 0 for s ≤ 3, P ∈ Pic⁰(C).

Consider the short exact sequence:

0 → O_X(K_X + L_m+ a^∗P − F₁− F₂) → O_X(K_X + L_m+ a^∗P )

→ O_F1(K_F1 + L_m|_F1) ⊕ O_F2(K_F2 + L_m|_F2) → 0.

One sees that |KX + Lm + a^∗P | separates general points on distinct general fibers F1 and F2.

Similarly, one has the surjection

H⁰(X, O_X(K_X + L_m+ a^∗P )) → H⁰(F, O_F(K_F + L_m)).

If |mK_F| is birational then |K_X + L_m+ a^∗P | separates general points on F and so does |mK_X + a^∗P |. This completes the proof. ¤

4. Varieties over elliptic curve

In this section, we consider varieties X of general type admitting a morphism to an elliptic curve a : X → C. We will use the results and the notation of [At].

Lemma 4.1. For any integer m ≥ 2, let I := O_X(−bB_mc) be as in (2.7). Then a_∗(ω_X^m) = a_∗(ω_X^m⊗I) is an IT⁰ vector bundle of rank P_m(F ).

Proof. Clearly, we have an inclusion a_∗(ω^m_X⊗I) ,→ a_∗(ω^m_X). As in (2.7), one sees that they have the same rank. Let Q be the coker- nel. Then Q has generic rank 0 and hence is supported on finitely many points. If Q 6= 0, then h⁰(C, Q) 6= 0. However, a_∗(ω_X^m⊗I) is IT⁰ and h⁰(a_∗(ω_X^m⊗I)) = h⁰(a_∗(ω^m_X)). This is a contradiction. Therefore,

Q = 0. ¤

For simplicity, we will write E_m := a_∗(ω^m_X).

Definition 4.2. Let E be a vector bundle on an elliptic curve C. We define

d(E) := max{deg(L)|L ⊂ E is a rank one subbundle }.

Recall that a maximal splitting of a vector bundle E of rank r (on an elliptic curve C) is a sequence of vector bundles (L1, L2, ..., Lr) such that L₁ is a line bundle of maximum degree contained in E and (L₂, ..., L_r) is a maximal splitting of E/L₁. Therefore d(E) = deg(L₁).

Recall also that by a result of Atiyah [At], given an indecomposable vec- tor bundle E of rank r on an elliptic curve, there is a maximal splitting (L₁, L₂, ..., L_r) such that L₁ ≤ L₂ ≤ ... ≤ L_r are line bundles, where L_i ≤ L_i+1 denotes that there is an injection L_i ,→ L_i+1. Moreover, if E is a vector bundle of rank r and degree 0 ≤ s ≤ r, then E has a maxi- mal splitting with L₁ = ... = L_r−s= O, and L_r−s+1 = ... = L_r = L for some line bundle L of degree 1. Notice that

bµ(E)c = d(E) = deg(L₁).

We also remark, that by a result of Tu [Tu], any indecomposable vector bundle is semistable.

Lemma 4.3. Let E be an indecomposable vector bundle on an elliptic curve, then the following are equivalent

(1) deg(E) > 0, (2) E is IT⁰, (3) E is ample.

Proof. (1) ⇒ (2). If deg(E) > 0 and h¹(E ⊗ P ) 6= 0 for some P ∈ Pic⁰(E), then h⁰(E^∗ ⊗ P^∗) 6= 0 and hence there is a non-zero homomorphism P → E^∗. This is impossible as by [Tu], E^∗ is semi- stable of negative slope.

(2) ⇒ (3). Let E be an IT⁰ vector bundle. By a result of Hartshorne (cf. [Ha] Theorem 1.3), it suffices to show that every quotient line bundle has degree ≥ 1. Let L be a quotient line bundle of E. It is easy to see that E being IT⁰ implies that L is IT⁰ and hence has degree

≥ 1.

(3) ⇒ (1). If E is ample, then det(E) is an ample line bundle. So

deg(E) = deg(det(E)) > 0. ¤

Lemma 4.4. Let E be an indecomposable vector bundle on an elliptic curve with d(E) ≥ 1. Then E is generically generated by its global sections.

Proof. This can be easily proven by induction on rank. If the rank of E is 1, then the statement follows from the fact that h⁰(E) > 0. Suppose that the statement is true when the rank is < r. Given E of rank r, the quotient bundle F = E/L₁ has a decomposition F = ⊕F_i where each F_i is indecomposable with d(F_i) ≥ 1. It follows from the inductive hypothesis that each F_i and hence also F is generically generated by its global sections. Since H¹(L₁) = 0, it follows that 0 → H⁰(L₁) → H⁰(E) → H⁰(F ) → 0 is exact. Therefore we have a commutative

diagram

0 −−−→ H⁰(L1)⊗OC −−−→ H⁰(E)⊗OC −−−→ H⁰(F )⊗OC −−−→ 0

 y^α

 y^β

 y^γ

0 −−−→ L₁ −−−→ E −−−→ F −−−→ 0

Since α and γ are surjective on an open set then so is β. ¤ Definition 4.5. For any vector bundle E, we write E = ⊕E_i for its decomposition in to indecomposable vector bundles. We define ν(E) :=

min µ(E_i).

Theorem 4.6. Let a : X → C be a morphism from a smooth projective variety of general type to an elliptic curve, F a general fiber and P ∈ Pic⁰(C). If |mKF| is birational for some m ≥ 2 and ν(a∗ω^m_X) ≥ 1, then

|mKX + a^∗P | separates points on F .

Proof. This follows easily from the above Lemma 4.4. ¤ Lemma 4.7. Let E, F be IT⁰ vector bundles, then E⊗F is IT⁰. Proof. By Lemma 4.3, one sees that for a vector bundle, being IT⁰ is equivalent to being ample. Moreover, if E, F are ample, then E⊗F is

ample. The Lemma now follows easily. ¤

Lemma 4.8. Let E₁, E₂ be indecomposable vector bundles. Then Hom (E₁, E₂) 6= 0 implies that µ(E₂) ≥ µ(E₁).

Proof. See [Tu]. ¤

Lemma 4.9. Let E be an IT⁰ vector bundle which admits a short exact sequence

0 → F → E → Q → 0

such that Q has generic rank ≤ 1. Then ν(E) ≥ min{1, ν(F )}.

Proof. We distinguish the follows cases:

1. Q is torsion.

For every indecomposable component of E, say E_i, there exists a non- zero map from an indecomposable component F_j of F . By Lemma 4.8, µ(E_i) ≥ µ(F_j). So ν(E) ≥ ν(F ). Notice that this step did not require E to be IT⁰.

2. Q is torsion-free.

Then Q is a line bundle. Since E is IT⁰, it follows that Q is IT⁰, hence µ(Q) ≥ 1. For any E_i, there exists a non-zero map from a component F_j of F unless E_i ∼= Q. In either case, we have µ(Ei) ≥ min{1, ν(F )}.

3. In general, let L := Q^∗∗, and K := ker(E → L). Then it is clear that K fits into a short exact sequence

0 → F → K → Q_tor → 0.

Thus ν(K) ≥ ν(F ) by (1). By (2), ν(E) ≥ min{1, ν(F )}. ¤

5. Proof of the main Theorem

Proof of Theorem 1.1. We distinguish the follows cases according to their Albanese fiber dimension:

1. f := dim X − dim alb(X) = 0.

It follows from generic vanishing theorem (cf. [GL1], [GL2]) that if χ(ωX) > 0, then KX is full. Therefore, by Proposition 2.9, we are done.

2. f = 1.

By [Ko2] Corollary 3.2, χ(ω_X) = χ(a_∗ω_X) − χ(R¹a_∗ω_X) and by [Ko1]

Proposition 7.6, R¹a_∗ω_X = ω_W. Since W is of maximal Albanese dimension, by Theorem 1 of [GL1], χ(ω_W) ≥ 0. Hence χ(a_∗ω_X) ≥ χ(ω_X) > 0. By the generalized generic vanishing theorem of [CH2], Proposition 2.2, one sees that a_∗ω_X is full and hence so is K_X. There- fore, by Proposition 2.9, we are done with this case.

3. f = 2.

Let a : X → C be the Albanese map. Note that a has connected fibers. Consider in its Stein factorization X → W → C. Since W is normal, it is a smooth curve. There are induced maps of abelian varieties Alb(X) → Alb(W ) → Alb(C) which are easily seen to be isomorphisms, so that W = C.

3.a. If g(C) ≥ 2.

This case follows from Proposition 3.1.

3.b. g(C) = 1.

We may assume that q(X) = 1 and thus p_g(X) ≥ 1. We have that χ(X, ω_X) = χ(C, a_∗ω_X) − χ(C, R¹a_∗ω_X) + χ(C, R²a_∗ω_X).

Moreover, by [Ko1] Proposition 7.6, R²a_∗ω_X = ω_C = O_C and so χ(R²a_∗ω_X) = 0. Now, h¹(C, R¹a_∗ω_X) = q(X) − h⁰(C, R²a_∗ω_X) = 0 implies that χ(C, R¹a_∗ω_X) ≥ 0. Hence, one has χ(C, a_∗ω_X) > 0. So by the generalized generic vanishing of [CH2], Proposition 2.2, we see that KX is full as in Case 2.

Let F be the general fiber of a and F0 be its minimal model. If

|4K_F| is birational, then we are done by Theorem 2.8. So by Bombieri’s classification [Bo], it remains to consider the case that (K_F²₀, p_g(F₀)) = (1, 2). We recall that for this surface, q = 0, |K_F0| has a unique base point and |2K_F0| is base point free.

Claim. Let E_m = a_∗(ω_X^m). Then ν(E_m) ≥ 1 for m ≥ 5.

Grant this for the time being. Then, by Lemma 4.4, for any P ∈ Pic⁰(C) ∼= Pic⁰(X), every indecomposable component of E_m⊗P for m ≥ 5 is generically generated by global sections. That is, there exists an open set U ⊂ C and a surjection

H⁰(E_m⊗P )⊗O_U → (E_m⊗P )|_OU. In particular, there is a surjection

H⁰(X, ω_X^m⊗a^∗P ) → H⁰(E_m⊗P )⊗O_U → E_m⊗P ⊗k(y) ∼= H⁰(ω_F^m_y),

for general y ∈ U. If follows that |mKX + P | separates points on a general fiber Fy for m ≥ 5. Since |mKX + P | also separates points on different general fibers for m ≥ 3 (cf. Theorem 2.8), we are done.

Therefore it remains to verify the Claim.

Let R_m := H⁰(ω_F^m), we first show that R₁R_m ⊂ R_m+1 has codimen- sion ≤ 1 for all m ≥ 0. To see this, we may assume that F = F₀. Let Y ∈ |K_F| be a general curve. We have that Y is non-singular, for oth- erwise Y is singular at the base point of |K_F| and hence 1 = Y⁰· Y ≥ 2 for Y⁰ 6= Y ∈ |K_F|.

Let σ₁ ∈ R₁ be a section defining Y , then we have the short exact sequence

0 → ω^m_F −→ ω^σ1 _F^m+1−→ω^m+1_F |_Y → 0.

By the Kawamata-Viehweg vanishing theorem and since q(F ) = 0, we have an exact sequence on global sections

0 → R_m −→ R^σ1 _m+1−→H⁰(ω_F^m+1|_Y) → 0, for m ≥ 0. By Riemann-Roch, one sees that

h⁰(ω_F^m|_Y) = { m if m = 1, 2, m − 1 if m ≥ 3.

We claim that the cokernel of the homomorphism H⁰(ω_F^m)⊗H⁰(ω_F) → H⁰(ω_F^m+1) has dimension 1 for m = 1 and m ≥ 3 and dimension 0 for m = 2. Consider in fact the exact sequence:

0 → ω_F^m−1 → ω^m_F ⊗ H⁰(ω_F) → ω_F^m+1 → C(z) → 0

where z is the unique base point of |K_F|. If m ≥ 3, all higher coho- mologies vanish and so we get an exact sequence of global sections. An easy spectral sequence argument shows that the sequence

0 → H⁰(ω_F^m−1) → H⁰(ω_F^m) ⊗ H⁰(ω_F) → H⁰(ω^m+1_F )

is also exact for m = 1, 2. By an easy dimension count, the above statement follows.

We now consider the multiplication map

ϕ_m : a_∗ω_X⊗a_∗ω_X^m → a_∗ω_X^m+1.

By the above claim, one sees that ϕ_m has cokernel of generic rank ≤ 1 for all m ≥ 1 and it is generically surjective for m = 2.

We now distinguish the following cases:

1. E₁ := a_∗ω_X is indecomposable.

Since K_X is full, so is E₁. By the semipositivity of a_∗ω_X, we have that deg(E₁) ≥ 0. If deg(E₁) = 0, then E₁ ∼= U⊗P for some vector bundle U with maximal splitting (O_C, O_C) and some P ∈ Pic⁰(C).

However, U⊗P is not full. Thus we have deg(E₁) > 0 and hence µ(E₁) ≥ _rk(E)¹ = ¹₂. According to the ring structure given in [At], one sees that ν(E₁⊗E₁) ≥ ν(E₁) + ν(E₁) ≥ 1 and hence ν(imϕ₁) ≥ 1 by Lemma 4.8. Moreover, the cokernel of ϕ₁ is IT⁰ and has rank ≤ 1,

so ν(E2) ≥ 1 by Lemma 4.9. Repeatedly applying this argument, one sees that ν(Em) ≥ 1 for m ≥ 5.

2. E₁ is decomposable.

Then it is of the form L ⊕ P with deg(L) ≥ 1, deg(P ) ≥ 0. Note that ϕ₁ factors through Sym²(E₁) = P² ⊕ L⊗P ⊕ L². Moreover, im(ϕ₁) has rank ≥ 3 because ϕ₁ has cokernel of rank ≤ 1. It follows that Sym²(E₁) ∼= im(ϕ1).

Before we move on, notice that E_m is IT⁰ for m ≥ 2 by Lemma 4.1.

2.a. E₂ is indecomposable.

By Lemma 4.8, µ(E₂) ≥ 2 for there is a non-zero map from L² to E₂. By considering ϕ₂, ..., ϕ_m−1 and Lemma 4.9, one has that ν(E_m) ≥ 1 for m ≥ 5.

2.b. E2 = E1⊕ E2 with ranks 3, 1 respectively.

Then µ(E1) ≥ 1 for there is a non-zero map from L or L² in to im(ϕ1) to E1. Since E2 is IT0, so is E2. Thus µ(E2) ≥ 1 for it is a line bundle of IT⁰. Hence we have ν(E₂) ≥ 1. Proceeding as above, it follows that ν(E_m) ≥ 1 for m ≥ 5.

2.c. E₂ = ⊕E_i factors into 4 bundles of rank 1.

Since E₂ is IT⁰, so are the bundles E_i. Thus µ(E_i) ≥ 1 and hence ν(E₂) ≥ 1. Proceeding as above, it again follows that ν(E_m) ≥ 1 for m ≥ 5.

2.d. E₂ decomposes into bundles of rank (2, 1, 1) or (2, 2).

Since each component is IT⁰, we have ν(E₂) ≥ ¹₂. We then consider the multiplication map

ψ : a_∗ω²_X⊗a_∗ω_X² → a_∗ω_X⁴.

One can check that R₂R₂ ⊂ R₄ has codimension ≤ 1 (its image in fact contains the image of R₁R₁R₂ and hence it contains the image of R₁R₃ which has codimension at most 1). So we conclude that ν(E₂⊗E₂) ≥ ν(E₂) + ν(E₂) ≥ 1 and hence ν(E₄) ≥ 1 by Lemma 4.9. Then, by considering ϕ4, · · · , ϕm−1, it follows that ν(Em) ≥ 1 for m ≥ 5. This

completes the proof. ¤

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Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan.

National Center for Theoretical Sciences, Taipei Office.

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

Department of Mathematics, University of Utah, 155 South 1400 East, JWB 233, Salt Lake City, UT 84112-0090, USA

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