Pluricanonical systems on irregular 3-folds of general type

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DOI 10.1007/s00209-006-0028-9

Mathematische Zeitschrift

Pluricanonical systems on irregular 3-folds of general type

Jungkai A. Chen · Christopher D. Hacon

Received: 2 February 2006 / Accepted: 19 May 2006 / Published online: 23 August 2006

Abstract In this paper we prove that if X is an irregular 3-fold withχ(ωX) > 0, then

|mKX| is birational for all m ≥ 5.

1 Introduction

Given a non-singular variety of general type, by definition, the pluricanonical system

|mKX| defines a birational map ϕmto a projective space, for all sufficiently big inte- gers 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 ϕmis an embedding for m≥ 3.

For surfaces, by a result of [2],ϕmis birational for m ≥ 5. Recently, following ideas of Tsuji (cf. [9, 14, 17]), it has been shown that for any positive integer n> 0, there exists a positive integer rn> 0 such that ϕmis birational for m≥ rnfor all varieties of general type and of dimension n. Tsuji [18] has also shown, that r3 ≤ 18(2⁹· 3⁷)!.

Recently Todorov [15], 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 irregular varieties, i.e.

varieties with q(X) := h⁰(X, X) ≥ 0. In view of recent results on irregular varieties,

The first author was partially supported by NSC and NCTS of Taiwan. The second author was partially supported by NSF research grant no: 0456363.

J. A. Chen

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

J. A. Chen (

B

⁾

National Center for Theoretical Sciences, Taipei Office, Taipei, Taiwan e-mail: jkchen@math.ntu.edu.tw

C. D. Hacon

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

e-mail: hacon@math.utah.edu

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one might hope that their geometry 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 gen- eral type such that |4KS| is not birational and a curve C of general type. For all P∈ Pic⁰(C × S) ∼= Pic⁰(C) one sees that |4KC×S+ P| is not birational.

The main technical ingredient is the technique developed in [3], where the Fourier–Mukai transform and vanishing theorems are used to study adjoint linear series on irregular varieties. For the reader’s convenience, in Sect. 2 we recall the main result from [3]. Its application to pluricanonical system is summarized in The- orem 2.8. One finds a very sharp bound in Proposition 2.9 when the Albanese fiber dimension is small. Starting in Sect. 3, we will work on irregular varieties over curves.

The case when X maps to a curve with genus g≥ 2 is handled in Sect. 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 Sect. 4, we are able to prove our main theorem for threefolds in Sect. 5.

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 twoQ-divisors D₁, D₂ areQ-linearly equivalent (D₁∼_QD2) if there exists an integer m> 0 such that mDi

are linearly equivalent. Numerical equivalence is denoted≡. And we write D1≥ D2if D₁− D2is 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 ˆA with Pic⁰(A). We will denote byP, the normalized Poincaré 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 OA-modules to the category ofO_ˆA-modules defined by

Sˆ(M) = (p_ˆA)_∗(P⊗ p^∗_AM).

Here p_A and p_Â denote the projections of A× Â to A and Â. Similarly we define S(N) = (pA)∗(P⊗ p^∗_ÂN). Let RS (resp. R ˆS) be the derived functor ofS (resp. ˆS) between the corresponding derived categories. By a result of Mukai (cf. [13] Theorem 2.2), we have:

Theorem 2.1 There exist isomorphisms of functors

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

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A coherent sheafFon an abelian variety A is said to be IT⁰if Hⁱ(A,F⊗ P) = 0 for all i> 0 and all P ∈ Pic⁰(A).

Lemma 2.2 LetF be a coherent IT⁰ sheaf on an abelian variety A. IfF = 0, then H⁰(A,F⊗ P) = 0 for all P ∈ Pic⁰(A).

Proof Please see [3] Lemma 2.1.

Lemma 2.3 Let_F= 0 be an IT⁰coherent sheaf on an abelian variety A. Suppose that there is a non-zero mapF →C(z). Then the induced map H⁰(F⊗P) → H⁰(C(z)) is non-zero for general P∈ Pic⁰(A).

Proof Please see [3] 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 = 0. Suppose furthermore that a_∗OX(KX+ L) = 0. Then we have:

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

a_∗OX(KX+ L) ⊗C(z) ∼= H⁰(F,OF(KX+ L)).

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

(∗) 0 →OX(KX+ L) ⊗Ix →OX(KX+ L) →C(x) → 0.

Pushing forward to Z, we get

0→ a∗(OX(KX+ L) ⊗Ix) → a∗OX(KX+ L) →C(z) → · · · Since x∈ Bs|(KX+ L)|F| and

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

a_∗_O_X(KX+ L) → a∗OX(KX+ L) ⊗C(z) →C(z) is non-zero, whence surjective.

By Corollary 10.15 of [12], a_∗_O_X(KX+ L) is IT⁰. Since a_∗_O_X(KX + L) = 0 by assumption, applying Lemma 2.3 to the sheaf a_∗OX(KX+ L), we get (1).

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Before we move forward to the proof of the next two statements, we remark that the assumption that x₁∈ Bs|KX+L+a^∗P| for all P implies that a∗(OX(KX+L)⊗Ix₁) 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∗(OX(KX+ L)⊗Ix₁,x2) → a∗(OX(KX+ L)⊗Ix₁) →C(z) → · · · The last map factors as

a_∗(OX(KX+ L)⊗Ix₁) → a_∗(OX(KX+ L)⊗Ix₁)⊗C(z)

→ a∗OX(KX+ L)⊗C(z) →C(z).

It is obtained by evaluating at x₂. The assumption that|(KX+ L)|F| separates x1, x₂ shows that this is surjective. Again, by applying Lemma 2.3 to the sheaf a_∗(OX(KX+ L)⊗Ix₁), we are done.

Finally we consider(2). Again we have (assuming now that x2lies over z):

0→ a∗(OX(KX+ L)⊗Ix1,x2) → a∗(OX(KX+ L)⊗Ix1) →C(z) → · · · But now we look at the map

a_∗(OX(KX+ L)⊗Ix₁) → a∗(OX(KX+ L)⊗Ix₁)⊗C(z)

→ a∗OX(KX+ L)⊗C(z) →C(z)

which is obtained by evaluating at x₂. Since a(x1) = a(x2), it follows that a∗(OX(KX+ L)⊗Ix₁)⊗C(z) ∼= a∗OX(KX + 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 defineBm,Pto be the fixed component of|mKX+a^∗P| andBm(resp._B_m,F) to be the fixed component of|mKX| (resp. |mKF|). After replacing X by an appropriate birational model, we may and we usually do assume thatBm,P,Bm,Fhave 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 KXis W-big, i.e. sK_X ≥ g^∗L for some ample divisor L and some integer s 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)KX− g^∗bH| such that

B ab

|F≤Bm,F,

B ab

≤Bm,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 [3], 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,OX(D) ⊗ a^∗P) = 0 . We say that D is full if V⁰(D) = Pic⁰(A).

Note that if a_∗OX(D) is IT⁰and non-zero, then by Lemma 2.2, D is full.

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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 Pm(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)KX− Bm ≡ 1

ag^∗H+ {Bm}

where B_m = Bm + {Bm}. By [12] 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_∗OX(KX + Lm) is IT⁰. Recall that by Lemma 2.5, one has Bm|F ≤ Bm,F. Thus, H⁰(F,OF((KX + Lm)|F)) ∼= H⁰(F,OF(mKF)). In particular, a∗OX(KX+ Lm) = 0. By Lemma 2.2, KX+ Lmis full and hence so is mK_X.

Assume that for some integers m, n≥ 1, mKX and nK_Xare full and consider the morphism

|mKX + a^∗P₁| + |nKX+ a^∗P₂| → |(m + n)KX+ a^∗(P1+ P2)|

∼= |KX + Lm+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 mKX + a^∗P nor a base point of nKX+ a^∗P. Therefore x is not a base point of|(m + n)KX + a^∗P| for all P ∈ Pic⁰(A). And it follows that x is not a base point of|KX+ Lm+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)KX+ 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)KFis 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)KFis birational, then|(m+n+t)KX+a^∗P| separates two general points on a general fiber for all P∈ Pic⁰(A).

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

|(m + n)KX+ a^∗P₁| + |tKX+ a^∗P₂| → |(m + n + t)KX+ a^∗(P1+ P2)|.

separates x₁, x₂for 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) |mKX+ 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|mKX+ P| is birational for a general (resp. all) P∈ Pic⁰(X) for m ≥ 2 (resp. m ≥ 3).

(2) If f = 1, then

(a) |mKX+ 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|mKX+ P| is birational for a general (resp. all) P∈ Pic⁰(X) and for m ≥ 3 (resp. m ≥ 4).

(3) If f = 2, then

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

(b) If moreover K_X 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) follow from Theorem 2.8, letting m = n = t = 2. If we assume that KX

is full, then we let m= n = t = 1.

If f ≥ 1, one sees that for m ≥ 2, mKXis full if and only if P_m(F) = 0. Since F is of general type, for f ≤ 2 it is well known that Pm(F) > 0 for all m ≥ 2. Therefore mKX

is full for m≥ 2.

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

Finally, if f= 2, then mKFis birational for m≥ 5. By Theorem 2.8, we have (a) 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.

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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. Sup- pose that for some m≥ 4, |mKF| is birational. Then |mKX+ a^∗P| is birational for all P∈ Pic⁰(C).

Proof Fix an integer k 0 such that P2k(F) > 0. We write

2kKX − 3kF = 2kKX/C+ 2ka^∗KC− 3kF = 2kKX/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. [19,20]), one has that the restriction map

H⁰(X,OX(2nKX− 3nF)) → H⁰(F,OF(2nKF)),

is surjective for all n  k sufficiently large and divisible by k and by m. There- fore, a_∗(ω²ⁿ_X(−3nF)) is a vector bundle of rank P2n(F) and degree at least P2n(F) (4g(C) − 7)n. Since F is of general type, by Riemann-Roch, one sees that 2KX− 3F is big. We write |2nKX − 3nF| = |Nn| + Bn where B_n is the fixed part. We have 2nK_X ∼ Mn+ Bn= (Nn+ 3nF) + Bn. After replacing X by an appropriate birational model, we may assume that|Nn|, |Mn| are free and that Bn is a divisor with normal crossing support. Note that Mnis nef and big. We write|2nKF| = |(Mn)|F| +B2n,F. By the above surjection, we have thatB2n,F= (Bn)|F.

We have(m − 3)KX ≡ ^(m−3)M_2n ⁿ+^(m−3)B_2n ⁿ and 2K_X − 3F ≡^N_nⁿ+^B_nⁿ. Let Lm:= (m − 1)KX−

(m − 1)Bn

2n

. Since m divides n, we have that Bn≤ _mⁿBm. Therefore,

K_X + Lm≥ mKX−

m− 1 2n

n mB_m

≥ mKX−

1 2B_m

. Similarly,

K_F+ Lm|F≥ mKF−

m− 1 2n ^B^2n,F

≥ mKF−

1 2^B^2m,F

, and|KF+ Lm|F| ∼= |mKF| = ∅. Note that

Lm≡ (m − 3)Mn

2n +Nn

n +

(m − 1)Bn

2n

+ 3F.

By Kawamata–Viehweg vanishing, one has that

H¹(X,OX(KX + Lm+ a^∗P− sF)) = 0 for s≤ 3, P ∈ Pic⁰(C).

Consider the short exact sequence:

0→OX(KX+ Lm+ a^∗P− F1− F2) →OX(KX+ Lm+ a^∗P)

→OF₁(KF₁+ Lm|F₁) ⊕OF2(KF2+ Lm|F2) → 0.

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

and F2.

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Similarly, one has the surjection

H⁰(X,OX(KX+ Lm+ a^∗P)) → H⁰(F,OF(KF+ Lm)).

If|mKF| is birational then |KX+ Lm+ a^∗P| separates general points on F and so does

|mKX+ 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 [1].

Lemma 4.1 For any integer m≥ 2, letI:=OX(−Bm) be as in (2.7). Then a_∗(ω^m_X) = a_∗(ω^m_X⊗I) is an IT⁰vector bundle of rank P_m(F).

Proof Clearly, we have an inclusion a_∗(ω^m_X⊗I) → a∗(ω_X^m). As in (2.7), one sees that they have the same rank. Let_Q be the cokernel. Then _Q has generic rank 0 and hence is supported on finitely many points. If_Q= 0, then h⁰(C,Q) = 0. However, a_∗(ω^m_X⊗I) is IT⁰and h⁰(a_∗(ω_X^m⊗I)) = h⁰(a_∗(ω^m_X)). This is a contradiction. Therefore,

Q= 0.

For simplicity, we will write Em:= 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,. . . , Lr) is a maximal splitting of E/L1. Therefore d(E) = deg(L1). Recall also that by a result of Atiyah [1], given an inde- composable vector bundle E of rank r on an elliptic curve, there is a maximal splitting (L1, L₂,. . . , Lr) such that L1 ≤ L2 ≤ · · · ≤ Lr are line bundles, where L_i ≤ Li+1

denotes that there is an injection L_i → Li+1. Moreover, if E is a vector bundle of rank r and degree 0≤ s ≤ r, then E has a maximal splitting with L1= · · · = Lr−s=O, and L_r−s+1= · · · = Lr= L for some line bundle L of degree 1. Notice that

µ(E) = d(E) = deg(L1).

We also remark, that by a result of Tu [16], 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) = 0 for some P ∈ Pic⁰(E), then h⁰(E^∗⊗ P^∗) = 0 and hence there is a non-zero homomorphism P → E^∗. This is impossible as by Tu [16], E^∗is semi-stable of negative slope.

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(2) ⇒ (3). Let E be an IT⁰vector bundle. By a result of Hartshorne [8] (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 IT⁰implies that L is IT⁰and 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/L1has a decom- position F= ⊕Fiwhere each F_iis indecomposable with d(Fi) ≥ 1. It follows from the inductive hypothesis that each F_iand 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

^α ^β ^γ

0 −−−−→ L₁ −−−−→ E −−−−→ F −−−−→ 0

Sinceα and γ are surjective on an open set then so is β.

Definition 4.5 For any vector bundle E, we write E= ⊕Eifor its decomposition in to indecomposable vector bundles. We defineν(E) := min µ(Ei).

Theorem 4.6 Let a : X→ C be a morphism from a smooth projective variety of gen- eral 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 IT⁰vector bundles on an elliptic curve, then E⊗F is IT⁰. Proof By Lemma 4.3, one sees that for a vector bundle, being IT⁰is 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 on an elliptic curve. Then Hom(E1, E₂) = 0 implies that µ(E2) ≥ µ(E1).

Proof See [16].

Lemma 4.9 Let E, F be vector bundles on an elliptic curve. Thenν(E⊗F) ≥ ν(E) + ν(F).

This is presumably well known, but as we could not find a reference, we include a sketch of the proof.

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Proof It suffices to show that if E and F are indecomposable, thenν(E⊗F) ≥ µ(E) + µ(F). Recall that by Lemma 24 of [1] we may write E ∼= E⊗Fh and F ∼= F⊗Fh

where(rank(E), deg(E)) = 1, (rank(F), deg(F)) = 1, h, h≥ 1 andFkdenotes the indecomposable vector bundle of degree 0 and rank k. By Lemma 18 of [1],_F_h⊗Fh

decomposes as a direct sum of indecomposable vector bundle of degree 0 say_F_h_i. It follows that

ν(E⊗F) = ν(E⊗ F⊗Fh⊗Fh) = min{ν(E⊗ F⊗Fhi)} = ν(E⊗ F).

Here we have used the fact that if G is an indecomposable vector bundle then ν(G⊗Fhi) = ν(G), which follows easily from Lemmas 24 and 18 of [1].

Sinceµ(E) = µ(E) and µ(F) = µ(F), it suffices to show that ν(E⊗F) ≥ µ(E) + µ(F).

If we write rank(E) = p^k₁¹· · · p^knⁿand rank(F) = q^l₁¹· · · q^ln^mwhere p_i(resp. q_i) are distinct primes and ki(resp. li) are positive integers, then By Lemma 29 of [1], we have E= E1⊗ · · · ⊗Enand F= F1⊗ · · · ⊗Fmwhere the Ei(resp. Fi) are indecomposable vector bundles of rank p^k_iⁱ(resp. q^l_iⁱ) and(deg(Ei), pi) = 1 (resp. (deg(Fi), qi) = 1).

By Theorems 13 and 14 of [1], one sees that if p_i= qjthen E_i⊗Fjsplits as a direct sum of irreducible vector bundles whose rank is a power of pi= qjand whose slope is also given byν(Ei⊗Fj) = µ(Ei) + µ(Fj). Repeatedly applying Lemma 28 of [1], one sees that E⊗ Fdecomposes as a product of indecomposable vector bundles of slope

µ(E) + µ(F).

Lemma 4.10 Let E be an IT⁰vector bundle on an elliptic curve 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 Fjof F. By Lemma 4.8,µ(Ei) ≥ µ(Fj). 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, henceµ(Q) ≥ 1.

For any E_i, there exists a non-zero map from a component F_jof F unless E_i∼= Q.

In either case, we haveµ(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 → Qtor→ 0.

Thusν(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:

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1. f := dim X − dim alb(X) = 0.

It follows from generic vanishing theorem (cf. [6, 7]) that ifχ(ωX) > 0, then KX

is full. Therefore, by Proposition 2.9, we are done.

2. f= 1.

By [11] Corollary 3.2,χ(ωX) = χ(a∗ωX) − χ(R¹a_∗ωX) and by [10] Proposition 7.6, R¹a_∗ωX = ωW. Since W is of maximal Albanese dimension, by Theorem 1 of [6],χ(ωW) ≥ 0. Hence χ(a_∗ωX) ≥ χ(ωX) > 0. By the generalized generic vanishing theorem of [4], Proposition 2.2, one sees that a_∗ωX is full and hence so is K_X. Therefore, by Proposition 2.9, we are done with this case.

3. f= 2.

Let a : X → C ⊂ Alb(X) be the Albanese map. Note that a has connected fibers. Consider in fact 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 pg(X) ≥ 1. We have that χ(X, ωX) = χ(C, a∗ωX) − χ(C, R¹a_∗ωX) + χ(C, R²a_∗ωX).

Moreover, by [10] Proposition 7.6, R²a_∗ωX= ωC=OCand soχ(R²a_∗ωX) = 0. Now, h¹(C, R¹a_∗ωX) = q(X) − h⁰(C, R²a_∗ωX) = 0 implies that χ(C, R¹a_∗ωX) ≥ 0. Hence, one hasχ(C, a∗ωX) ≥ 0. So by the generalized generic vanishing of [4], Proposition 2.2, we see that K_Xis full as in Case 2.

Let F be the general fiber of a and F₀be its minimal model. If|4KF| is birational, then we are done by Theorem 2.8. So by Bombieri’s [2] classification, it remains to consider the case that(K²_F₀, pg(F0)) = (1, 2). We recall that for this surface, q = 0,

|KF0| has a unique base point and |2KF0| is base point free.

Claim Let E_m= a∗(ω_X^m). Then ν(Em) ≥ 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 Em⊗P for m ≥ 5 is generically generated by global sections. That is, there exists an open set U⊂ C and a surjection

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

H⁰(X, ω^m_X⊗ a^∗P) → H⁰(Em⊗ P) ⊗OU → Em⊗ P ⊗ k(y) ∼= H⁰(ω^m_F_y), for general y∈ U. If follows that |mKX+ P| separates points on a general fiber Fyfor 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 Rm := H⁰(ω^m_F), we first show that R1Rm⊂ R_m+1has codimension≤ 1 for all m≥ 0. To see this, we may assume that F = F0. Let Y ∈ |KF| be a general curve. We have that Y is non-singular, for otherwise Y is singular at the base point of|KF| and hence 1= Y· Y ≥ 2 for Y= Y ∈ |KF|.

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

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the exact sequence:

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

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

0→ H⁰(ω^m−1_F ) → H⁰(ω^m_F) ⊗ H⁰(ωF) → 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

ϕm: a_∗ωX⊗ a∗ω_X^m→ a∗ω_X^m⁺¹.

By the above claim, one sees thatϕmhas cokernel of generic rank≤ 1 for all m ≥ 1 and it is generically surjective for m= 2. Notice that Em= a∗ω^m_Xis IT⁰for m≥ 2 by Lemma 4.1. By Lemma 4.7, deg Em> 0 for m ≥ 2

We now distinguish the following cases:

1. E₁ := a∗ωXis indecomposable.

Since KX is full, so is E₁. By the semipositivity of a_∗ωX (cf. [5]), we have that deg(E1) ≥ 0. If deg(E1) = 0, then E1 ∼= U ⊗ P for some vector bundle U with maximal splitting(OC,OC) 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. By Lemma 4.9, one sees thatν(E1⊗ E1) ≥ ν(E1)+ν(E1) ≥ 1 and hence ν(imϕ1) ≥ 1 by Lemma 4.8.

Moreover, the cokernel ofϕ1is IT⁰and has rank≤ 1, so ν(E2) ≥ 1 by Lemma 4.10. Repeatedly applying this argument, one sees thatν(Em) ≥ 1 for m ≥ 5.

2. E₁is 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 ϕ1has cokernel of rank≤ 1. It follows that Sym²(E1) ∼= im(ϕ1).

2.a. E₂is indecomposable.

By Lemma 4.8, µ(E2) ≥ 2 for there is a non-zero map from L² to E₂. By consideringϕ2, ...,ϕ_m−1and Lemma 4.9, one has thatν(Em) ≥ 1 for m ≥ 5.

2.b. E2 =E1⊕E2with ranks 3, 1 respectively.

Thenµ(E1) ≥ 1 for there is a non-zero map from L or L² in to im(ϕ1) toE1. Since E₂is IT⁰, so is_E₂. Thusµ(E2) ≥ 1 for it is a line bundle of IT⁰. Hence we haveν(E2) ≥ 1. Proceeding as above, it follows that ν(Em) ≥ 1 for m ≥ 5.

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

Since E2 is IT⁰, so are the bundlesEi. Thusµ(Ei) ≥ 1 and hence ν(E2) ≥ 1.

Proceeding as above, it again follows thatν(Em) ≥ 1 for m ≥ 5.

2.d. E₂decomposes into bundles of rank(2, 1, 1) or (2, 2).

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

ψ : a_∗ω²_X⊗ a_∗ω²_X → a_∗ω⁴_X.

One can check that R2R2 ⊂ R4has codimension≤ 1 (its image in fact contains the image of R₁R₁R2and hence it contains the image of R₁R3which has codimension at most 1). By Lemma 4.9,ν(E2⊗ E2) ≥ ν(E2) + ν(E2) ≥ 1 and hence ν(E4) ≥ 1 by Lemma 4.10. Then, by considering ϕ4,. . . , ϕm−1, it follows that ν(Em) ≥ 1 for m ≥ 5. This completes the proof.

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Acknowledgments We are indebted to Meng Chen, De-Qi Zhang, Yujiro Kawamata, I-Hsun Tsai, Hélène Esnault, Eckart Viehweg and Chin-Lung Wang for useful conversations and comments on this subject and to the referee for many helpful suggestions.

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