Digital Object Identifier (DOI) 10.1007/s002080100201
Math. Ann. 320, 367–380 (2001)
Mathematische Annalen
Pluricanonical maps of varieties of maximal Albanese dimension
Jungkai A. Chen· Christopher D. Hacon
Received July 7, 2000 / Published online April 12, 2001 – © Springer-Verlag 2001
Abstract. LetX be a smooth complex projective algebraic variety of maximal Albanese dimen- sion. We give a characterization ofκ(X) in terms of the set V0(X, ωX) := {P ∈ Pic0(X)|h0(X, ωX⊗ P ) = 0}. An immediate consequence of this is that the Kodaira dimension κ(X) is invariant under smooth deformations. We then study the pluricanonical mapsϕm: X P(H0(X, mKX)).
We prove that ifX is of general type, ϕm is generically finite form ≥ 5 and birational for m ≥ 5dim(X) + 1. More generally, we show that for m ≥ 6the image of ϕmis of dimension equal toκ(X) and for m ≥ 6κ(X) + 2, ϕmis the stable canonical map.
Mathematics Subject Classification (2000): 14C20, 14F17
Introduction
Generic vanishing theorems have recently proven to be a very effective tool in the study of the geometry of irregular varieties. In this paper we will show how these techniques can be used to answer a series of natural questions about varieties of maximal Albanese dimension.
In what follows, X will denote a smooth complex projective algebraic va- riety, Alb(X) denotes the Albanese variety of X and albX : X −→ Alb(X) the corresponding Albanese map. We will assume that X is of maximal Al- banese dimension and hence albXis a generically finite morphism. In particular q(X) ≥ dim(X). Koll´ar made the following
Conjecture [Ko3, (17.9.3)]. If X is of general type and maximal Albanese dimension. Thenχ(X, ωX) > 0.
As an immediate consequence of the generic vanishing theorems of Green and Lazarsfeld (cf. Theorem 1.3.1), one sees that this is equivalent toh0(X, ωX⊗P ) >
J.A. Chen
Department of Mathematics, National Chung Cheng University, Ming Hsiung, Chia Yi, 621, Taiwan (e-mail: jkchen@math.ccu.edu.tw)
C.D. Hacon
Department of Mathematics, University of Utah, 155 South 1400 East JWB 233, Salt Lake City, UT 84112, USA (e-mail: chhacon@math.utah.edu)
The author was partially supported by National Science Council, Taiwan (NSC-89-2115-M-194-016)
0 for allP ∈ Pic0(X). In [EL1, Example 1.13], Ein and Lazarsfeld produce a counterexample to this conjecture. However, we will show that a similar result does hold.
Theorem 1. Let X be a variety of maximal Albanese dimension. The trans- lates through the origin of the irreducible components ofV0(X, ωX) := {P ∈ Pic0(X)|h0(X, ωX⊗P ) = 0} generate a subvariety of Pic0(X) of dimension κ(X) − dim(X) + q(X). In particular, if X is of general type, V0(X, ωX) gen- erates Pic0(X).
This is a generalization of a result of Ein and Lazarsfeld [EL2] (see Theorem 1.3.2). An immediate consequence is the following:
Corollary 2. LetX be of maximal Albanese dimension. Then the Kodaira di- mension ofX is invariant under smooth deformations.
We next turn our attention to the study of the linear series|mKX| and the corresponding rational mapsϕm:= ϕ|mKX|. In [Ko3], Koll´ar shows that ifX is a smooth proper variety of general type with generically large algebraic fundamen- tal group, then the sections of|mKX| define a birational map for m ≥ 10dim(X). Following [AS], it is reasonable to expect a quadratic bound in dim(X).
Theorem 3. IfX is of maximal Albanese and Kodaira dimensions, then ϕ5 is generically finite, andϕ5dim(X)+1is birational.
Our proof is based on the techniques of Koll´ar [Ko3]. However, technical difficulties imply that whenX is not of general type, the bounds are somewhat weaker.
Theorem 4. IfX is of maximal Albanese dimension, then ϕ6(X) has dimension equal toκ(X) and ϕ6κ(X)+2 is the stable canonical map.
We do not expect these bounds to be optimal. It would be interesting to produce examples of varieties of general type and maximal Albanese dimension for whichϕmis not birational form ≥ 3.
Acknowledgements. We would like to thank P. Belkale, H. Clemens, Y. Kawamata, J. Koll´ar, R.
Lazarsfeld and I.-H. Tsai for valuable conversations.
Conventions and notations
(0.1) Throughout this paper, we work over the field of complex numbersC.
(0.2) ForD1, D2Q-divisors on a variety X, we write D1 ≺ D2ifD2− D1 is effective, andD1≡ D2ifD1andD2are numerically equivalent.
(0.3) |D| will denote the linear series associated to the divisor D, and Bs|D|
denotes the base locus of|D|.
(0.4) For a real number a, let a be the largest integer ≤ a and a be the smallest integer≥ a. For a Q-divisor D =
aiDi, let D =
aiDi and
D =
aiDi.
(0.5) Let F be a coherent sheaf on X, then hi(X, F) denotes the complex di- mension ofHi(X, F). In particular, the plurigenera h0(X, ωX⊗m) are denoted byPm(X) and the irregularity h0(X, ΩX1) is denoted by q(X).
(0.6) We will denote by Alb(X) the Albanese variety of X, by albX : X −→
Alb(X) the Albanese morphism. As usual Pic0(X) is the abelian variety dual to Alb(X) parametrazing all topologically trivial line bundles on X. Picτ(X) will denote the set of torsion elements in Pic0(X).
(0.7) Letf : X → Y be a generically finite morphism. Rf denotes the ramifica- tion divisor.
(0.8) Letf : X → Y be a morphism. An irreducible divisor E on X is said to be f -exceptional if dimf (E) < dimf (X) − 1. And E is f -vertical if dimf (E) ≤ dimf (X) − 1. An effective divisor is f -exceptional (resp. f -vertical) if every component is.
(0.9) We will say that aQ-divisor ∆ is klt (Kawamata log terminal) if ∆ has normal crossings support and∆ = 0. We refer to [Ko3, 10.1.5] for the general definition of klt divisor.
1 Preliminaries
1.1 Higher direct images of dualizing sheaves
For the convenience of the reader, we recall various results which will be fre- quently used in what follows.
Theorem 1.1.1. LetX, Y be projective varieties, X smooth and f : X −→ Y a surjective map. LetM be a line bundle on X such that M ≡ f∗L + ∆, where L is aQ-divisor on Y and (X, ∆) is klt. Then
(a)Rjf∗(ωX⊗M) is torsion free for j ≥ 0.
(b) ifL is nef and big, then Hi(Y, Rjf∗(ωX⊗M)) = 0 for i > 0, j ≥ 0.
(c)Rif∗ωXis zero fori > dim(X) − dim(Y ).
(d)R·f∗ωX∼=
Rif∗ωX[−i] and hp(X, ωX) =
hi(Y, Rp−if∗ωX).
(e) IfY is birational to an abelian variety and L is nef and big, it follows that H0(Y, Rjf∗(ωX⊗M)⊗P ) = 0 for all P ∈ Pic0(Y ) if and only if Rjf∗(ωX⊗M)
= 0.
Proof. See [Ko1], [Ko2], [Ko3], [Ka2], [EV1].
Theorem 1.1.2. Letf : X −→ Y be a surjective morphism from a smooth projective variety to a projective variety. LetQ ∈ Picτ(X). Then
R·f∗(ωX⊗Q) ∼=
Rif∗(ωX⊗Q)[−i].
In particularhp(X, ωX⊗Q) =
hi(Y, Rp−if∗(ωX⊗Q)).
Proof. [Ko2, Corollary 3.3].
1.2 The Iitaka fibration
LetX be a smooth complex projective variety of maximal Albanese dimension.
Leta : X −→ A be a generically finite map such that the image of X generates the abelian varietyA. A nonsingular representative of the Iitaka fibration of X is a morphism of smooth complex projective varietiesf : X −→ Y such that Xis birational toX, dim(Y ) = κ(X) and κ(Xy) = 0, where Xy is a generic geometric fiber off. Since our questions will be birational in nature, we may always assume thatX = X and f = f. The generic fiber Xy has maximal Albanese dimension as X does, hence q(Xy) ≥ dim(Xy). Since κ(Xy) = 0, it follows by [Ka1] that Xy is birational to an abelian variety. Therefore the image of the fiber Xy is the translate of an abelian subvarietyA, denoted Ky, anda|Xy : Xy −→ Ky is birationally equivalent to an ´etale map. SinceA can contain at most countably many abelian subvarieties, we may assume that the Ky are all translates of a fixed abelian subvarietyK of A. Since Xy −→ K is birationally ´etale,Xyis also birational to a fixed abelian variety which we denote by ˜K. Let p : A −→ S := A/K. Let Z (resp. W) denote the image of X in A (resp.S). By construction, the general fiber of f : X −→ Y maps to a closed point inS, therefore replacing X by an appropriate birational model, there exists a morphismq : Y −→ S such that q ◦ f = p ◦ a. We may assume that the above maps fit in the following commutative diagram
X −−−→ Za −−−→ A⊂
f p
Y −−−→ Wq −−−→ S.⊂
1.3 Cohomological support loci
Leta : X → A be a morphism from a smooth projective variety X to an abelian varietyA. If F is a coherent sheaf on X, then one can define the cohomological support loci by
Vi(X, T , F) := {P ∈ T ⊂ Pic0(A)|hi(X, F ⊗ a∗P ) = 0}.
In particular, ifa = albX: X → Alb(X), then we simply write Vi(X, F) := {P ∈ Pic0(X)|hi(X, F ⊗ P ) = 0}.
We say thatX has maximal Albanese dimension if dim(albX(X)) = dim(X).
The geometry of the lociVi(X, ωX) defined above is governed by the following:
Theorem 1.3.1 (Generic vanishing). (a) Any irreducible component ofVi(X, ωX) is a translate of a sub-torus of Pic0(X) and is of codimension at least i − (dim(X) − dim(albX(X))).
(b) LetP be a general point of an irreducible component T of Vi(X, ωX). Sup- pose thatv ∈ H1(X, OX) ∼= TPPic0(X) is not tangent to T . Then the sequence
Hi−1(X, ωX⊗ P )→ H∪v i(X, ωX⊗ P )→ H∪v i+1(X, ωX⊗ P ) is exact. Ifv is tangent to T , then the maps in the above sequence vanish.
(c) IfX is a variety of maximal Albanese dimension, then
Pic0(X) ⊃ V0(X, ωX) ⊃ V1(X, ωX) ⊃ ... ⊃ Vn(X, ωX) = {OX}.
(d) Every irreducible component ofVi(X, ωX) is a translate of a sub-torus of Pic0(X) by a torsion point.
Proof. See [GL1],[GL2],[EL1] and [S].
In [EL2], Ein and Lazarsfeld illustrate various examples in which the geometry ofX can be recovered from information on the loci Vi(X, ωX). In particular, they prove:
Theorem 1.3.2 [EL2]. IfX is a variety with maximal Albanese dimension and dim(V0(X, ωX)) = 0, then X is birational to an abelian variety.
Proposition 1.3.3 [EL2]. Leta : X −→ A be a generically finite map from a smooth projective variety to an abelian variety. LetP be any isolated point of V0(X, Pic0(A), ωX). Then a∗P = OX.
We will also need the following
Lemma 1.3.4 [CH, Lemma 2.1]. Let X be a variety of maximal Albanese dimension. Fix Q ∈ Picτ(X). Then h0(X, ω2X⊗Q⊗f∗P ) is constant for all torsionP ∈ Pic0(Y ).
2 Kodaira dimension of varieties of maximal Albanese dimension
Throughout this paper, we will assume thatX is of maximal Albanese dimension and hence of positive Kodaira dimensionκ(X) ≥ 0. We will frequently refer to the notation and results of Sect. 1. And we will need the following immediate consequence of [M, Theorem 2.2]:
Lemma 2.1. IfF is a coherent sheaf on an abelian variety A such that hi(A, F
⊗P ) = 0 for all i ≥ 0 and all P ∈ Pic0(A). Then F = 0.
Define
G := ker
Pic0(A) −→ Pic0(K) −→ Pic0( ˜K) .
We will frequently identify Pic0(S) with its image in Pic0(A). Let G := G/Pic0 (S). One sees that dim(G) = dim(A) − dim( ˜K) = dim(S) and Pic0(S) is contained inG. It follows that G is a finite group, hence G consists of finitely many translates of Pic0(S). Let Q1, ..., Qr ∈ G ⊂ Pic0(A) be a set of torsion line bundles representing lifts of the elements ofG.
Lemma 2.2. (a)V0(X, Pic0(A), ωX) ⊂ G.
(b) For everyQi, the lociV0(X, Qi + Pic0(S), ωX) are nonempty.
(c) IfOX∈ a∗(Qi+Pic0(S)), then there exists a positive dimensional component ofV0(X, Qi+ Pic0(S), ωX).
Proof. (a) If H0(X, ωX⊗a∗P ) = 0 for some P ∈ Pic0(A), then for general y ∈ Y , h0(Xy, ωXy⊗a∗P ) = h0( ˜K, OK˜⊗a∗P ) = 0. This is possible only if P is in the kernel of Pic0(A) −→ Pic0( ˜K) = Pic0(Xy).
To prove (b), consider π : X −→ W ⊂ S. Assume that V0(X, Q + Pic0(S), ωX) is empty, then by Theorem 1.3.1.c, Hi(X, ωX⊗a∗Q⊗π∗P ) = 0 for all i ≥ 0 and all P ∈ Pic0(S). By Theorem 1.1.2, Hk(W, Rjπ∗(ωX⊗ a∗Q)⊗P ) = 0 for all k, and all P ∈ Pic0(S). Therefore, by Lemma 2.1, Rjπ∗(ωX⊗ a∗Q) = 0 for all j. In particular, for a general fiber Xw of π : X −→ W,
h0(Xw, ωXw⊗a∗Q) = 0.
Notice thatXwis a finite union of general fibers ofX → Y hence a finite union of varieties birational to ˜K, therefore for Q ∈ G
h0(Xw, ωXw⊗a∗Q) = h0(Xw, ωXw) > 0,
which is a contradiction. We may therefore assume thatV0(X, Q+Pic0(S), ωX) is nonempty.
The assertion (c) now follows since by Proposition 1.3.3, any isolated point Q of V0(X, Pic0(A), ωX) must be such that a∗Q = OX. Theorem 2.3. Leta : X −→ A be a generically finite morphism from a smooth complex projective variety to an abelian variety. Then, the translates through the origin of the components ofV0(X, Pic0(A), ωX) generate Pic0(S).
Proof. Assume that translates through the origin of the components ofV0 :=
V0(X, Pic0(A), ωX) do not generate Pic0(S). Then there exists an abelian proper subvarietyT ⊂ Pic0(S) and a finite subgroup Gsuch thatV0⊂ T +G. Consider the induced morphism
π : X −→ T∗=: C, which factors throughX−→ Aa −→ S −→ C.p
Let Q ∈ Picτ(A) be any torsion element not contained in T + G. Then, H0(X, ωX⊗a∗Q⊗π∗P ) = 0 for all i ≥ 0 and all P ∈ Pic0(C). By Theorem 1.3.1.c,Hi(X, ωX⊗a∗Q⊗π∗P ) = 0 for all 0 ≤ i ≤ n and all P ∈ Pic0(C).
LetV ⊂ C be the image of X, and π : X −→ V be the induced map. By Theorem 1.1.2 and the projection formula, it follows that
Hj(V, Riπ∗(ωX⊗Q)⊗P ) = 0
for alli, j and P ∈ Pic0(C). By Lemma 2.1, Riπ∗(ωX⊗Q) = 0 for all i.
Ifπ : X −→ V is generically finite, then π∗(ωX⊗Q) is clearly non-zero.
We may therefore assume thatπ has positive dimensional generic fibers.
Letp be a point in V and Xp = π−1(p) be the inverse image of p. For general p ∈ V (not depending on Q), we have that
Hi(Xp, ωXp⊗Q) = Hi(Xp, (ωX⊗Q)⊗OXp) = 0.
LetB be the connected component through the origin of the kernel of A −→
C. The image of XpinA is contained in a translate of B which we denote by Bp. Sinceh0(Xp, ωXp⊗Q) = 0 for all but finitely many Q ∈ Pic0(B), it follows by Proposition 1.3.2 thatκ(Xp) = 0 and Xp −→ Bpis birationally an ´etale map of abelian varieties.
On the other hand, by the weak addition formula κ(Xp) + dim(V ) ≥ κ(X),
soκ(Xp) > 0 since κ(X) = dim(S) > dim(V ). This contradicts the assumption thatT is a proper subvariety of Pic0(S). Therefore, T = Pic0(S). Corollary 2.4. Leta : X −→ A be a generically finite morphism from a smooth complex projective variety to an abelian variety. The dimension of the subgroup of Pic0(A) generated by the translates through the origin of the components of V0(X, Pic0(A), ωX) is equal to κ(X) − dim(X) + q(X).
Example [Ko3, 17.9.5]. Letp : C −→ E be a degree 2 map from a genus 2 curve C to an elliptic curve E. We may assume that p∗OC = OE ⊕ L−1, for an appropriateL ∈ Pic(E) such that L⊗2 = OE(B) where B is the branch locus. Let e : ˜F −→ F be a degree 2 ´etale map of elliptic curves such that e∗O˜F = OF ⊕ P with P⊗2∼= OF. Define
X := ˜F × C/ < (i˜F × iC) > .
Herei˜F andiCdenote the involutions on ˜F and C respectively. We have that for a : X −→ F × E,
a∗(ωX) ∼= (OF ⊗ OE) ⊕ (P ⊗ L).
(Pull backs have been omitted.) It follows that Iitaka fibration has imageE and V0(X, Pic0(F × E), ωX) = {OF ×E} ∪ (P + Pic0(E)).
In particularV0(X, Pic0(E), ωX) does not generate Pic0(E).
Corollary 2.5. LetX be any variety of maximal Albanese dimension. κ(X) is invariant under smooth deformations.
Proof. Let∆ be an open neighborhood of a point 0 of a smooth projective curve, δ : X −→ ∆ be a smooth projective morphism with connected smooth fibers, X0:= δ−1(0) be a closed fiber of δ such that X0= X. Since q(X) is deformation invariant, for allt ∈ ∆, At := Alb(Xt) is an abelian variety of dimension q(X).
LetP ∈ Pic0(X) such that P⊗m = OX,P be a section of Pic0(X /∆) such thatP = P0:= P|X0,Pt := P|Xt ∈ Pic0(At) satisfies Pt⊗m= OXt. Let
X˜t := Spec
⊕m−1i=0Pt⊗i
be the corresponding ´etale cyclic cover of degreem. We have that the quantity h0( ˜Xt, ωX˜t) =
m−1
i=0
h0(ωXt⊗Pt⊗i)
is constant. However the functionsh0(Xt, ωXt⊗Pt⊗i) are upper semicontinuous int ∈ ∆, and hence also constant.
Let Ti denote the translates of the components Ti ofV0(X, Pic0(A), ωX) through the origin. The subvarietiesTiare determined by their torsion points (cf.
Theorem 1.3.1.d). In particular, recall that Pic0(Xt) ∼= H1(Xt, O∗Xt)/H1(Xt, Z) and the subvarietiesTit are determined by the corresponding vector subspaces ofH1(Xt, Q). We remark that given two Q vector subspaces Wti ⊂ H1(Xt, Q) continuous in the parameter t, then dim(Wt1∩ Wt2) and dim(Wt1 + Wt2) are constant.
From the above discussion, it follows that there exist subvarieties Tti of Pic0(At), which are smooth deformations of T0isuch thatV0(Xt, Pic0(At), ωXt)
= ∪Tti. Moreover, for any set of indicesI, the quantities
dim
i∈I
Tit
and dim
i∈I
Tit
are constant. In particular the quantity dim
V0(Xt, Pic0(At), ωXt)
= κ(Xt) + q(Xt) − dim(Xt)
is constant and henceκ(Xt) is also constant.
3 Pluricanonical maps of varieties of maximal Albanese dimension
We will keep the notation of the preceding sections. In particularX will be a smooth projective variety with maximal Albanese dimension anda : X −→ A will denote a generically finite morphism to an abelian variety.
Lemma 3.1. LetE be an a-exceptional effective divisor on X. If OX(E)⊗P is effective for someP ∈ Pic0(X), then P = OX.
Proof. If dim(X) = 1, then E = 0 and the assertion is clear. If dim(X) > 1, pick HA a sufficiently ample divisor onA and let H = a∗HA be the corresponding nef and big divisor onX. Choosing HAappropriately, we may assume thatH is a smooth divisor inX. We may also assume that h0(X, OX(E − H)⊗P ) = 0 for allP ∈ Pic0(X). If dim(X) ≥ 2, from the exact sequence of sheaves
0−→ OX(E − H)⊗P −→ OX(E)⊗P −→ OH(E)⊗P −→ 0, we have an injectionH0(X, OX(E)⊗P ) ;→ H0(H, OH(E)⊗P ) for all P ∈ Pic0(X). Similarly, there is an exact sequence of sheaves
0−→ OX(−H)⊗P −→ OX⊗P −→ OH⊗P −→ 0.
SinceH is nef and big,
hi(X, OX(−H)⊗P ) = hn−i(X, ωX(H )⊗P∗) = 0 f or all i < n.
Therefore,h0(X, OX⊗P ) = h0(H, OH⊗P ) = 0 for P = OXandh0(X, OX)
= h0(H, OH) = 1. Clearly, for a general choice of H , we have that H is of maximal Albanese dimension, andE|H is ana|H-exceptional divisor. Repeating the above procedure, by successively intersecting appropriate divisors pulled back fromA, one obtains a curve C ⊂ X such that
i) Ifh0(X, OX(E)⊗P ) > 0 then h0(C, OC(E)⊗P ) = h0(C, OC⊗P ) > 0, ii) Ifh0(C, OC⊗P ) > 0 then P = OX.
It follows that ifh0(X, OX(E)⊗P ) > 0, then P = OX. Lemma 3.2. LetD be an irreducible reduced divisor on X which is not a : X −→ A exceptional, H a Cartier divisor on X which is numerically trivial on the general fiber ofX −→ S. Then, D is not contained in the base locus of
|mKX+ H | for infinitely many values of m.
Proof. [Ko3, 17.6.1].
Let|mKX| = Fm+ |Mm|, where Bs|Mm| contains no divisors. We remark that, sinceX is of maximal Albanese dimension, we may assume that KXis effective.
Theorem 3.3. IfX is of general type, i.e. κ(X) = dim(X). Then for any integer s ≥ 3, Fs isa-exceptional. |5KX| defines a generically finite rational map and
|(5dim(X) + 1)KX| defines a birational map.
Proof. For any fixed divisorD on X, then there exists an m0such that|mKX− D| = ∅ for all m ≥ m0. Let R be an irreducible divisor of X which is not a-exceptional. By Lemma 3.2, R ⊂ Bs|mKX− D| for infinitely many m # 0.
Step 1. Fs is exceptional for alls ≥ 3.
It is easy to see thatFs beinga-exceptional is a condition which is independent of the particular birational model of X under consideration. F1 ⊂ Ra , the ramification divisor of a : X −→ A, contains at most finitely many non a- exceptional components, which we denote byRi. Fix positive integersmi such thatRi ⊂ Bs|miKX|, then Ri ⊂ Bs|λmiKX| for any integer λ > 0. It follows that form0=
mi,Fm0 isa-exceptional.
Next, fix an ample divisorH on A, then a∗H is nef and big on X. Let KX
be a canonical divisor. Letr be the multiplicity of KXatR. We distinguish two cases.
Case 1. r = 1.
By Lemma 3.2, there exists a positive integert such that the base locus of |tKX− a∗H| doesn’t contain R. Let B be an general element of |(s − 2)(tKX− a∗H)|.
By replacingX with an appropriate birational model, we may assume that R is smooth andB +R has normal crossing support and B does not contain R. Define
M := OX
(s − 2)KX− B t
≡
s − 2 t
a∗H + {B t }.
Consider the exact sequence
0→ ωX⊗M → ωX⊗M(R) → ωR⊗M → 0.
Since a∗H is nef and big and {Bt} is klt, one sees that H1(X, ωX⊗M) = 0.
The divisorR is not a-exceptional, so KR is effective and(a∗H)|R is nef and big.R is not contained in the support of B, so the Q divisor {1t(B|R)} is also klt. By Theorem 1.1.1.e,H0(R, ωR⊗M) = 0. Therefore, there is a divisor in
|KX+ M + R| not containing R. This gives a divisor in |sKX| = |KX+ M + R + (KX− R) + Bt | not containing R.
Case 2. r ≥ 2.
There exists positive integerst, m0such thatR ⊂ Bs|tKX− a∗H|, and R ⊂
Bs|m0KX|. For any integer s ≥ 2 let K := (s − 1)KX − R. Consider the following linear series
|mK−m0a∗H| = |
(s−1)m−m r −m0t
KX+m
r (KX−rR)+m0(tKX−a∗H)|.
It follows that form divisible by m0r, R is not in the base locus of |mK−m0a∗H|.
Choose a generalB ∈ |mK− m0a∗H| and define M := OX
K− B m
≡ m0
ma∗H + {B m}.
An argument similar to the one in the previous case again shows thatR is not in the base locus of|sKX|.
Step 2. |5KX| defines a generically finite map.
Replacing X by an appropriate birational model, we may assume that |M3| is base point free. Let D be a general member of M3 and∆ be the image of D inA. If ∆ is not of general type, then there exists an ample line bundle H, a semipositive line bundleL and a positive integer s such that ∆ · Ls· Hn−s−1= 0 andLs· Hn−s > 0. It follows that also D · (a∗H)n−s−1· (a∗L)s = 0.
Since X is of general type, there exists a rational number ? > 0 such that KX− ?(a∗H) is an effective Q-divisor. Therefore,
KX· (a∗H)n−s−1· (a∗L)s ≥ ?(a∗H)n−s· (a∗L)s > 0.
SinceKX− D ≡ F3isa-exceptional, it follows that
D · (a∗H)n−s−1· (a∗L)s = KX· (a∗H)n−s−1· (a∗L)s > 0.
This is a contradiction. We may therefore assume that ∆ and hence D are of general type. It is well known that for an appropriate desingularization∆of∆, the linear series|K∆| defines a generically finite rational map. Therefore, we may assume that|KD| defines a generically finite rational map. Let B ∈ |mKX−a∗H|, andM := OX(KX− Bm) ≡ a∗mH + {Bm}. We may assume that B has normal crossings support and does not containD. Consider the exact sequence of sheaves
0−→ ωX⊗M −→ ωX⊗M(D) −→ ωD⊗M −→ 0, which is also exact on global sections.
Claim. There existm, B such that M is effective.
Fixm1such that|m1KX− a∗p∗HS| is non empty. Let B1∈ |m1KX− a∗p∗HS|.
Letm := m1+ m2andB := B1+ m2KX ∈ |mKX− a∗p∗HS|. Let Γ be any prime divisor, andb1, k be the multiplicities of B1, KXalong Γ . For m2 # 0, we have that
b1+ km2
m = b1
m1+ m2
+ k m2
m1+ m2
≤ k.
Since there are only finitely many components ofB + KX, form2# 0, we have
Bm ≺ KX, and the claim follows.
It is easy to see thatM|Dis also effective. Therefore,|KX+D +M| restricted toD defines a generically finite rational map. If |KX+ D + M| does not define a generically finite rational map, then the closure of the image ofX which we denote byY ⊂ P = P(H0(X, ωX⊗M(D))), must be dominated by D. This is however impossible since by Theorem 1.1.1.eh0(X, ωX⊗M) > 0, and hence
|KX + D + M| contains non-trivial sections vanishing on D, i.e. there is a
hyperplane section ofP containing the image of D but not containing Y . Finally, the assertion follows from the inclusion of sheaves
ωX⊗M(D) ;→ ωX⊗5.
By [Ko1, Theorem 8.1],|(5dim(X) + 1)KX| defines a birational map. Theorem 3.4. |6KX| defines a rational map with image of dimension κ(X), and
|(6κ(X) + 2)KX| defines a stable canonical map.
Proof. Let HA, HS be the pull backs of sufficiently ample divisors on A, S respectively. For anyP ∈ Pic0(X), let
|mKX+ P | = |Mm,P| + Fm,P, whereBs|Mm,P| contains no divisor.
Step 1. Let S −→ S be any surjective map of abelian varieties such that y := dim(S) > y := dim(S) and s : X → S be the induced map. Then, for any divisorD ∈ |M4|, D is not s-vertical.
By Lemma 1.3.4, the linear series|2KX+ P | is nonempty for all P ∈ Pic0(S).
LetHS be the pull back toX of a sufficiently ample divisor on S. Let Z be an irreducible component of HSy, i.e. a general fiber ofs : X → S. For all P ∈ Pic0(S) one has a map of linear series
|2KX+ P | × |2KX− P | −→ |4KX|.
It follows that
F4⊂
P ∈Pic0(S)
(F2,P + F2,−P).
Claim 1. F24 ≺ F2,P for generalP ∈ Pic0(S).
LetV ⊂ Pic0(S) be an open set such that h0(X, ω⊗2X ⊗P ) is constant. Let R be any component ofF4of multiplicityr, then r2R ≺ F2,P for allP ∈ U ⊂ Pic0(S) where U is a Zariski dense subset such that U ∪ −U = Pic0(S). Since the condition r2R ≺ Bs|2KX + P | is Zariski closed in V , it follows that OX(2KX− r2R)⊗P is effective for all P ∈ V ⊂ Pic0(S). Since this holds for all components ofF4, the claim follows.
Claim 2. ker(Pic0(S) → Pic0(Z)) is a proper closed subvariety of Pic0(S).
LetV be the image of Z in S. V is a translate of ker(S → S). There exists infinitely manyP ∈ Pic0(S) such that P |V = OV. Hence pulling back toZ, we haveP |Z = OZ.
Claim 3. (2KX− F24)|Zis nota|Z-exceptional.
By Claim 1,(2KX− F24 + P )|Zis effective and for generalP ∈ Pic0(S).
By Claim 2, we may assume that P |Z = OZ for generalP ∈ Pic0(S). Since a|Z: Z −→ a(Z) is generically finite, the claim follows by Lemma 3.1.
Claim 3 implies that 2KX−F24 is not s-vertical (otherwise(2KX−F24)|Z
= OZ). From the inclusion of linear series
|2KX− F4
2 | ×|2KX− F4
2 | −−−−−−→ |+2F42 −F4 4KX− F4| it follows that 4KX− F4= M4is also nots-vertical.
Step 2. LetD be a general member of M4, and∆ be its image in S. Then ∆ is of general type.
If ∆ is not of general type, then ∆ is vertical for an appropriate projection S −→ S. By Step 1 this is impossible.
Step 3. |6KX| defines a rational map whose image is of dimension κ(X).
Let ˜∆ be an appropriate desingularization of ∆. The linear series |K∆˜| defines a generically finite rational map. ReplacingX by an appropriate birational model, we may assume that|M4| is free and hence D is a smooth subvariety that maps onto ˜∆.
Fix an ample divisor HS on S. Let B ∈ |mKX − a∗p∗HS| be a general member. ReplacingX by an appropriate birational model, we may assume that B has normal crossings support. Define
M := OX
KX− B m
≡ a∗p∗HS
m + {B m}.
As in the proof of the previous theorem, we may assume thatM, M|Dare effec- tive, andB does not contain D. Consider the exact sequence:
0−→ ωX⊗M −→ ωX⊗M(D) −→ ωD⊗M −→ 0.
By [Ko3, Theorem 10.19], this is exact on global sections. Sections ofωD⊗M lift to sections ofωX⊗M(D). Since OD(KD/ ˜∆)⊗M is effective. It follows that ωD⊗M also defines a rational map with image of dimension at least κ(X) − 1 = dim(W) − 1. By Theorem 1.1.1.e, |KX+ M| is non empty. An argument similar to the one in the proof of Theorem 3.3 shows that |KX + M + D| defines a rational map with image of dimension at leastκ(X). The assertion follows from the inclusion of sheaves
ωX⊗M(D) ;→ ωX⊗6. Step 4. |(6κ(X) + 2)KX| is the stable canonical map.
We may assume that|6KX| = |M6| + F6andM6is free. LetM be defined as in step 3.ϕ6factors as X −→ Y −→ Y := ϕ6(X). Pick D1, ..., Dκ general
sections ofM6, withκ = κ(X). Let Xi := D1∩...∩Di. ThenXκ= D1∩...∩Dκ is the union of deg(Y −→ Y) fibers of X −→ Y which we denote by Ft. We must show that sections of|(6κ(X) + 2)KX| separate these fibers. Let gi : Xi −→ ¯Xi
be the maps induced byX −→ Y. By [Ko3, Theorem 10.19], we have that H1(Xi, ωXi⊗M⊗M6⊗κ−i−1) −→ H1(Xi, ωXi⊗M⊗M6⊗κ−i) is injective for alli ≤ κ − 1. Therefore, the exact sequences
0→ ωXi⊗M⊗M6⊗κ−i−1→ ωXi⊗M⊗M6⊗κ−i → ωXi+1⊗M⊗M6⊗κ−i−1→ 0, are exact on global sections. It follows that
H0(X, ωX⊗M⊗M6⊗κ(X)) −→ H0(Xκ, ωXκ⊗M)
is surjective. The assertion now follows sinceωX⊗M⊗M6⊗κ(X)is a subsheaf of ωX⊗(6κ(X)+2),H0(Xκ, ωXκ⊗M) = ⊕H0(Ft, ωFt⊗M), and by Theorem 1.1.1.e,
H0(Ft, ωFt⊗M) = 0.
References
[AS] U. Angehrn, Y.-T. Siu, Effective freeness and point separation for adjoint bundles, Invent.
Math. 122(2) (1995), 291–308
[CH] J. A. Chen, C. D. Hacon, Characterization of Abelian Varieties. Invent. Math. 143 (2001), 435–447
[GL1] M. Green, R. Lazarsfeld, Deformation theory, generic vanishing theorems„ Invent. Math.
90 (1987), 389–407
[GL2] M. Green, R. Lazarsfeld, Higher obstruction to deforming cohomology groups of line bundles, Jour. Amer. Math. Soc. 4 (1991), 87–103
[EL1] L. Ein, R. Lazarsfeld, Singularities of theta divisors, and birational geometry of irregular varieties, Jour. Amer. Math. Soc. 10(1) (1997), 243–258
[EL2] L. Ein, R. Lazarsfeld, Unpublished personal communication
[EV1] H. Esnault, E. Viehweg, Revˆetements cyclique II, G´eom´etrie Alg`ebrique et Applications II, La R´abida. Herman, Paris (1987), 81–94
[Ka1] Y. Kawamata, Characterization of Abelian Varieties, Comp. Math. 43 (1981), 253–276 [Ka2] Y. Kawamata, Minimal models and the Kodaira dimension of algebraic fiber spaces, Jour.
Reine Angew. Math. 363 (1985), 1–46
[Ko1] J. Koll´ar, Higher direct images of dualizing sheaves I, Ann. Math. 123 (1986), 11–42 [Ko2] J. Koll´ar, Higher direct images of dualizing sheaves II, Ann. Math. 124 (1987), 171-202 [Ko3] J. Koll´ar, Shafarevich Maps and Automorphic Forms, Princeton University Press, 1995 [M] S. Mukai, Duality betweenD(X) and D( ˆX), with application to Picard sheaves, Nagoya
math. J. 81 (1981), 153–175
[S] C. Simpson, Subspaces of moduli spaces of rank one local systems, Ann. Sci. ´Ecole Norm.
Sup. (4) 26(3) (1993), 361–401
