OF MAXIMAL ALBANESE DIMENSION
Jungkai A. Chen, Christopher D. Hacon
Abstract. Let X be a smooth complex projective algebraic variety of maximal Al- banese dimension. We give a characterization of κ(X) in terms of the set V0(X, ωX) := {P ∈ Pic0(X)|h0(X, ωX⊗ P ) 6= 0}. An immediate consequence of this is that the Kodaira dimension κ(X) is invariant under smooth deformations. We then study the pluricanonical maps ϕm: X 99K P(H0(X, mKX)). We prove that if X is of general type, ϕmis generically finite for m ≥ 5 and birational for m ≥ 5dim(X) + 1. More generally, we show that for m ≥ 6 the image of ϕm is of dimension equal to κ(X) and for m ≥ 6κ(X) + 2, ϕmis the stable canonical map.
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 variety, Alb(X) denotes the Albanese variety of X and albX : X −→ Alb(X) the corre- sponding Albanese map. We will assume that X is of maximal Albanese dimension and hence albX is 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 dimen- sion. Then χ(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 to h0(X, ωX⊗P ) >
0 for all P ∈ 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 of V0(X, ωX) := {P ∈ Pic0(X)|h0(X, ωX⊗P ) 6= 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) generates 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:
The first author was partially supported by National Science Council, Taiwan (NSC-89-2115- M-194-016)
1991 Mathematics Subject Classification. Primary 14C20, 14F17.
Typeset by AMS-TEX
1
Corollary 2. Let X be of maximal Albanese dimension. Then the Kodaira dimen- sion of X is invariant under smooth deformations.
We next turn our attention to the study of the linear series |mKX| and the corresponding rational maps ϕm:= ϕ|mKX|. In [Ko3], Koll´ar shows that if X is a smooth proper variety of general type with generically large algebraic fundamental 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. If X is of maximal Albanese and Kodaira dimensions, then ϕ5 is generically finite, and ϕ5dim(X)+1 is birational.
Our proof is based on the techniques of Koll´ar [Ko3]. However, technical diffi- culties imply that when X is not of general type, the bounds are somewhat weaker.
Theorem 4. If X is of maximal Albanese dimension, then ϕ6(X) has dimension equal to κ(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 which ϕmis not birational for m ≥ 3.
Acknowledgment. We would like to thank P. Belkale, H. Clemens, Y. Kawa- mata, 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 numbers C.
(0.2) For D1, D2 Q-divisors on a variety X, we write D1 ≺ D2 if D2− D1 is effective, and D1≡ D2 if D1 and D2 are 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 bac be the largest integer ≤ a and dae be the smallest integer ≥ a. For a Q-divisor D = P
aiDi, let bDc = P
baicDi and dDe =P
daieDi.
(0.5) Let F be a coherent sheaf on X, then hi(X, F) denotes the complex di- mension of Hi(X, F). In particular, the plurigenera h0(X, ωX⊗m) are denoted by Pm(X) and the irregularity h0(X, Ω1X) 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) Let f : X → Y be a generically finite morphism. Rf denotes the ramifica- tion divisor.
(0.8) Let f : 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 a Q-divisor ∆ is klt (Kawamata log terminal) if ∆ has normal crossings support and b∆c = 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. Let X, Y be projective varieties, X smooth and f : X −→ Y a surjective map. Let M be a line bundle on X such that M ≡ f∗L + ∆, where L is a Q-divisor on Y and (X, ∆) is klt. Then
(a) Rjf∗(ωX⊗M ) is torsion free for j ≥ 0.
(b) if L is nef and big, then Hi(Y, Rjf∗(ωX⊗M )) = 0 for i > 0, j ≥ 0.
(c) Rif∗ωX is zero for i > dim(X) − dim(Y ).
(d) R·f∗ωX ∼=P
Rif∗ωX[−i] and hp(X, ωX) =P
hi(Y, Rp−if∗ωX).
(e) If Y 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. Let f : X −→ Y be a surjective morphism from a smooth projec- tive variety to a projective variety. Let Q ∈ Picτ(X). Then
R·f∗(ωX⊗Q) ∼=X
Rif∗(ωX⊗Q)[−i].
In particular hp(X, ωX⊗Q) =P
hi(Y, Rp−if∗(ωX⊗Q)).
Proof. [Ko2, Corollary 3.3]. ¤
1.2 The Iitaka Fibration.
Let X be a smooth complex projective variety of maximal Albanese dimension.
Let a : X −→ A be a generically finite map such that the image of X generates the abelian variety A. A nonsingular representative of the Iitaka fibration of X is a morphism of smooth complex projective varieties f0 : X0 −→ Y such that X0 is birational to X, dim(Y ) = κ(X) and κ(X0y) = 0, where X0y is a generic geometric fiber of f0. Since our questions will be birational in nature, we may always assume that X = X0 and f = f0. 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 subvariety A, denoted Ky, and a|Xy : Xy −→ Ky is birationally equivalent to an ´etale map. Since A can contain at most countably many abelian subvarieties, we may assume that the Kyare all translates of a fixed abelian subvariety K 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 in S, therefore replacing X by an appropriate birational model, there exists a morphism q : 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
y
yp Y −−−−→ Wq −−−−→ S.⊂
1.3 Cohomological Support Loci.
Let a : X → A be a morphism from a smooth projective variety X to an abelian variety A. 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 ) 6= 0}.
In particular, if a = albX : X → Alb(X), then we simply write Vi(X, F) := {P ∈ Pic0(X)|hi(X, F ⊗ P ) 6= 0}.
We say that X has maximal Albanese dimension if dim(albX(X)) = dim(X).
The geometry of the loci Vi(X, ωX) defined above is governed by the following:
Theorem 1.3.1 (Generic vanishing).
(a) Any irreducible component of Vi(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) Let P be a general point of an irreducible component T of Vi(X, ωX). Suppose that v ∈ 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. If v is tangent to T , then the maps in the above sequence vanish.
(c) If X is a variety of maximal Albanese dimension, then
Pic0(X) ⊃ V0(X, ωX) ⊃ V1(X, ωX) ⊃ ... ⊃ Vn(X, ωX) = {OX}.
(d) Every irreducible component of Vi(X, ωX) is a translate of a sub-torus of Pic0(X) by a torsion point.
Proof. See [GL1],[GL2],[EL1] and [S]. ¤
In [EL1], Ein and Lazarsfeld illustrate various examples in which the geometry of X can be recovered from information on the loci Vi(X, ωX). In particular, they prove:
Theorem 1.3.2 [EL2]. If X 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]. Let a : X −→ A be a generically finite map from a smooth projective variety to an abelian variety. Let P 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 di- mension. Fix Q ∈ Picτ(X). Then h0(X, ωX2⊗Q⊗f∗P ) is constant for all torsion P ∈ Pic0(Y ).
2. Kodaira dimension of varieties of maximal Albanese dimension Throughout this paper, we will assume that X is of maximal Albanese dimension and hence of positive Kodaira dimension κ(X) ≥ 0. We will frequently refer to the notation and results of §1. And we will need the following immediate consequence of [M, Theorem 2.2]:
Lemma 2.1. If F 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
³
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 in G. 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 of G.
Lemma 2.2.
(a) V0(X, Pic0(A), ωX) ⊂ G.
(b) For every Qi the loci V0(X, Qi+ Pic0(S), ωX) are nonempty.
(c) If OX 6∈ a∗(Qi+Pic0(S)), then there exists a positive dimensional component of V0(X, Qi+ Pic0(S), ωX).
Proof. (a) If H0(X, ωX⊗a∗P ) 6= 0 for some P ∈ Pic0(A), then for general y ∈ Y , h0(Xy, ωXy⊗a∗P ) = h0( ˜K, OK˜⊗a∗P ) 6= 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 that Xwis a finite union of general fibers of X → 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 that V0(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. Let a : 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 of V0(X, Pic0(A), ωX) generate Pic0(S).
Proof. Assume that translates through the origin of the components of V0 :=
V0(X, Pic0(A), ωX) do not generate Pic0(S). Then there exists an abelian proper subvariety T ⊂ Pic0(S) and a finite subgroup G0 such that V0⊂ T + G0. Consider the induced morphism
π : X −→ T∗=: C, which factors through X −→ Aa −→ S −p → C.
Let Q ∈ Picτ(A) be any torsion element not contained in T + G0. 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⊗ Q⊗π∗P ) = 0 for all 0 ≤ i ≤ n and all P ∈ Pic0(C).
Let V ⊂ 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 all i, j and P ∈ Pic0(C). By Lemma 2.1, Riπ∗(ωX⊗Q) = 0 for all i.
If π : X −→ V is generically finite, then π∗(ωX⊗Q) is clearly non-zero. We may therefore assume that π has positive dimensional generic fibers.
Let p 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.
Let B be the connected component through the origin of the kernel of A −→ C.
The image of Xp in A is contained in a translate of B which we denote by Bp. Since h0(Xp, ωXp⊗Q) = 0 for all but finitely many Q ∈ Pic0(B), it follows by Proposition 1.3.2 that κ(Xp) = 0 and Xp −→ Bp is birationally an ´etale map of abelian varieties.
On the other hand, by the weak addition formula κ(Xp) + dim(V ) ≥ κ(X),
so κ(Xp) > 0 since κ(X) = dim(S) > dim(V ). This contradicts the assumption that T is a proper subvariety of Pic0(S). Therefore, T = Pic0(S). ¤ Corollary 2.4. Let a : 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]. Let p : 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 appropriate L ∈ Pic(E) such that L⊗2= OE(B) where B is the branch locus. Let ˜F −→ F be a degree 2 ´etale map of elliptic curves such that e∗OF˜ = OF⊕ P with P⊗2∼= OF. Define
X := ˜F × C/ < (iF˜× iC) > .
Here iF˜ and iC denote 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 image E and V0(X, Pic0(F × E), ωX) = {OF ×E} ∪ (P + Pic0(E)).
In particular V0(X, Pic0(E), ωX) does not generate Pic0(E).
Corollary 2.5. Let X 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 all t ∈ ∆, At:= Alb(Xt) is an abelian variety of dimension q(X).
Let P ∈ Pic0(X) such that P⊗m= OX, P be a section of Pic0(X /∆) such that P = P0:= P|X0, Pt:= P|Xt ∈ Pic0(At) satisfies Pt⊗m= OXt. Let
X˜t:= Spec¡
⊕m−1i=0 Pt⊗i¢
be the corresponding ´etale cyclic cover of degree m. We have that the quantity
h0( ˜Xt, ωX˜t) =
m−1X
i=0
h0(ωXt⊗Pt⊗i)
is constant. However the functions h0(Xt, ωXt⊗Pt⊗i) are upper semicontinuous in t ∈ ∆, and hence also constant.
Let Tidenote the translates of the components Tiof V0(X, Pic0(A), ωX) through the origin. The subvarieties Ti are determined by their torsion points (cf. Theo- rem 1.3.1.d). In particular, recall that Pic0(Xt) ∼= H1(Xt, O∗Xt)/H1(Xt, Z) and the subvarieties Titare determined by the corresponding vector subspaces of H1(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 Ttiof Pic0(At), which are smooth deformations of T0isuch that V0(Xt, Pic0(At), ωXt) = ∪Tti. More- over, for any set of indices I, the quantities
dim Ã\
i∈I
Tit
!
and dim ÃX
i∈I
Tit
!
are constant. In particular the quantity dim¡
V0(Xt, Pic0(At), ωXt)¢
= κ(Xt) + q(Xt) − dim(Xt)
is constant and hence κ(Xt) is also constant. ¤
3. Pluricanonical maps of varieties of maximal Albanese dimension We will keep the notation of the preceding sections. In particular X will be a smooth projective variety with maximal Albanese dimension and a : X −→ A will denote a generically finite morphism to an abelian variety.
Lemma 3.1. Let E be an a-exceptional effective divisor on X. If OX(E)⊗P is effective for some P ∈ 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 on A and let H = a∗HA be the corresponding nef and big divisor on X. Choosing HA appropriately, we may assume that H is a
smooth divisor in X. We may also assume that h0(X, OX(E − H)⊗P ) = 0 for all P ∈ 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 injection H0(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.
Since H 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 6= OX and h0(X, OX) = h0(H, OH) = 1. Clearly, for a general choice of H, we have that H is of maximal Albanese dimension, and E|H is an a|H-exceptional divisor. Repeating the above procedure, by successively intersecting appropriate divisors pulled back from A, one obtains a curve C ⊂ X such that
i) If h0(X, OX(E)⊗P ) > 0 then h0(C, OC(E)⊗P ) = h0(C, OC⊗P ) > 0, ii) If h0(C, OC⊗P ) > 0 then P = OX.
It follows that if h0(X, OX(E)⊗P ) > 0, then P = OX. ¤ Lemma 3.2. Let D 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 of X −→ 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, since X is of maximal Albanese dimension, we may assume that KX is effective.
Theorem 3.3. If X is of general type, i.e. κ(X) = dim(X). Then for any integer s ≥ 3, Fs is a-exceptional. |5KX| defines a generically finite rational map and
|(5dim(X) + 1)KX| defines a birational map.
Proof. For any fixed divisor D on X, then there exists an m0such that |mKX−D| 6=
∅ for all m ≥ m0. Let R be an irreducible divisor of X which is not a-exceptional.
By Lemma 3.2, R 6⊂ Bs|mKX− D| for infinitely many m À 0.
Step 1. Fs is exceptional for all s ≥ 3.
It is easy to see that Fs being a-exceptional is a condition which is indepen- dent 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 by Ri. Fix positive integers misuch that Ri 6⊂ Bs|miKX|, then Ri 6⊂ Bs|λmiKX| for any integer λ > 0. It follows that for m0=Q
mi, Fm0 is a-exceptional.
Next, fix an ample divisor H on A, then a∗H is nef and big on X. Let KX be a canonical divisor. Let r be the multiplicity of KX at R. We distinguish two cases.
Case 1. r = 1.
By Lemma 3.2, there exists a positive integer t 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 replacing X with an appropriate birational model, we may assume that R is smooth and B + R has normal crossing support and B does not contain R. Define
M := OX
µ
(s − 2)KX− bB tc
¶
≡ µs − 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 divisor R 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 ) 6= 0. Therefore, there is a divisor in |KX+ M + R|
not containing R. This gives a divisor in |sKX| = |KX+ M + R + (KX− R) + bBtc|
not containing R.
Case 2. r ≥ 2.
There exists positive integers t, m0 such that R 6⊂ Bs|tKX− a∗H|, and R 6⊂
Bs|m0KX|. For any integer s ≥ 2 let K0:= (s − 1)KX− R. Consider the following linear series
|mK0− m0a∗H| = |¡
(s − 1)m −m
r − m0t¢
KX+m
r(KX− rR) + m0(tKX− a∗H)|.
It follows that for m divisible by m0r, R is not in the base locus of |mK0− m0a∗H|.
Choose a general B ∈ |mK0− m0a∗H| and define M := OX
µ
K0− bB mc
¶
≡ m0
m a∗H + {B m}.
An argument similar to the one in the previous case again shows that R 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 in A. If ∆ is not of general type, then there exists an ample line bundle H, a semipositive line bundle L and a positive integer s such that ∆ · Ls· Hn−s−1= 0 and Ls· 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 ² > 0 such that KX− ²(a∗H) is an effective Q-divisor. Therefore,
KX· (a∗H)n−s−1· (a∗L)s≥ ²(a∗H)n−s· (a∗L)s> 0.
Since KX− D ≡ F3is a-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 ∆0 of ∆, the linear series |K∆0| defines a generically finite rational map. Therefore, we may assume that |KD| defines a generically finite rational map. Let B ∈ |mKX− a∗H|, and M := OX(KX − bmBc) ≡ a∗mH + {mB}. We may assume that B has normal crossings support and does not contain D. Consider the exact sequence of sheaves
0 −→ ωX⊗M −→ ωX⊗M (D) −→ ωD⊗M −→ 0, which is also exact on global sections.
Claim. There exist m, B such that M is effective.
Fix m1 such that |m1KX− a∗p∗HS| is non empty. Let B1∈ |m1KX− a∗p∗HS|.
Let m := m1+ m2 and B := B1+ m2KX ∈ |mKX− a∗p∗HS|. Let Γ be any prime divisor, and b1, k be the multiplicities of B1, KX along Γ. For m2 À 0, we have that
bb1+ km2
m c = b b1
m1+ m2
+ k m2
m1+ m2
c ≤ k.
Since there are only finitely many components of B + KX, for m2 À 0, we have bmBc ≺ KX, and the claim follows.
It is easy to see that M |D is also effective. Therefore, |KX+ D + M | restricted to D defines a generically finite rational map. If |KX+ D + M | does not define a generically finite rational map, then the closure of the image of X which we denote by Y ⊂ P = P(H0(X, ωX⊗M (D))), must be dominated by D. This is however impossible since by Theorem 1.1.1.e h0(X, ωX⊗M ) > 0, and hence |KX+ D + M | contains non-trivial sections vanishing on D, i.e. there is a hyperplane section of P 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 respec- tively. For any P ∈ Pic0(X), let
|mKX+ P | = |Mm,P| + Fm,P, where Bs|Mm,P| contains no divisor.
Step 1. Let S −→ S0 be any surjective map of abelian varieties such that y :=
dim(S) > y0 := dim(S0) and s0: X → S0 be the induced map. Then, for any divisor D ∈ |M4|, D is not s0-vertical.
By Lemma 1.3.4, the linear series |2KX+ P | is nonempty for all P ∈ Pic0(S).
Let HS0 be the pull back to X of a sufficiently ample divisor on S0. Let Z be an irreducible component of HSy00, i.e. a general fiber of s0 : X → S0. 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. dF24e ≺ F2,P for general P ∈ Pic0(S).
Let V ⊂ Pic0(S) be an open set such that h0(X, ω⊗2X ⊗P ) is constant. Let R be any component of F4 of multiplicity r, then dr2eR ≺ F2,P for all P ∈ U ⊂ Pic0(S) where U is a Zariski dense subset such that U ∪ −U = Pic0(S). Since the condition dr2eR ≺ Bs|2KX+ P | is Zariski closed in V , it follows that OX(2KX− dr2eR)⊗P is effective for all P ∈ V ⊂ Pic0(S). Since this holds for all components of F4, the claim follows.
Claim 2. ker(Pic0(S) → Pic0(Z)) is a proper closed subvariety of Pic0(S).
Let V be the image of Z in S. V is a translate of ker(S → S0). There exists infinitely many P ∈ Pic0(S) such that P |V 6= OV. Hence pulling back to Z, we have P |Z 6= OZ.
Claim 3. (2KX− dF24e)|Z is not a|Z-exceptional.
By Claim 1, (2KX − dF24e + P )|Z is effective and for general P ∈ Pic0(S).
By Claim 2, we may assume that P |Z 6= OZ for general P ∈ Pic0(S). Since a|Z : Z −→ a(Z) is generically finite, the claim follows by Lemma 3.1.
Claim 3 implies that 2KX− dF24e is not s0-vertical (otherwise (2KX− dF24e)|Z
= OZ). From the inclusion of linear series
|2KX− dF4
2 e| × |2KX− dF4
2 e|−−−−−−−−→ |4K+2dF42e−F4 X− F4| it follows that 4KX− F4= M4 is also not s0-vertical.
Step 2. Let D 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 −→ S0. 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. Replacing X 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.
Replacing X by an appropriate birational model, we may assume that B has normal crossings support. Define
M := OX
µ
KX− bB mc
¶
≡ a∗p∗HS
m + {B m}.
As in the proof of the previous theorem, we may assume that M , M |Dare effective, and B 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 of ωD⊗M lift to sections of ωX⊗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 least κ(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| + F6 and M6is free. Let M be defined as in step 3. ϕ6 factors as X −→ Y −→ Y0 := ϕ6(X). Pick D1, ..., Dκ general sections of M6, with κ = κ(X). Let Xi := D1∩ ... ∩ Di. Then Xκ = D1∩ ... ∩ Dκ is the union of deg(Y −→ Y0) 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 by X −→ Y0. 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 all i ≤ 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 since ωX⊗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 ) 6= 0. ¤
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Jungkai Alfred Chen, Department of Mathematics, National Chung Cheng Uni- versity, Ming Hsiung, Chia Yi, 621, Taiwan
E-mail address: jkchen@math.ccu.edu.tw
Christopher Derek Hacon, Department of Mathematics, University of Utah, 155 South 1400 East JWB 233, Salt Lake City, UT 84112, USA
E-mail address: chhacon@math.utah.edu
