OF MAXIMAL ALBANESE DIMENSION
Jungkai A. Chen, Christopher D. Hacon
Abstract. We study algebraic fiber spaces f : X −→ Y where Y is of maximal Albanese dimension. In particular we give an effective version a theorem of Kawa- mata: If Pm(X) = 1 for some m ≥ 2, then the Albanese map of X is surjective.
Combining this with [CH] it follows that X is birational to an abelian variety if and only if P2(X) = 1 and q(X) = dim(X).
Introduction
In this paper we combine the generic vanishing theorems of [GL1], [GL2], the techniques of [EL1] and the results of [Ko1] and [Ko2] to answer a number of natural questions concerning the geometry and birational invariants of irregular complex algebraic varieties.
Throughout the paper, we are motivated by the following:
Conjecture K (Ueno). Let X be a nonsingular projective algebraic variety such that κ(X) = 0 and let albX : X −→ Alb(X) be the Albanese map. Then
(1) albX is surjective and has connected fibers, i.e. albX is an algebraic fiber space.
(2) Let F be a general fiber of albX. Then κ(F ) = 0.
(3) There is an ´etale covering B −→ Alb(X) such that X×Alb(X)B is birationally equivalent to F × B over B.
The main evidence towards this conjecture is given by Theorem (Kawamata).
(1) Conjecture K (1) is true [Ka1, Theorem 1].
(2) If F has a good minimal model (i.e. a model with only canonical singularities, and such that mK ∼ 0 for some positive integer m). Then Conjecture K is true [Ka2].
We remark here that in the proofs of our statements, we will make use [Ka1, Theorem 1] in an essential way. As a consequence of [Ka1, Theorem 1], one sees that:
The first author was partially supported by National Science Council, Taiwan (NSC-89-2115- M-194-029 )
1991 Mathematics Subject Classification. Primary 14J10, 14F17; Secondary 14D99.
Typeset by AMS-TEX 1
Corollary (Kawamata, [Ka1]). If κ(X) = 0 then q(X) ≤ dim(X). Moreover, if q(X) = dim(X) then albX: X −→ Alb(X) is a birational morphism.
There has also been considerable interest in effective versions of this result.
Koll´ar has shown that:
Theorem (Koll´ar, [Ko4]).
(1) If P3(X) = 1 then the Albanese map is surjective.
(2) Moreover, if P4(X) = 1 and q(X) = dim(X), then albX : X −→ Alb(X) is a birational morphism.
Conjecture (Koll´ar, [Ko3, 18.13]). X is birational to an abelian variety if and only if q(X) = dim(X) and Pm(X) = 1 for some m ≥ 2.
Our first result is:
Theorem 1. If for some integer m ≥ 2 the m-th plurigenera Pm(X) equals 1, then the Albanese map of X is surjective.
Combining Theorem 1 with [CH] we obtain:
Corollary 2. Koll´ar’s conjecture above [Ko3, 18.13] is true.
We are also able to generalize the corollary of Kawamata’s Theorem above:
Corollary 3. If Pm(X) = P2m(X) = 1 for some m ≥ 2, then q(X) ≤ dim(X) − κ(X). If q(X) = dim(X) − κ(X), then the general fiber of the Iitaka fibration of X is birationally equivalent to a fixed ´etale cover ˜A of A := Alb(X).
Next, we study algebraic fiber spaces f : X −→ Y where X, Y are smooth projective varieties and Y is of maximal Albanese dimension. The generic vanishing theorems of Green and Lazarsfeld are a very effective technique in the study of irregular varieties. In §2, we prove a more general version of Theorem 2.1.3 below, which applies to irregular varieties not necessarily of maximal Albanese dimension.
Using this result we are able to show:
Theorem 4. If κ(X) = 0, let a := albX: X → Alb(X) be the Albanese map, then (1) Either a∗ωX is a zero sheaf or a torsion line bundle.
(2) P1(FX/Alb(X)) ≤ 1.
(3) If P1(X) = 1, then a∗ωX= OAlb(X).
(4) There is a generically finite cover ˜X −→ X with κ(X) = κ( ˜X), such that albX ∗˜ (ωX˜) = OAlbX˜.
We remark that the statement (2) can be regarded as an effective version of Conjecture K (2). Using a well known result of Fujita [Mo, (4.1)], for any X with κ(X) = 0, there exists a generically finite cover ˜X −→ X such that κ( ˜X) = 0 and P1( ˜X) = 1 as in (3).
We explain the significance of (4). For any algebraic fiber space f : X −→ Y the rank at a generic point of the sheaves f∗(ω⊗mX/Y) corresponds to the plurigenera Pm(FX/Y) of a generic geometric fiber. It is expected that the positivity of the sheaves f∗(ωX/Y⊗m ) measures the birational variation of the geometric fibers. Our methods unluckily cannot be applied to the case m ≥ 2 because of the lack of a suitable geometric interpretation of the sheaves f∗(ω⊗mX/Y).
If F = FX/Alb(X)has a good minimal model, then by Kawamata’s result, there is an ´etale covering B −→ Alb(X) such that X ×Alb(X)B is birationally equiva- lent to F × B over B. Since κ(F ) = 0, by Fujita’s lemma again, there exists a generically finite cover ˜F such that κ( ˜F ) = 0 and P1( ˜F ) = 1. Since F has a good minimal model, we may assume in fact that (for an appropriate birational model with canonical singularities) KF˜= 0. Let ˜X = ˜F × B. Then, for all m ≥ 1
πB ∗(ω⊗mX˜ ) = OB,
where πB: ˜X −→ B is the projection to the second factor.
The following lemma is useful:
Lemma 5. Let f : X −→ Y be an algebraic fiber space, Y of maximal Albanese dimension. If κ(X) = 0, then Y is birational to an abelian variety.
Theorem 6. Let f : X −→ Y be an algebraic fiber space, Y of maximal Albanese dimension.
(1) If κ(X) = κ(Y ), then P1(FX/Y) ≤ 1.
(2) If P1(FX/Y) ≥ 1, then κ(X) ≥ κ(Y ).
The above statements are closely connected to the following well known conjec- ture
Conjecture Cn,m. Let f : X −→ Y be an algebraic fiber space, dim(X) = n and dim(Y ) = m. Then
κ(X) ≥ κ(Y ) + κ(FX/Y).
This conjecture is true when FX/Y has a good minimal model [Ka2]. If one could generalize the generic vanishing theorem to the sheaves of the form ωY⊗f∗ωX/Ym for m ≥ 2, then using the same techniques, the Cn,m conjecture would follow for all algebraic fiber spaces f : X −→ Y , with Y of maximal Albanese dimension.
Finally, we prove a generalization of [Ka1, Theorem 15]:
Theorem 7. Suppose X is a variety with κ(X) = 0 and dim(X) ≤ 2q(X). If P1(X) > 0, then h0(X, Ωn−qX ) > h0(Alb, Ωn−qAlb). In particular, if dim(X) = q(X) + 1, then P1(X) = 0.
We then illustrate how one can use this result to recover the Conjecture K, in the case dim(X) = q(X) + 1 [Ka1, Theorem 15].
Acknowledgment. We are in debt to P. Belkale, A. Bertram, L. Ein, J. Koll´ar and R. Lazarsfeld for valuable coversations.
0. Conventions and Notations
(0.1) Throughout this paper, we work over the field of complex numbers C. X will denote a smooth projective algebraic variety.
(0.2) A general point of X denotes a point in the complement of a countable union of proper Zariski closed subsets of X.
(0.3) Let f : X −→ Y be an algebraic fiber space, i.e. a surjective morphism with connected fibers. Then FX/Y will denote the fiber over a general geometric point of Y .
(0.4) For D1, D2 Q-divisors on a variety X, we write D1≡ D2 if D1and D2are numerically equivalent.
(0.5) 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. 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.
(0.6) 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).
1. Preliminaries 1.1 The Iitaka Fibration.
Let X be a smooth complex projective variety with κ(X) > 0. Then a nonsin- gular representative of the Iitaka fibering of X is a morphism of smooth complex projective varieties f0 : X0−→ V such that X0 is birational to X, dim(V ) = κ(X) and κ(X0v) = 0, where X0vis a generic geometric fiber of f0. Since our questions will be birational in nature, we may always assume that X = X0. Let A := Alb(X) and let Z denote the image of X in A. Let Z0 denote an appropriate desingularization of Z, we may assume that X −→ Z factor s through Z0. By [Ka1] the images a(Xv) = Kv are translates of abelian subvarieties of A. Since A contains at most countably many abelian subvarieties, we may assume that Kv are all translates of a fixed abelian subvariety K ⊂ A. Let S := A/K and W denote the image of Z in S. Let W0 be an appropriate desingularization of W . We may assume that the induced morphism π : X −→ W factors through a morphism π0 : X −→ W0. Consider now a birational model of the Iitaka fibration f : X −→ V .
Claim. We may assume that the map π0 factors through f and a morphism q0 : V −→ W0.
To see this, note that by construction, there is an open dense subset U of V and a map U −→ S. However by a standard argument this must complete to a rational map V −→ S (see e.g. [Ka1, Lemma 14]). Since the problem is birational, we may assume that V −→ S is infact a morphism that factors through W0. The above maps fit in the following comutative diagram
X −−−−→ Z0 −−−−→ Z −−−−→ A⊂
f
y
yp V −−−−→ Wq0 0 −−−−→ W −−−−→ S.⊂
1.2 Fourier Mukai Transforms.
Let A be an abelian variety, and denote the corresponding dual abelian variety by ˆA. Let P be the normalized Poincar´e bundle on A × ˆA. For any point y ∈ ˆA
let Pydenote the associated topological trivial line bundle. Define the functor ˆS of OA-modules into the category of OAˆ-modules by
S(M ) = πˆ A,∗ˆ (P⊗π∗AM ).
The derived functor R ˆS of ˆS then induces an equivalence of categories between the two derived categories D(A) and D( ˆA). In fact, by [M]: There are isomorphisms of functors:
RS ◦ R ˆS ∼= (−1A)∗[−g]
and
R ˆS ◦ RS ∼= (−1Aˆ)∗[−g], where [−g] denotes ”shift the complex g places to the right”.
The index theorem (I.T.) is said to hold for a coherent sheaf F on A if there exists an integer i(F) such that for all j 6= i(F), Hj(A, F⊗P ) = 0 for all P ∈ Pic0(A).
The weak index theorem (W.I.T.) holds for a coherent sheaf F if there exists an integer which we again denote by i(F) such that for all j 6= i(F), RjS(F) = 0. Itˆ is easily seen that the I.T. implies the W.I.T. We will denote the coherent sheaf Ri(F)S(F) on ˆˆ A by ˆF.
It follows, for example that given any coherent sheaf F, if hi(A, F⊗P ) = 0 for all i and all P ∈ Pic0(A) then F = 0. Consequently, if F −→ G is an injection of sheaves which induces isomorphisms in cohomology
Hi(A, F⊗P )−→ H∼= i(A, G⊗P ), for all i and all P ∈ Pic0(A), then F ∼= G.
Proposition 1.2.1. If F is a coherent sheaf on A such that for all P ∈ Pic0(A) we have h0(A, F⊗P ) = 1 and hi(A, F⊗P ) = 0 for all i > 0. Then F is supported on an abelian subvariety of A.
Proof. F satisfies the WIT and ˆF is a line bundle M on Pic0(A) = ˆA such that M has index i(M ) = dim(A), and ˆM = (−1A)∗F. Any line bundle with i(M ) = dim(A), is negative semidefinite. It is well known that there exists a morphism of abelian varieties b : Pic0(A) −→ A0 such that M = b∗M0 for some negative definite line bundle M0 on A0. It follows that ˆM and hence F are supported on the image
of b∗: ˆA0−→ A. ¤
2. Relative Generic Vanishing Theorems
We start by recalling some facts on cohomological support loci. Let π : 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, A, F) := {P ∈ Pic0(A)|hi(X, F ⊗ π∗P ) 6= 0}.
In particular, if π = 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 2.1 (Generic Vanishing Theorem).
(1) 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))).
(2) Let P ∈ T 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.
(3) If X is a variety of maximal Albanese dimension, then
Pic0(X) ⊃ V0(X, ωX) ⊃ V1(X, ωX) ⊃ ... ⊃ Vn(X, ωX) = {OX}.
(4) Every irreducible component of Vi(X, ωX) is a translate of a sub-torus of Pic0(X) by a torsion point.
Proof. For (1),(2), see [GL1],[GL2]. For (3), see [EL1] and for (4), see [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 this section we prove a relative version of Theorem 2.1. Let π : X −→ Y be a surjective map of smooth projective varieties. Assume that Y has maximal Albanese dimension. Let n and m be the dimension of X and Y respectively. We wish to study the geometry of the loci Vi(Y, Rjπ∗ωX). By a result of Simpson [S], the irreducible components of these loci are torsion translates of subtori of Pic0(Y ).
Therefore, their geometry is completely determined by their torsion points. Recall [Ko2, Corollary 3.3] that for any torsion Q ∈ Pic0(X), one has
R·π∗(ωX⊗Q) ∼=X
Rif∗(ωX⊗Q)[−i].
In particular hp(X, ωX⊗Q) =P
hi(Y, Rp−if∗(ωX⊗Q)).
Proposition 2.2. For π : X −→ Y as above
Pic0(Y ) ⊃ V0(Y, Rjπ∗ωX) ⊃ V1(Y, Rjπ∗ωX) ⊃ ... ⊃ Vm(Y, Rjπ∗ωX).
Proof. We will prove that hi(Y, Rjπ∗ωX⊗P ) > 0 implies hi−1(Y, Rjπ∗ωX⊗P ) > 0 for all i > 0. By Simpson’s result mentioned above, it suffices to prove the assertion for torsion elements P ∈ Pic0(Y ). Fix a very ample line bundle L on Y and for 1 ≤ α ≤ m, let Dα be sufficiently general divisors in |L|. Denote the preimage by D˜α = π−1(Dα). Consider the sequence of varieties defined by X0 = X, Xα+1 = Xα∩ ˜Dα+1. Let Yα := π(Xα). We may assume that the Dα are chosen so that Xα, Yαare smooth for 0 ≤ α ≤ m. The corresponding morphisms π|Xα : Xα−→ Yα
will also be denoted simply by π.
By [Ko 2, Corollary 3.3], for any torsion P ∈ Pic0(Y ), we may identify Hi(Yα, Rj π∗ωXα⊗P ) as a subgroup of Hi+j(Xα, ωXα⊗π∗P ).
Claim. There exists a surjective map
H0(Yi, Rjπ∗ωXi⊗P ) −→ Hi(Y, Rjπ∗ωX⊗P ).
Proof. We will prove the assertion for P = OY. In the general case the proof proceeds analogously. Consider the exact sequence of sheaves
0 −→ ωXt −→ ωXt( ˜Dt+1) −→ ωXt+1−→ 0 this induces sequences of sheaves
0 −→ Rjπ∗ωXt −→ Rjπ∗ωXt( ˜Dt+1) −→ Rjπ∗ωXt+1 −→ 0.
By step 4 of the proof of [Ko1, Theorem 2.1 iii], for appropriately chosen Dα, the above sequence is exact and equivalent to
Rjπ∗ωXt⊗¡
0 −→ OYt−→ OYt(Dt+1) −→ OYt+1(Dt+1) −→ 0¢ .
By [Ko 1], Hk(Yt, Rjπ∗ωXt(Dt+1)) = 0 for all k > 0. We therefore have an exact sequence
0 −→ H0(Yt, Rjπ∗ωXt) −→ H0(Yt, Rjπ∗ωXt(Dt+1)) −→
H0(Yt+1, Rjπ∗ωXt+1) −→ H1(Yt, Rjπ∗ωXt) −→ 0, and isomorphisms Hk(Yt+1, Rjπ∗ωXt+1) ∼= Hk+1(Yt, Rjπ∗ωXt) for k = 1, ..., m − t − 1. The claim now follows.
Moreover, the map
H0(Yi, Rjπ∗ωXi) −→ Hi(Y, Rjπ∗ωX) −→ Hi+j(X, ωX)
is induced by the inclusion H0(Yi, Rjπ∗ωXi) −→ Hj(Xi, ωXi) and by the cobound- ary maps
δ : Hj(Xi, ωXi) δ
j
−−−→ Hi−1 j+1(Xi−1, ωXi−1) δ
j+1
−−−→ ...i−2 δ j+i
−−−→ H0 i+j(X, ωX).
The map δ is dual to the map
δ∗: Hn−i−j(X, OX) −→ Hn−i−j(X1, OX1) −→ ... −→ Hn−i−j(Xi, OXi) induced by successive restrictions. In turn, this map is complex conjugate to the map
δ¯∗: H0(X, Ωn−i−jX ) −→ H0(X1, Ωn−i−jX1 ) −→ ... −→ H0(Xi, Ωn−i−jXi ) induced by successive restrictions. Let V be the subspace of H0(X, Ωn−i−jX ) corre- sponding to the complex conjugate of Hi(Y, Rjπ∗ωX)∗. It follows from the above claim that ¯δ∗ induces an injection V ,→ H0(Xi, Ωn−i−jX
i ).
Fix a general smooth point p of π(Xi). Let a : Y −→ A be the Albanese map.
For a general point p ∈ Yi. One may assume that z1, ..., zmare local holomorphic coordinates for a(Y ) at a(p), where z1, ..., zm−i are local holomorphic coordinates for a(Yi) at a(p). We may assume furthermore that the pull backs of the zi, which we denote by xi, give rise to local holomorphic coordinates for Y and Yi at p. We may assume that dxl(p) = 0 in Ω1Yi⊗C(p) for all m − i + 1 ≤ l ≤ m (i.e. dxl is conormal to Yi at p). Fix ˜p ∈ π−1(p). Let ω ∈ H0(A, Ω1A) such that ω(p) = dzm. Let y1, ..., yn be local holomorphic coordinates on X centered at ˜p ∈ π−1(p) such that yi = xi◦ π for all 1 ≤ i ≤ m. Let v be any element in V ⊂ H0(X, Ωn−j−iX ).
Then,
v(˜p) =X aKdyK
where the sum runs over all multiindices of length n − i − j ie K = (k1, ..., kn−i−j) and dyK = dyk1∧ ... ∧ dykn−i−j. Since V injects in H0(Xi, Ωn−i−jXi ), for any fixed v ∈ V , by genericity of the choice of the point p, we may assume that v(˜p) 6= 0 in Ωn−i−jXi ⊗C(˜p). In particular, there exists a multiindex ¯K such that aK¯(˜p) 6= 0, and for all 1 ≤ l ≤ n − i − j, ¯kl doesn’t belong to {m − i + 1, ..., m}. Therefore v ∪ ω ∈ H0(X, Ωn−j+1X ) is non-zero since the coefficient of dyK∪{m}¯ is non-zero.
Composing again with complex conjugation and Serre duality we see that there is a non-zero element in Hi−1(Y, Rjπ∗ωX) ⊂ Hj+i−1(X, ωX). ¤ The following notation will be convenient. For any line bundle L on Y and v ∈ H1(Y, OY), we will denote by KiL,Y,v and BL,Y,vi+1 the kernel and the image of the map
Hi(Y, L)−→ H∧v i+1(Y, L).
Let HiL,Y,v := KiL,Y,v/BL,Y,vi . The subscripts Y and v will be dropped when no confusion is likely.
Let τ : H1(X, OX) −→ Pic0(X) be the map induced by the exponential sheaf sequence. Let ∆ ⊂ SpecC[t] be a neigborhood of 0. For P ∈ Pic0(X) and v ∈ H1(X, OX), let P a line bundle on X × ∆ such that P|X×t ∼= P ⊗τ (tv). Let pX: X × ∆ −→ X and p∆: X × ∆ −→ ∆ be the projections to the first and second factors. We will need the following result due to Green and Lazarsfeld.
Theorem 2.3 [GL2]. There is a neighborhood of 0 for which Rip∆∗(p∗XωX⊗P) ∼=¡
KiωX⊗P⊗O∆/m
¢⊕¡
HiωX⊗P⊗O∆
¢.
Proof. This is a generalization of [GL2, Theorem 3.2], which follows from the com- ments preceeding [GL2, Theorem 6.1]. See also [ClH]. ¤ Corollary 2.4. Let φ ∈ KiωX⊗P ⊂ Hi(X, ωX⊗P ). Let Φ ∈ Rip∆∗(p∗XωX⊗P) be a section such that (Φ)0 = φ. Assume that Φ|(∆−0) = 0. Then φ = γ ∪ v for an appropriate γ ∈ Hi−1(X, ωX⊗P ).
Proof. Since Φ|(∆−0) = 0, there exists an integer k ≥ 0 such that tkΦ = 0 ∈ Rip∆∗(p∗XωX⊗P). Therefore,
φtk+ φ1tk+1+ φ2tk+2+ . . . = 0 ∈¡
KωiX⊗P⊗O∆/m¢
⊕¡
HiωX⊗P⊗O∆
¢.
If k = 0 then φ = 0 ∈ KiωX⊗P, and there is nothing to show. If k ≥ 1 then φ = 0 ∈ HiωX⊗P, and hence φ = γ ∪ v for an appropriate γ ∈ Hi−1(X, ωX⊗P ). ¤ Proposition 2.5. Let π : X −→ Y be an algebraic fiber space, Y of maximal Albanese dimension. If Hi(Y, π∗ωX⊗P ) = 0 for all i and all P in a punctured neighborhood of a torsion P0∈ Pic0(Y ), then, for any v ∈ H1(Y, OY), the complex
Hi−1(Y, π∗ωX⊗P0)−→ H∪v i(Y, π∗ωX⊗P0)−→ H∪v i+1(Y, π∗ωX⊗P0) (∗) is exact.
Proof. We will use the notation of the proof of Proposition 2.2. Let us first consider the folowing diagram, which is commutative for each square.
p∆∗(p∗XωXi⊗P) −−−−→ Rδ∆ ip∆∗(p∗XωX⊗P)
rest=0
y rest=0
y
H0(Xi, ωXi⊗π∗P0) −−−−→ HδX i(X, ωX⊗π∗P0) −−−−→ H∪π∗v i+1(X, ωX⊗π∗P0)
k
x
∪
x
∪
x
H0(Yi, π∗ωXi⊗P0) −−−−→ HδY i(Y, π∗ωXi⊗P0) −−−−→ H∪v i+1(Y, π∗ωXi⊗P0) Let f ∈ Hi(Y, π∗ωX⊗P0), such that f ∪v = 0 ∈ Hi+1(Y, π∗ωX⊗P0). By abuse of notation, we will also denote by f the corresponding element in Hi(X, ωX⊗ π∗P0).
Since δX is surjective, we take ˜f ∈ H0(Xi, ωXi⊗π∗P0) a lift of f . We remark that also ˜f ∪ π∗v = 0. Let ˜F ∈ p∆∗(p∗XωXi⊗P) be the section corresponding to f + 0t + 0t˜ 2+ ... and F ∈ Rip∆∗(p∗XωX⊗P) be the corresponding section under the coboundary map. One has F |t=0= f ∈ Hi(X, ωX⊗π∗P0)
By the claim in the proof of 2.2, the condition Hi(Y, π∗ωX⊗P ) = 0 is equivalent to the vanishing of the following map
H0(Yi, π∗ωXi⊗P ) ∼= H0(Xi, ωXi⊗π∗P ) −→ Hi(X, ωX⊗π∗P ) (∗∗).
For all P in a punctured neighborhood of P0 the map (∗∗) vanishes. Therefore F |(∆−0)= 0. By Corollary 2.4, f = γ ∪π∗v for an appropriate γ ∈ Hi−1(X, ωX⊗π∗ P0).
Following [Ko2], write γ =P
gj where gj ∈ Hi−1−j(Y, Rjπ∗ωX⊗P0), then gj∪ v ∈ Hi−j(Y, Rjπ∗ ωX⊗P0∗), so f = g0∪ v, and therefore (∗) is exact. ¤
3. Proof of Theorems
Proof of Theorem 1. By [Ka1, Theorem 1], we may assume that κ(X) > 0. Let f : X −→ V be a birational model of the Iitaka fibration, π : X −→ W and S be as in §1.1.
Assume that m = 2 (the proof proceeds analogously for any m ≥ 2). Fix H an ample divisor on S. For a fixed r À 0, after replacing X by an appropriate birational model, we may assume that
|rKX− π∗H| = |Mr| + Fr
where |Mr| is non-empty and free, and Fr has simple normal crossings. Let B be a general divisor of the linear series |rKX − π∗H|. We may assume again that B has normal crossing support. Define L := OX(KX − bBrc). We have that L ≡ (π∗H/r) + {Br} is numerically equivalent to the sum of the pull back of a nef and big Q-divisor on W and a klt Q-divisor on X.
As in the proof of [CH, Lemma 2.1], it is possibile to arrange that |2KX| =
|KX+ L| + bBrc. In particular h0(X, ωX⊗L) = 1.
Since π : X −→ W is a surjective map, by [Ko3, Corollary 10.15], π∗(ωX⊗L) is a torsion free coherent sheaf on W , and hi(W, π∗(ωX⊗L)⊗P ) = 0 for all i > 0 and P ∈ Pic0(S). It follows that
h0(W, π∗(ωX⊗L)⊗P ) = χ(W, π∗(ωX⊗L)⊗P )
= χ(W, π∗(ωX⊗L)) = h0(W, π∗(ωX⊗L)) = 1 for all P ∈ Pic0(S).
By Proposition 1.2.1, π∗(ωX⊗L) is supported on an abelian subvariety S0 of S.
Since π∗(ωX⊗L) is torsion free and the image of X generates S, we see that S0= S,
and hence X −→ A is surjective. ¤
Proof of Corollary 3. If κ(X) = 0, this is a result of Kawamata. We may therefore assume that κ(X) > 0. By Theorem 1, X −→ Alb(X) is surjective. By [CH, Lemma 2.1] h0(X, ωX⊗m⊗ π∗P ) > 0 for all P ∈ Pic0(S). From the map of linear series
|mKX+ Q| × |mKX− Q| −→ |2mKX|
it follows that P2m(X) ≥ dim(S) + 1, and therefore dim(S) = 0. The general geometric fiber Xv of the Iitaka fibration has dimension dim(X) − κ(X). Since dim(S) = 0, by construction, it follows that albX(Xv) = Alb(X) and hence dim(X) − κ(X) ≥ q(X).
Suppose now that dim(X) − κ(X) = q(X). Then the map Xv −→ Alb(X) is birationally ´etale, and we may assume that Xv is birational to ˜A for a fixed abelian
variety ˜A. ¤
Remark. Under the same hypothesis one can show that in fact q(V ) = 0. In particular if Pm(X) = P2m(X) = 1 for some m ≥ 2 and q(X) = dim(X) − 1, κ(X) = 1, then V = P1.
Lemma 3.1. If κ(X) = 0, then V0(X, ωX) contains at most one point.
Proof. Assume that there are 2 distinct points P, Q in V0(X, ωX). By Theorem 2.1, we may assume that P, Q ∈ Pic0(X) are torsion elements. Pick any m > 0 such that P⊗m = Q⊗m = OX, then if P 6= Q, we have h0(X, ωX⊗m) > 1 which is
impossible. Therefore P = Q. ¤
Lemma 3.2. Let a : X −→ A be an algebraic fiber space from a smooth projective variety X to an abelian variety A. Then
K := ker¡
Pic0(A) −→ Pic0(X)¢
= {OA}.
Proof. The map a factors as X −−−→ Alb(X) −albX → A. The map Alb(X) −→ A is surjective, therefore K consists of a discrete subgroup of Pic0(A). Define A0 = (Pic0(A)/K)∗. Then there are maps
X −→ Alb(X) −→ A0 −→ A.
Since X −→ A has connected fibers, then A = A0. ¤ Proof of Theorem 4. (1) Let A := Alb(X). By [Mu] a∗ωX is zero if and only if Vi(A, a∗ωX) is empty for all i. By Proposition 2.2, this is equivalent to V0(A, a∗ωX) being empty. Thus if a∗ωX 6= 0, by Lemma 3.1, we may assume that V0(A, a∗ωX) consists of exactly one (torsion) point say P and h0(A, a∗ωX⊗P ) = 1. We shall prove that the injection OA −→ a∗ωX⊗P is in fact an isomorphism of sheaves.
Therefore a∗ωX∈ Pic0(A).
To this end, we consider complex D(v)
. . . Hi−1(A, a∗ωX⊗P )−→ H∪v i(A, a∗ωX⊗P )−→ H∪v i+1(A, a∗ωX⊗P ) . . . By Proposition 2.5, this is exact for all v ∈ H1(A, OA).
Step 1. Let V∗= H1(A, OA) and P = P(V ). There is an exact sequence of vector bundles on P:
0 −→ H0(A, a∗ωX⊗P )⊗OP(−q) −→ H1(A, a∗ωX⊗P )⊗OP(−q + 1) −→ ...
... −→ Hq(A, a∗ωX⊗P )⊗OP−→ 0. (K•)
To see this, it is enough to check exactness on each fiber (see [EL1] for a similar argument). A point in P corresponds to a line in H1(A, OA) containing a point say v. On the fibers above [v], the sequence of vector bundles corresponds to the complex D(v) which is exact. Similarly, there is an exact sequence of vector bundles on P:
0 −→ H0(A, OA)⊗OP(−q) −→ H1(A, OA)⊗OP(−q + 1) −→ ...
... −→ Hq(A, OA)⊗OP−→ 0. (K•0)
There is an injection of sheaves i : OA−→ a∗ωX⊗P determined by the choice of a section of a∗ωX⊗P . This induces a map of complexes i•: K0•−→ K•.
Step 2. i∗ : Hi(A, OA) −→ Hi(A, a∗ωX⊗P ) is an isomorphism for 0 ≤ i ≤ q.
We will proceed by induction. By assumption H0(A, OA) ∼= H0(A, a∗ωX⊗P ).
Assume now that i∗ : Hi(A, OA) ∼= Hi(A, a∗ωX⊗P ) for all i < r, we must show that i∗ : Hr(A, OA) −→ Hr(A, a∗ωX⊗P ) is also an isomorphism. Twisting the complexes K0•, K• by OP(−r) and taking cohomology, we have:
... ...
y
y Hr−1(A, OA)⊗Hq−1(OP(−q − 1)) −→i∗
∼= Hr−1(A, a∗ωX⊗P )⊗Hq−1(OP(−q − 1))
y
y
Hr(A, OA)⊗Hq−1(OP(−q)) −→i∗ Hr(A, a∗ωX⊗P )⊗Hq−1(OP(−q)).
y
y
0 0
An easy spectral sequence argument implies that for r > 1, the vertical lines are exact. By the five lemma, we obtain the required isomorphism.
Step 3. a∗ωX⊗P = OA.
Let R be the cokernel of i : OA,→ a∗ωX⊗P . For all i ≥ 0 and Q ∈ {Pic0(A) − OA}, hi(A, OA⊗Q) = hi(A, a∗ωX⊗P ⊗Q) = 0. By Step 2, it follows that hi(A, R⊗
Q) = 0 for all i ≥ 0 and Q ∈ Pic0(A). By §1.2, R = 0. This completes the proof of (1).
(2) follows from (1) since the generic rank of a∗ωXcorresponds to P1(FX/Alb(X)).
(3) also follows immediately from (1).
(4) By Fujita’s lemma [Mo, (4.1)] there is a smooth projective variety ˜X, and a generically finite surjective morphism ν : ˜X −→ X such that κ( ˜X) = κ(X) = 0 and P1( ˜X) > 0. Hence P1( ˜X) = 1. The Albanese map of ˜X is surjective by [Ka1],
and (4) now follows from (3). ¤
Proof of Lemma 5. Let f∗ : Alb(X) −→ Alb(Y ) be the map induced from f . Since κ(X) = 0, by [Ka1, Theorem 1] albX : X −→ Alb(X) is an algebraic fiber space.
It follows easily that f∗ and f∗ ◦ albX are surjective maps. Consider the Stein factorization Alb(X) −→ A0 −→ Alb(Y ) of the map f∗. Then Alb(X) −→ A0 is an algebraic fiber space and A0 −→ Alb(Y ) is an ´etale map of abelian varieties. It follows that also a0 : X −→ A0 is an algebraic fiber space and the fibers FX/Y are contracted by a0. By [Ka1], there exists an induced (generically finite) rational map Y 99K A0. It follows that the generic degree of the surjective map Y −→ A0 is 1.
Therefore, Y −→ Alb(Y ) is birationally ´etale. ¤
Proof of Theorem 6. To prove (1), we consider the generically finite map albY : Y −→ albY(Y ) ⊂ Alb(Y ).
Step 1. If κ(X) ≥ 0, then the Iitaka model of X dominates the Iitaka model of Y . Therefore κ(X) ≥ κ(Y ).
Let X −→ V and Y −→ W be appropriate birational models of the Iitaka fibrations of X and Y respectively. Since κ(FX/V) = 0, it follows by Lemma 5 that f (FX/V) is birational to an abelian variety. And the map
f (FX/V) −→ albY(f (FX/V))
is birationally ´etale. Therefore, it is easy to see (following the proof of [Ka1, Theorem 13]) that f (FX/V) is contained in the fibers of the Iitaka fibration Y −→
W . Therefore, there exists a rational map V 99K W . By changing birational models, we may assume that it is a morphism.
Step 2. If κ(X) = κ(Y ), then P1(FX/Y) ≤ 1.
Since X −→ Y and Y −→ W are algebraic fiber spaces and dim(V ) = κ(X) = κ(Y ) = dim(W ), it follows that V −→ W is birational. FX/V = FX/W −→ FY /W is also an algebraic fiber space with generic fiber FX/Y. One sees that κ(FX/V) = 0 and FY /W is of maximal Albanese dimension and hence FY /W is birational to an abelian variety by Lemma 5. It follows by Theorem 4 that P1(FX/Y) ≤ 1.
We now prove (2). Assume that κ(Y ) = 0. Since h0(FX/Y, ωX|FX/Y) > 0, then f∗ωX 6= 0. If h0(X, ωX⊗f∗P ) = 0 for all P ∈ Pic0(Y ), then by Proposition 2.2, hi(Y, f∗ωX⊗P ) = 0 for all i and all P ∈ Pic0(Y ) and hence f∗ωX = 0 a contra- diction. Therefore h0(X, ωX⊗f∗P ) > 0 for some P ∈ Pic0(Y ). By Theorem 2.1.4, we may assume that P⊗r = OY for some appropriate integer r > 0. Therefore, Pr(X) > 0 and κ(X) ≥ 0.
If κ(Y ) > 0 then following [Ka1, Theorem 13], there exists an ´etale cover ˜Y −→ Y which is birational to Γ × P with Γ of general type (and maximal Albanese dimension) and P an abelian variety. Consider the corresponding ´etale cover ˜X :=
X ×YY −˜ → X and the induced algebraic fiber space ˜f : ˜X −→ ˜Y . Since κ( ˜X) = κ(X) and κ( ˜Y ) = κ(Y ), it suffices to show that κ( ˜X) ≥ κ( ˜Y ) = κ(Γ). Since Γ is of general type, by a theorem of Viehweg (see e.g. [Mo, 6.2.d]), we have that κ( ˜X) ≥ κ(Γ) + κ(FX/Γ˜ ). We then consider the fiber space FX/Γ˜ −→ FY /Γ˜ . Since κ(FY /Γ˜ ) = κ(P ) = 0, one sees that κ(FX/Γ˜ ) ≥ κ(FY /Γ˜ ) = 0 by the preceeding case.
The assertion now follows. ¤
Proof of Theorem 7. If dim(X) = q(X) + 1, this is [Ka1, Theorem 15]. Let A = Alb(X) and q := q(X) = dim(A). We have already seen that OA is an isolated point of V0(X, A, ωX). Therefore, proceeding as in the proof of Theorem 4, we have that hq(A, a∗ωX) = h0(A, a∗ωX) = 1. By [Ko1, Proposition 7.6], Rn−qa∗ωX= ωA. Therefore, by Hodge symmetry, Serre duality and [Ko2]
h0(X, Ωn−qX ) = hq(X, ωX) ≥ hq(A, a∗ωX) + hq−(n−q)(A, Rn−qa∗ωX) >
h2q−n(A, ωA) = hn−q(A, OA) = h0(A, Ωn−qA ).
If n = q + 1 and P1(X) = 1, then we would have h0(X, Ω1X) > h0(A, Ω1A) which is
impossible. ¤
Corollary 3.4 ([Ka1, Theorem 15]). Let κ(X) = 0, dim(X) = q(X) + 1. Then Conjecture K holds
Proof. By Fujita’s lemma [Mo, (4.1)] there is a smooth projective variety Y , and a generically finite surjective morphism ν : Y −→ X such that κ(Y ) = κ(X) = 0 and P1(Y ) = 1. One sees that Alb(Y ) −→ Alb(X) is surjective and hence q(Y ) ≥ q(X) = dim(X) − 1. Therefore, by Theorem 7, q(Y ) ≥ dim(Y ) and hence by Kawamata’s Theorem, Y −→ Alb(Y ) is birational. The fibers of Y −→ Alb(X) are translates of a fixed elliptic curve E. The corollary now follows. ¤
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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, University of California Riverside, Department of Mathematics, Riverside, CA 92521, USA
E-mail address: chhacon@math.utah.edu
