Cohomological support loci of varieties of Albanese fiber dimension one

0.22 MB·English

by  Zhi Jiang

COHOMOLOGICAL SUPPORT LOCI OF VARIETIES OF
ALBANESE FIBER DIMENSION ONE
2
1
ZHI JIANGAND HAOSUN
0
2
r Abstract. Let X bea smooth projective variety of Albanese fiberdi-
p mension 1 and of general type. We prove that the translates through
A 0 of all components of V0(ωX) generate Pic0(X). We then study the
pluricanonical maps of X. We show that |4KX| induces a birational
1
map.
1

]
G
A 1. Introduction
.
h In [CH3], Chen and Hacon proved that if X is of maximal Albanese
t
a dimension and of general type, then the translates through 0 of all compo-
m nents of V0(ω ) generate Pic0(X). This is a fundamental result to study
X
[ the pluricanonical maps of varieties of maximal Albanese dimension (cf.

2 [J1, Ti, JLT]). Here we provide a similar result for varieties of general type
v and of Albanese fiber dimension one:
8
4 Theorem 1.1. Let X be a smooth projective variety of dimension ≥ 2,
4
of Albanese fiber dimension one and of general type. Then the translates
2
through 0 of all irreducible components of V0(ω ) generates Pic0(X).
. X
9
0 We notice that this kind of result applies only for varieties of Albanese
1
fiber dimension ≤ 1. Indeed, we take X = Y ×Z, where Y is a variety of
1
: maximalAlbanesedimensionandofgeneraltypeandZ isavarietyofgeneral
v
type of dimension l ≥ 2 with p (Z) = q(Z) = 0. Then X is of Albanese
i g
X fiber dimension l but V0(ω ) is empty. As an application of Theorem 1.1,
X
r we give an improvement of a result of Chen and Hacon [CH4]:
a
Theorem1.2. LetX be asmooth projective varietyofAlbanese fiberdimen-
sion one and of general type. Then the 4-canonical map ϕ is birational.
|4KX|
In Section 2, we recall some definitions and results which are usually used
in the study of irregular varieties. Then we generalize a result of Pareschi
and Popa on the generation property of M-regular sheaves (see Theorem
2.8). We will prove Theorem 1.1 in Section 3 and extend it to non general
Date: April12, 2012.
2000 Mathematics Subject Classification. 14E05.
Key words and phrases. Irregular variety,pluricanonical map, M-regularity.
ThesecondauthorwaspartiallysupportedbytheMathematicalTianyuanFoundation
of China (No. 11126192).
1
2 ZHIJIANGANDHAOSUN
type case in Section 4. We then study the pluricanonical maps of a variety
of Albanese fiber dimension 1 and prove Theorem 1.2 in Section 5.
Notation. In this note, X will always be a smooth complex projective
variety and we denote by a : X → A the Albanese morphism of X. If
X X
l = dimX−dima (X), then we say X is of Albanese fiber dimension l. In
X
particuler, X is of Albanese fiber dimension 0 if and only if X is of maximal
Albanese dimension.
For an abelian variety A, we will frequently denote by A the dual abelian
b
variety. Moreover, for a morphism t : A → B between abelian varieties,
we will denote by t : B → A the dual morphism between the dual abelian
varieties. b b b
Lett :X → AbeamorphismfromX toanabelianvarietyandletF bea
coherentsheafonX,wewilldenotebyVi(F,t) = {P ∈ A|hi(X,F⊗t∗P) >
b
0} the i-th cohomological support loci of F. In particular, if t = a , we
X
will simply denote Vi(F,a ) by Vi(F).
X
Acknowledgements. The first author thanks Prof. Jungkai A. Chen for
various comments and pointing out an error in an earlier version of this
paper. The second author wish to thank Dr. Lei Zhang for many useful
discussions.
2. Fourier-Mukai transform and M-regularity
In this section, we recall some important techniques that will be needed
throughout the paper.
Let A be a complex abelian variety of dimension g. We denote by P
the normalized Poincar´e bundle on A× A. Let p and p be the canonical
projections to A and A. There is a funcbtor S from thbe category of O -
A
modulestothecategorybofOAb-modulesdefinedcbyS(M) = p∗(p∗(M)⊗P).
Similarly we define S(N) = p (p∗(N)⊗P). Let RcS (respb. RS) be the
∗
derived functor of S (resp. S)bbetween the two derived categorcies D(A)
c
and D(A).
b
Theorem 2.1. [Mu, Theorem 2.2] There are isomorphisms of functors
RS ◦RS ∼= (−1Ab)∗[−g] and RS ◦RS ∼= (−1A)∗[−g].
c c
Definition 2.2. Let F be a coherent sheaf on a smooth projective variety
Y.
(1) We say F is full if V0(F) = Pic0(Y).
(2) F is said to be a GV-sheaf if codimVi(F) ≥ i for all i≥ 0.
(3) F is called M-regular if codimVi(F) > i for every i > 0 and Y is
an abelian variety.
Lemma 2.3. Let F be a GV-sheaf on a smooth projective variety Y, let W
be an irreducible component of V0(F), and let k = codimPic0(Y)W. Then
W is also a component of Vk(F). Hence dimX ≥ k.
VARIETIES OF ALBANESE FIBRE DIMENSION ONE 3
Proof. Please see [PP2, Proposition 3.15]. (cid:3)
Lemma 2.4. If F is a M-regular sheaf on an abelian variety A, then F is
full.
Proof. If F is not full, then V0(F) is a proper subvariety of Pic0(A). By
Lemma 2.3, there is i> 0 such that codimVi(F) = i. This contradicts the
M-regularity assumption on F. (cid:3)
Definition 2.5. Let F be a coherent sheaf on a smooth projective variety
Y.
(1) We say F is continuously globally generated at y ∈ Y (in briefCGG
at y) if the nature map
MH0(F ⊗α)⊗α∨ → F ⊗C(y)
α∈U
is surjective for any non-empty open subset U ⊂ Pic0(Y).
(2) F issaidtohavenoessentialbasepointaty ∈Y ifforanysurjective
map F → O , there is a non-empty open subset U ⊂ Pic0(Y) such
y
that for all α ∈ U, the induced map H0(F ⊗α) → H0(O ⊗α) is
y
surjective.
Remark2.6. Ourdefinitionofessentialbasepointiscompatiblewith[CH4,
Definition 2.1]. Actually, thefollowing lemmashows thatthetwo conditions
of Definition 2.5 are equivalent.
Lemma 2.7. If F is a coherent sheaf on a smooth projective variety Y and
y is a point on Y, then F is CGG at y if and only if F has no essential
base point at y.
Proof. Firstly, we assume that F is CGG at y. For any surjective map
F → O , the induced map F⊗C(y) → O is also surjective. The definition
y y
of CGG implies that the composition
MH0(F ⊗α)⊗α∨ → F ⊗C(y)→ Oy
α∈U
is surjective for any non-empty open subsetU ⊂ Pic0(Y). It follows that for
any non-empty open subset U ⊂ Pic0(Y), there is an α ∈ U such that the
induced map H0(F ⊗α) → H0(O ⊗α) is surjective. By semi-continuity,
y
one sees that for a general α ∈ Pic0(Y), the induced map H0(F ⊗α) →
H0(O ⊗α) is surjective. Thus F has no essential base point at y.
y
Conversely, suppose that F has no essential base point at y. One can
write F ⊗ C(y) = k V , where V ∼= C(y), i = 1,2,...,k. Let p :
Li=1 i i i
F ⊗C(y) → V be the canonical projection and ϕ : F → F ⊗C(y) → V
i i i
be the composition. Since F has no essential base point at y, we know that
there exists non-empty open subsets U (1 ≤ i ≤ k) of Pic0(Y) such that for
i
any α ∈ U the induced map ϕ : H0(F ⊗α) → V ⊗α is surjective. For
i i i
e
4 ZHIJIANGANDHAOSUN
any non-empty open subset U ⊂ Pic0(Y), since ∩k U is also a non-empty
0 i=0 i
open subset, the map
k
M H0(F ⊗α)⊗α∨ −→ MVi = F ⊗C(y)
α∈U0 i=1
is surjective. It follows that F is CGG at y. (cid:3)
A coherent sheaf F on an abelian variety A is said to be IT0 if Hi(F ⊗
α) = 0 for all i ≥ 1 and all α ∈ Pic0(A). The following theorem generalizes
[PP1, Proposition 2.13]
Theorem 2.8. Let F and H be coherent sheaves on a complex abelian
variety A. Suppose that either F is M-regular and H is IT0 or F is IT0
and H is full. Then for any non-zero morphism F → H there exists a
non-empty open subset U of Pic0(A) such that for all α ∈ U, the induced
map H0(F ⊗α) → H0(H ⊗α) is non-zero.
Proof. By Theorem 2.1, we have
Hom(F,H )= HomD(A)(F,H ) ∼= HomD(Ab)(RS(F),RS(H )).
c c
If F is M-regular and H is IT0, as in the proof of [PP1, Theorem 2.5],
since RS(H )= S(H ), there is a nature inclusion
c c
HomD(Ab)(RS(F),S(H )) ֒→ Hom(S(F),S(H )).
c c c c
Hence we obtain a nature injective map
Φ: Hom(F,H ) → Hom(S(F),S(H )).
c c
If F is IT0 and H is full, then RS(F) = S(F). Note that
c c
HomD(Ab)(S(F),RS(H )) = Hom(S(F),S(H )),
c c c c
so we also obtain the nature injective map
Φ: Hom(F,H ) → Hom(S(F),S(H )).
c c
Thus for any non-zero morphism ϕ: F → H , the induced morphism
Φ(ϕ) : S(F) → S(H ),
c c
is also non-zero.
We choose a non-empty open subset V such that both h0(F ⊗ α) and
h0(H ⊗α) are constant for all α ∈ V. From the base change theorem, it
follows that S(F)⊗C(α) ∼=H0(F ⊗α) and S(H )⊗C(α) ∼= H0(H ⊗α)
c c
for all α ∈ V. By Lemma 2.4, we know that our assumptions on F and H
guarantee both F and H are full. Hence Φ(ϕ)| : S(F)| → S(H )| is
V V V
c c
a non-zero morphism between vector bundles. Let Z be the loci where the
rank of Φ(ϕ)| vanishes, and let U = V \Z. One sees that for all α ∈ U,
V
the induced map H0(F ⊗α) → H0(H ⊗α) is non-zero. (cid:3)
VARIETIES OF ALBANESE FIBRE DIMENSION ONE 5
Applying Theorem 2.8 to the case H = O for some y ∈ A, we obtain
y
the following corollary.
Corollary 2.9. [PP1, Proposition 2.13] If F is a non-zero M-regular sheaf
on a complex abelian variety A, then F has no essential base point at any
y ∈ A.
The following proposition is a slight generalization of [GL, Theorem 0.1
(2)].
Proposition 2.10. Let α : X → A be a morphism from a smooth porjective
variety to an abelian variety such that dimX−dimα(X) = f. Assume that
V0(ω ,α) has an irreducible component P +B of codimension 0 ≤ k <∞
X 0
b
in A. Then there exists a commutative diagram
b
X α // A
f π
(cid:15)(cid:15) (cid:15)(cid:15)
X g // B
B
where π is the natural projection, f is a fibration and X is a normal variety
B
of dimension dimX −f −k.
Proof. Weknowthata ω isaGV-sheaf onA(see[H]). HencebyLemma
X∗ X
2.3, P +B is an irreducible component of Vk(α ω ). By Simson’s result
0 ∗ X
b
[Si], we may take P to be a torsion line bundle.
0
h t
Let X −→ X −→ A (resp. X → X → B) be the Stein factorization of α
A B
(resp. the Stein factorization of the natural morphism from X to B). We
then have
X(cid:15)(cid:15)
❇
❇
❇
❇❇α
(cid:15)(cid:15) h ❇❇❇!!
f XA t // A
hB π
(cid:31)(cid:31) (cid:15)(cid:15) (cid:15)(cid:15)
X g // B
B
Since t is finite, Hk(A,α ω ⊗Q) = Hk(X ,h ω ⊗t∗Q) 6= 0, for any
∗ X A ∗ X
Q ∈ P +B. On the other hand, by Kolla´r’s theorem ([Kol2, Theorem 3.4])
0
b
we have
hk(XA,h∗ωX ⊗t∗(P0 ⊗P) = X hi(XB,RjhB∗(h∗(ωX ⊗P0))⊗g∗P)
i+j=k
6= 0,
forallP ∈ B. Forallj ≥ 0,thesheaves Rjh (h (ω ⊗P ))areGV sheaves
B∗ ∗ X 0
(see [H]), hbence Rkh (h (ω ⊗P )) 6= 0. We conclude again by Kolla´r’s
B∗ ∗ X 0
theorem ([Kol2, Theorem 3.4]) that dimX −dimX ≥ k and hence the
A B
equality holds and dimX = dimX −f −k. (cid:3)
B
6 ZHIJIANGANDHAOSUN
3. Proof of Theorem 1.1
Lemma 3.1. Let Y be a smooth projective variety of general type and of
maximal Albanese dimension. Let t : Y → A be a morphism to an abelian
variety A of dimension ≥ 1 such that dimY −dimt(Y) = 1. Then
dimV0(ω ,t) ≥ 1,
Y
where
V0(ω ,t) := {P ∈ A| H0(Y,ω ⊗t∗P) 6= 0},
Y Y
b
and dimV0(ω ,t) is by definition the maximal dimension of an irreducible
Y
component of V0(ω ,t).
Y
Proof. We have the following commutative diagram
Y aY // A
❇ Y
❇
❇
❇
❇❇ µ
t ❇❇ (cid:15)(cid:15)
A
We may assume that t(Y) generates the whole abelian variety A and µ
is surjective, otherwise K is an effective divisor ([CH2]), and hence ker(µ :
Y
A→ A ) is contained in V0(ω ,t) and is of dimension ≥ 1. b
Y X
b b
When t(Y) generates the whole abelian variety A, µ : A → A is an
X
isogeny onto its image and b b b
dimV0(ω ,t) =dim(µ(A)∩V0(ω )).
Y Y
b b
Hence if V0(ω ) = A , Lemma 3.1 is clear. Otherwise, χ(Y,ω ) = 0 and
Y Y Y
V0(ω ) is a union ofbtorsion translates of proper abelian sub-varieties of A
Y Y
and dimV0(ω ) ≥ 1 (see [CH3]). b
Y
µ
Let K be the neutral component of the kernel of A −→ A. We then have
Y
the exact sequence
A −→µb A ։ K.
Y
b b b
Because an irreducible component of a general fiber of t is a smooth curve
of genus ≥ 2, hence for any α∈ A , t (ω ⊗P ) is a non-zero GV-sheaf on
Y ∗ X α
b
A. Therefore for any α ∈ A , there exists τ ∈A such that
Y α
b b
H0(Y,ω ⊗P ⊗t∗P )≃ H0(A,t (ω ⊗P )⊗P )6= 0.
Y α τα ∗ Y α τα
Hence the composition of morphisms V0(ω ) ֒→ A → K is surjective.
Y Y
ThereexistsanirreduciblecomponentQ+B ofV0(ω b),wherbeQisatorsion
Y
b
pointofA andB isanabelian subvarietyof A ,suchthatthecomposition
Y Y
b b b
of the natural morphisms
Q+B ֒→ A → K
Y
b b b
is surjective and then (Q+B)∩µ(A) is not empty.
If dimV0(ω ,t) = 0, thenb b b
Y
(1) dim(µ(A)∩V0(ω )) = 0.
Y
b b
VARIETIES OF ALBANESE FIBRE DIMENSION ONE 7
Hence
dim((Q+B)∩µ(A)) = 0,
b b b
and the projection Q + B → K is finite. Therefore, codimb (Q + B) =
b b AY b
dimA ≥ dimY −1.
b
Ontheotherhand,weknowbytheproofofTheorem3in[EL]thatQ+B
b b
is also an irreducible component of VdimA(ω ). Hence dimA = dimY −1
Y
b
(p,µ)
and the natural morphism A −−−→ B×A is an isogeny. We then consider
Y
Y aY // A µ // A
◆◆◆◆◆p◆B◆◆◆◆◆◆◆◆'' (cid:15)(cid:15)Yp
B
By the proof of Theorem 3 in [EL] (or see Proposition 2.10), a general
fiber of p is generically finite and surjective over the fiber of p, and hence
B
generically finite and surjective over A via µ◦a . Hence the image p (Y)
Y B
is a curve on B. We take the Stein factorization of p :
B
γ
Y −→ Z → B.
Since a general fiber F of γ is of general type and generically finite over A,
by [CH3, Theorem 1], there exists a positive dimensional torus T ⊂ A such
that H0(F,ω ⊗a∗µ∗P)6= 0, for any P ∈ T. Hence γ (ω ⊗a∗ µ∗Pb) is a
F Y ∗ Y/Z Y
non-zero nef vector bundle on Z.
If Z is a smooth curve of genus ≥ 2, Then by Riemann-Roch, we deduce
that
H0(Y,ω ⊗a∗µ∗P)6= 0.
Y Y
Hence
µ(T) ⊂ µ(A)∩V0(ω )֒→ A ,
Y Y
which contradicts (1).b b b b
If Z is an elliptic curve, then since Z generates B, Z is isogenous to B.
Then Q+B is an irreducible component of V0(ω ) of dimension 1. But it
Y
b
is impossible by Proposition 3.6 in [CDJ].
We then conclude the proof of Lemma 3.1. (cid:3)
Proposition 3.2. Let X be a smooth projective variety of dimension of gen-
eraltype. AssumethatX isofAlbanesefiberdimension1. ThendimV0(ω )≥
X
1.
Proof. We take the Stein factorization of a :
X
X
❈
❈❈
g ❈❈a❈❈X
(cid:15)(cid:15) ❈!!
W f // A
X
We denote dimX = n≥ 2 and denote by m ≥ 2 the genus of a general fiber
of g.
8 ZHIJIANGANDHAOSUN
We argue by contradiction. Assume that V0(ω ) = V0(a ω ) is a
X X∗ X
unionoffinitepoints. Sincea ω isaGV-sheaf(see[H,Corollary4.2])and
X∗ X
V0(a ω )isaunionoffinitepoints,thenby[Mu,Example3.2],a ω isa
X∗ X X∗ X
homogeneous vector bundle on A . Hence VdimAX(a ω ) 6= ∅, therefore
X X∗ X
dimA = dima (X) = n−1.
X X
There are three steps to deduce a contradiction.
First step, we claim that a is a fibration. Hence f is an isomorphism
X
and R1a ω = O .
X∗ X AX
We take n − 2 general very ample divisors H , 1 ≤ i ≤ n − 2 on A .
i X
Let the smooth curve C be the intersection of H on A . Take base change
i X
C ֒→ A to the above commutative diagram, we have
X
(2) X
C❇
❇
gC ❇❇❇h❇C
(cid:15)(cid:15) ❇❇
W fC // C
C
Then h ω = a ω | .
C∗ XC/C X∗ X C
If degf > 1, since a is the universal morphism to an abelian variety, f
X
is ramified in codimension 1. Hence f : W → C is a ramified cover. We
C C
know that g ω is a nef vector bundle on W (see [V2]). Then by
C∗ XC/WC C
Riemann-Roch, we conclude that
degh ω = degf (g ω )⊗ω > 0
C∗ XC/C C∗(cid:0) C∗ XC/WC WC/C(cid:1)
which contradicts the fact that deg(a ω | ) = 0. We finish the proof of
X∗ X C
the first step.
We then write a ω = ⊕s V , where V is the tensor productof P ∈
X∗ X i=1 Pi Pi i
Pic0(A ) with a unipotent vector bundle on A and s rank(V ) = m.
X X Pi=1 i
Notice that P is torsion (see [Si]).
i
Second step, we claim that P = O for all i. In other words, all V are
i X i
unipotent vector bundles.
We still argue by contradiction to prove this claim.
If there exists a non-trivial P , then Hn−1(A ,V ⊗P∗) 6= 0 and hence
i X Pi i
Hn−1(X,ω ⊗P∗) 6= 0. Take the ´etale cover π : A → A induced by the
X i X X
e
group generated by all P . Let π : X → X be the induced ´etale cover. We
j X
e
notice that
q(X) = hn−1(X,ωXe)≥ hn−1(X,ωX)+hn−1(X,ωX ⊗Pi∗) > q(X).
e e
VARIETIES OF ALBANESE FIBRE DIMENSION ONE 9
We have the following commutative diagram:
X πX // X
aXe ||①①①①①τ①①①① e
(cid:9)(cid:9) ||②②②②②②②②a②XMme(Xe)❊❊❊❊❊❊υ❊❊❊"" (cid:15)(cid:15) eaX (cid:15)(cid:15) aX
AXe µ // AX π // AX
e
Sincea isafibration,eitherτ isgenerically finiteandX isthenofmaximal
X
Albaneesedimension,orτ isafibrationandυisanisomorephism. Thelatteris
impossible,sinceae(X)generates thewholeabelian variety Ae. Inthefirst
X X
e
case, X is of maximal Albanese dimension and a is surjective of relative
X
dimenseion 1. We then apply Lemma 3.1 and deduece that dimV0(ωXe,aX) ≥
1. This is impossible because dimV0(ωXe,aX) = dimV0(ωX)= 0. e
We then conclude the proof of the seconed step.
In the last step, we are going to deduce a contradiction. Since V is a
i
unipotent vector bundle, hn−1(A ,V )≥ 1. We then have ([Kol2, Theorem
X i
3.1])
n−1 = hn−1(X,ω ) = hn−1(A ,a ω )+hn−2(A ,R1a ω )
X X X∗ X X X∗ X
s
= Xhn−1(AX,Vi)+hn−2(AX,OAX)
i=1
≥ s+n−1,
which is impossible.
We then conclude the proof of Proposition 3.2. (cid:3)
We are now able to prove a slightly more general version of Theorem 1.1:
Theorem 3.3. Let X be a smooth projective variety of general type and
let α : X → A be a morphism to an abelian variety such that dimα(X) =
dimX −1. Then the translates through 0 of all irreducible components of
V0(ω ,α) generate A.
X
b
Proof. Let T ֒→ A be the abelian subvariety generated by all the translate
through 0 obf all irbreducible components of V0(ω ,α).
X
10 ZHIJIANGANDHAOSUN
Assume that T is a proper subvariety of A. We consider the dual and get
b b
F(cid:127)_ h // K(cid:127)_
X(cid:15)(cid:15) // α(X)(cid:31)(cid:127) // A(cid:15)(cid:15)
❉❉❉f❉❉❉❉❉❉!! (cid:15)(cid:15) (cid:15)(cid:15) p
f(X)(cid:31)(cid:127) // T
Let F be an irreducible component of a general fiber of f, K be the kernel
of p, and h : F → K be the restriction of α to the fibers. Notice that
dimF −dimh(F) = 1. We now study the group
V0(ω ,h) := {P ∈K | H0(F,ω ⊗h∗P) 6= 0}.
F F
b
We know that for any P ∈A , f (ω ⊗P) is a GV-sheaf (possible 0) on
X ∗ X
T. Hence V0(ω ,α) surjectivebover V0(ω ,h) via the morphism A → K.
X F X
From the construction of T, we know that V0(ω ,h) is a unionbof finibte
F
b
points.
On the other hand, we consider
h
!!
F // A // K.
aF F π
Since dimV0(ω ,h) = 0, then π(K)∩V0(ω ) is a union of finite points in
F F
A and π is an isogeny to its imbagbe. By Proposition 3.2, π is not surjective.
F
Abs we habve seen in the proof of Lemma 3.1, the composibtion of morphisms
V0(ω ) → A → A /π(K) is surjective. Hence there exists a positive
F F F
dimensional ibrreducibble cbombponent P +B of V0(ω ) of codimension equal
0 F
b
to dimπ(K)≥ dimF −1.
If a bisbgenerically finite. We can simply apply Lemma 3.1 to get a
F
contradiction. If F is of Albanese fiber dimension 1, we apply Proposition
2.10 to conclude that the image of the natural morphism F −a−→F A ։ B is
F
a point, which is impossible since a is the Albanese morphism of F.
F
Hence T = A and we are done. (cid:3)
b b
For surfaces X of Albanese fiber dimension 1, because χ(X,ω ) > 0, we
X
always have V0(ω ,a ) = Pic0(X). In higher dimensions, we can construct
X X
varietiesofAlbanesefiberdimension1andofgeneraltypewithV0(ω ,a )a
X X
propersubsetofA inthesamewayasintheconstructionofEin-Lazarsfeld
X
b
threefold [EL, Example 1.13].
Example 3.4. Let C → E be a ramified double cover over elliptic curve
i i
E for i = 1,2, with associated involution ι . Let C → P1 be a ramified
i i 3
double cover, with associated involution ι . Let X be a desingularization of
3