# Cohomological support loci and Pluricanonical systems on irregular varieties

Zhi Jiang  Zhi Jiang, Shanghai Center for Mathematical Sciences, China
<EMAIL_ADDRESS>
###### Abstract.

For an irregular variety $X$ of general type, we show that if a general fiber
$F$ of the Albanese morphism of $X$ satisfies certain Hodge theoretic
condition, the $0$-th cohomological support loci of $K_{X}$ generates
$\operatorname{Pic}^{0}(X)$. We then show that the condition that the $0$-th
cohomological support loci of $K_{X}$ generates $\operatorname{Pic}^{0}(X)$
can often be applied to prove the birationality of certain pluricanonical maps
of $X$.

###### Key words and phrases:

cohomological support loci, pluricanonical maps, generic vanishing

###### 2010 Mathematics Subject Classification:

14E05, 14D07, 14F10

##  1\. Introduction

In this article, we study the pluricanonical systems of irregular varieties.
This topic was initiated in a series of article of Jungkai Chen and Hacon
[CH1, CH2, CH3].

Let $X$ be a smooth projective variety of general type with
$q(X)=h^{1}(X,\mathcal{O}_{X})>0$. Fix the Albanese morphism
$a_{X}:X\rightarrow A_{X}$ from $X$ to its Albanese variety $A_{X}$. We denote
by $F$ a connected component of a general fiber of $a_{X}$. We say that $X$ is
of Albanese fiber dimension $m$ if $\dim F=m$. The properties of the
pluricanonical maps of $X$ are closely related to the properties of the
pluricanonical systems of $F$. When $F$ or $X$ is of low dimensions, the
general picture is well-understood.

######  Theorem 1.1.

[J.A.Chen-Hacon [CH1]; Jiang-Lahoz-Tirabassi [JLT]] Let $X$ be a smooth
projective variety of general type, of maximal Albanese dimension. Then the
linear system $|mK_{X}+P|$ induces a birational map of $X$ for any integer
$m\geq 3$ and $P\in\operatorname{Pic}^{0}(X)$.

Note that this is an optimal result, which can be easily deduced from the
properties of pluricanonical systems on curves of genus $2$. A similar result
was proved by the author and Hao Sun when $F$ is a curve.

######  Theorem 1.2.

[Jiang-Sun [JS]] Let $X$ be a smooth projective variety of general type, of
Albanese fiber dimension $1$. Then the linear system $|mK_{X}+P|$ induces a
birational map of $X$ for any integer $m\geq 4$ and
$P\in\operatorname{Pic}^{0}(X)$.

It is worth noting that the structure of

$V^{0}(K_{X}):=\\{P\in\operatorname{Pic}^{0}(X)\mid H^{0}(X,K_{X}\otimes
P)\neq 0\\}$

plays an important role in the proof of both above theorems. We know from
generic vanishing theory (see for instance [GL] and [Sim]) that

$V^{j}(K_{X}):=\\{P\in\operatorname{Pic}^{0}(X)\mid H^{j}(X,K_{X}\otimes
P)\neq 0\\},$

for $j\geq 0$, is a union of torsion translates of abelian subvarieties of
$\operatorname{Pic}^{0}(X)$. We say that  $V^{j}(K_{X})$ generates
$\operatorname{Pic}^{0}(X)$ if the irreducible components of $V^{j}(K_{X})$
generate $\operatorname{Pic}^{0}(X)$ as a group. One of the main points in the
proof of Theorem 1.1 and 1.2 is to show that $V^{0}(K_{X})$ always generates
$\operatorname{Pic}^{0}(X)$ in both cases.

The first main result of this paper is a general criteria for the property
that $V^{0}(K_{X})$ generates $\operatorname{Pic}^{0}(X)$.

Recall that we say a smooth projective variety $V$ satisfies the infinitesimal
Torelli condition if the natural cup product

$\displaystyle
H^{1}(V,T_{V})\rightarrow\mathrm{Hom}(H^{0}(V,K_{V}),H^{1}(V,\Omega_{V}^{\dim
V-1}))$

is injective. The infinitesimal Torelli condition is of course closely related
with the variation of Hodge structures of $V$.

Given a smooth projective variety of general type $V$. We know that the
birational transformation group $\mathrm{Bir}(V)$ is a finite group and acts
naturally on $H^{0}(V,K_{V})$.

######  Theorem 1.3.

Assume that $X$ is of general type and $F$ satisfies the following conditions:

  * (C1)

the canonical model of $F$ is a smooth projective variety $V$ which satisfies
the infinitesimal Torelli condition;

  * (C2)

$\mathrm{Bir}(F)$ acts faithfully on $H^{0}(F,K_{F})$,

then $V^{0}(K_{X})$ generates $\operatorname{Pic}^{0}(X)$.

This is a further extension of the corresponding results in [CH3] and [JS].
Indeed, it is well known that a non-hyperelliptic curve of genus $\geq 3$
satisfies both conditions in the assumption of Theorem 1.3. One of the main
ingredient of the proof of Theorem 1.3 is a decomposition theorem of Hodge
modules on abelian varieties proved in [PPS]. Actually our proof works also
for higher cohomology support loci $V^{0}(R^{j}f_{*}K_{X})$ (see Theorem 2.1).

We also explore the relation between the structure of $V^{0}(K_{X})$ and the
properties of the pluricanonical systems of $X$.

######  Theorem 1.4.

Assume that $V^{0}(K_{X})$ generates $\operatorname{Pic}^{0}(X)$ and $|K_{F}|$
induces a birational map of $X$. Then the linear system $|3K_{X}+P|$ induces a
birational map of $X$ for $P\in\operatorname{Pic}^{0}(X)$ general.

The conditions in Theorem 1.3 and Theorem 1.4 apply to varieties whose
Albanese fibers $F$ has sufficiently positive canonical bundles. It is well
known that surfaces of general type does not satisfy the infinitesimal Torelli
condition in general. Besides surfaces with $p_{g}=0$, there are surfaces with
$p_{g}>0$, which does not satisfy the infinitesimal Torelli condition (see for
instance [To]). Hence Theorem 1.3 and Theorem 1.4 do not apply to varieties of
Albanese fiber dimension $2$. It is then surprising to the author that by
taking hyperplane sections, the structure of cohomological support loci of
varieties of Albanese fiber dimension $1$ has implications on the
pluricanonical systems of varieties of Albanese fiber dimension $2$.

######  Theorem 1.5.

Assume that $X$ is a smooth projective variety of general type and a connected
component of a general fiber of the Albanese morphism of $X$ is a surface $F$
with $p_{g}(F)>0$ and $q(F)=0$. Then $|5K_{X}+P|$ induces a birational map of
$X$ for all $P\in\operatorname{Pic}^{0}(X)$.

######  Remark 1.6.

In [CCCJ], the authors shows that $|5K_{X}+P|$ always induces a birational map
for any irregular $3$-fold $X$, which is an optimal result. The essential
difficulty in that result is to deal with irregular $3$fold which is of
Albanese fiber dimension $2$. Hence Theorem 1.5 can be regarded as a partial
higher dimensional generalization of the main result in [CCCJ]. The proofs of
the two results are quite different. The authors applied repeatedly Reider
type arguments in [CCCJ] and this kind of result is out of reach in higher
dimensions.

### Acknowledgements

Parts of the work was finished during the author’s visit to National Center
for Theoretical Sciences in Taipei in January 2019. The author thanks the warm
hospitality of Jungkai and Jhengjie. The author is supported by the Natural
Science Foundation of China (No. 11871155 and No. 11731004), by the National
Key Research and Development Program of China (No. 2020YFA0713200), and by the
Natural Science Foundation of Shanghai (No. 21ZR1404500).

##  2\. The structure of $V^{0}(K_{X})$

###  2.1. Hodge Modules

The proof of the main result relies heavily on the language of Hodge modules.
We recall some results about Hodge Modules on abelian varieties. All results
can be found in [PPS], where Pareschi, Popa, and Schnell applied the mechinary
of Hodge modules to prove the general Chen-Jiang decomposition for higher
direct image of canonical bundles on abelian varieties generalizing the main
result in [CJ]. Indeed, Pareschi, Popa, and Schnell proved the decomposition
of certain Hodge modules which is crucial for our purpose.

Let $X$ be a complex manifold. We denote $\mathbf{HM}_{\mathbb{R}}(X,w)$ be
the category of real Hodge module on $X$ of weight $w$, which is a semi-simple
$\mathbb{R}$-linear abelian category,endowed with a faithful functor to the
category of real perverse sheaves. An object $M$ of
$\mathbf{HM}_{\mathbb{R}}(X,w)$ consists of a regular holonomic left
$\mathcal{D}_{X}$-module $\mathcal{M}$, a good filtration $F_{\cdot}M$ by
coherent $\mathcal{O}_{X}$-modules, a perverse sheaf $M_{\mathbb{R}}$ with
coefficients in $\mathbb{R}$ and an isomorphism
$M_{\mathbb{R}}\otimes_{\mathbb{R}}\mathbb{C}\simeq\mathrm{DR}(\mathcal{M})$.
The support of $M$ is defined to be the support of the perverse sheaf
$M_{\mathbb{R}}$. One can also define polarizable Hodge modules. One can check
[PPS] or [Sai] for more details about this category.

  * (A)

Every object $M\in\mathbf{HM}_{\mathbb{R}}(X,w)$ admits a locally finite
decomposition by strict support:

$M\simeq\bigoplus_{i=1}^{n}M_{i},$

where each $M_{i}\in\mathbf{HM}_{\mathbb{R}}(X,w)$ has strict support equal to
an irreducible analytic subvariety $Z_{i}\subset X$.

  * (B)

The category of polarizable real Hodge modules of weight $w$ with strict
support $Z\subset X$ is equivalent to the category of generically defined
polarizable real variations of Hodge structure of weight $w-\dim Z$ on $Z$.

  * (C)

Denote by $\mathbb{R}_{X}[\dim X]\in\mathbf{HM}_{\mathbb{R}}(X,\dim X)$ the
polarizable real Hodge module corresponding to the constant real variation of
Hodge structure on $X$. For any morphism $f:X\rightarrow A$ from $X$ to an
abelian variety and an integer $j\geq 0$,
$\mathcal{H}^{f}_{*}\mathbb{R}_{X}[\dim X]$ is a polarizable real Hodge module
of weight $j+\dim X$ on $A$. Let
$M=(\mathcal{M},F_{\cdot}\mathcal{M},M_{\mathbb{R}})$ be the direct summand of
$\mathcal{H}^{j}f_{*}\mathbb{R}_{X}[\dim X]$ with strict support $f(X)$. Then
the first non-trivial grading piece of $\mathcal{M}$ is
$R^{j}f_{*}\omega_{X}$.

  * (D)

Let $M$ be as in $(C)$ above. Then we associate a complex polarizable Hodge
module $(M\oplus M,J_{M})$ as in [PPS], whose underlying perverse sheaf is
simply $M\otimes_{\mathbb{R}}\mathbb{C}$. Then by [PPS, Theorem 7.1 and
Corollary 7.3], we know that

$(M\oplus
M,J_{M})\simeq\bigoplus_{j=1}^{n}q_{j}^{-1}(N_{j},J_{j})\otimes_{\mathbb{C}}\mathbb{C}_{\rho_{j}},$

where $q_{j}:A\rightarrow A_{j}$ is a surjective morphism with connected
fibers between abelian varieties, $\rho_{j}\in\mathrm{Char}(A)$ is a unitary
character, and $(N_{j},J_{j})\in\mathbf{HM}_{\mathbb{C}}(T_{j},\dim X-\dim
q_{j})$ is a simple polarizable complex Hodge module with
$\chi(T_{j},N_{j},J_{j})>0$.

  * (E)

Under the assumption of $(D)$, then by [PPS, Theorem D], for any
$k\in\mathbb{Z}$,

$\mathrm{gr}_{k}^{F}\mathcal{M}\simeq\oplus_{j=1}^{n}q_{j}^{*}\mathcal{F}_{j}\otimes
L_{j},$

where $\mathcal{F}_{j}$ is an M-regular sheaf on $T_{j}$ and $L_{j}$ is a
torsion line bundle on $T$.

###  2.2. The proof of Theorem 1.3

We work in a more general setting.

######  Theorem 2.1.

Assume that $X$ is a smooth projective variety of general type. Let
$f:X\rightarrow A$ be a morphism from $X$ to an abelian variety and let $F$ be
a connected component of a general fiber of $f$. Fix a positive integer $j\geq
0$. Assume that $F$ satisfies the following conditions:

  * (C1’)

the canonical model of $F$ is a smooth projective variety $V$ which satisfies
the following infinitesimal Torelli condition: the map induced by cup-product

$H^{1}(V,T_{V})\rightarrow\mathrm{Hom}(H^{j}(V,K_{V}),H^{j+1}(V,\Omega_{V}^{\dim
V-1}))$

is injective;

  * (C2’)

$\mathrm{Bir}(F)$ acts faithfully on $H^{j}(F,K_{F})$,

then $V^{0}(R^{j}f_{*}\omega_{X})$ generates $\operatorname{Pic}^{0}(X)$.

###### Proof.

We argue by contradiction. Assume that the translates through the origin of
the irreducible components of $V^{0}(R^{j}f_{*}\omega_{X})$ generate a proper
abelian subvariety ${\widehat{B}}$ of $\operatorname{Pic}^{0}(A)$. Consider
the dual $B$ of ${\widehat{B}}$, we then have

Let $K$ be the fiber of of $q:A\rightarrow B$ over a general point $b\in B$
and $Y$ the corresponding fiber of $X\rightarrow B$. We then have a morphism
$f_{b}:Y\rightarrow K$. Note that $R^{j}f_{*}\omega_{X}|_{K}\simeq
R^{j}f_{b*}\omega_{Y}$. We then consider the natural restriction morphism
$\pi:\operatorname{Pic}^{0}(A)\rightarrow\operatorname{Pic}^{0}(K)$. It is
clear that

$\pi(V^{0}(R^{j}f_{*}\omega_{X}))\subset V^{0}(R^{j}f_{b*}\omega_{Y}),$

since $H^{0}(A,R^{j}f_{*}\omega_{X}\otimes P)\neq 0$ implies that
$H^{0}(K,R^{j}f_{b*}\otimes P|_{K})\neq 0$ for $\in\operatorname{Pic}^{0}(A)$.
We claim that $\pi(V^{0}(f_{*}\omega_{X}))=V^{0}(f_{b*}\omega_{Y})$. Indeed,
for $Q\in V^{0}(R^{j}f_{*}\omega_{Y})$. We choose
$P\in\operatorname{Pic}^{0}(A)$ such that $P|_{K}=Q$. Then

$q_{*}(R^{j}f_{*}\omega_{X}\otimes P)$

is non-trivial, since its fiber at $b$ is exactly
$H^{0}(K,R^{j}f_{b*}\omega_{Y}\otimes Q)\neq 0$. On the other hand, it is
known that $q_{*}(R^{j}f_{*}\omega_{X}\otimes P)$ is a GV sheaf by Hacon’s
criterion [H] and Kollár’s vanishing theorem [K]. Thus there exists
$P^{\prime}\in\operatorname{Pic}^{0}(B)$ such that
$H^{0}(B,q_{*}(f_{*}\omega_{X}\otimes P)\otimes P^{\prime})\neq 0$. Hence

$P\otimes P^{\prime}\in V^{0}(R^{j}f_{*}\omega_{X})$

and $\pi(P\otimes P^{\prime})=Q$.

By the construction of ${\widehat{B}}$, we then know that $\dim
V^{0}(R^{j}f_{b*}\omega_{Y})=0$ and hence $V^{0}(R^{j}f_{b*}\omega_{Y})$
consists of finite many torsion points. Repeating the above argument, we can
assume furthermore that $K$ is a simple abelian variety. We now work with
$f_{b}:Y\rightarrow K$.

Let $m=\dim Y$. Since $\dim V^{0}(R^{j}f_{b*}\omega_{Y})=0$, we conclude from
the decomposition theorem [PPS] that $R^{j}f_{b*}\omega_{Y}$ is a direct sum
of torsion line bundles on $K$. In particular, $f_{b}$ is surjective.

Let $M$ be the direct summand of $\mathcal{H}^{j}f_{b*}\mathbb{R}[m]$ with
strict support $f_{b}(Y)$. Note that the first non-trivial grading piece of
the D-module $\mathcal{M}$ of $M$ is $R^{j}f_{b*}\omega_{Y}$.

By Subsection 2.1 (D), we know that

$\displaystyle(M\oplus M,J_{M})$ $\displaystyle=$
$\displaystyle\mathcal{H}^{\prime}\bigoplus\mathcal{H}^{\prime\prime}$
$\displaystyle=$
$\displaystyle\big{(}\bigoplus_{j}q^{-1}V_{j}\otimes\mathbb{C}_{\rho_{j}}\big{)}\bigoplus\mathcal{H}^{\prime\prime},$

where $q:A\rightarrow\mathrm{Spec}\;\mathbb{C}$ is the constant morphism,
$V_{j}$ are Hodge structures of weights $\dim F+j$,
$\rho_{j}\in\mathrm{Char}(K)$ are unitary characters, and
$\mathcal{H}^{\prime\prime}$ is a direct sum of simple complex polarizable
Hodge modules with positive holomorphic Euler characteristic. The fact that
$\dim V^{0}(R^{j}f_{b*}\omega_{Y})=0$ implies that $R^{j}f_{b*}\omega_{Y}$ is
indeed the first non-trivia grading piece of the underlying D-module of
$\mathcal{H}^{\prime}$. Indeed, since the holomorphic Euler characteristic of
$\mathcal{H}^{\prime\prime}$ is positive, any Hodge grading piece of
$\mathcal{H}^{\prime\prime}$ has positive holomorphic Euler characteristic
(see [PPS, Lemma 15.1]).

By Subsection 2.1 (B), $\mathcal{H}^{j}f_{b*}\mathbb{R}[m]$ corresponds to the
variation of Hodge structure of $H^{j+\dim F}(Y_{t})$, where $t\in K$ belongs
to the smooth locus of $f_{b}$ and $Y_{t}$ is the fiber of $f_{b}$ over $t$.
Then $\mathcal{H}^{\prime}$ is a polarizable variation of complex Hodge
structure, whose fiber over a general point $t\in K$ is a sub-Hodge structure
of $H^{j+\dim F}(Y_{t},\mathbb{C})$ containing $H^{j}(Y_{t},K_{Y_{t}})$. Note
that $\mathcal{H}^{\prime}$ is trivial up to an étale cover of $K$.

Since $F$ is a connected component of $Y_{t}$, the composition of the Kodaira-
Spencer map with the cup product of variation of Hodge structures:

$T_{K,t}\rightarrow
H^{1}(F,T_{F})\rightarrow\mathrm{Hom}(H^{j}(F,K_{F}),H^{j+1}(F,\Omega_{F}^{\dim
F-1}))$

is zero.

We now consider the relative canonical model of $f_{b}$:

${\widetilde{f}}_{b}:{\widetilde{Y}}\rightarrow K.$

By the assumption, we know that a connected component of the fiber
${\widetilde{f}}_{b}$ over $t$ is the the canonical model $V$ of $F$. By
assumption, $V$ is a smooth projective variety which satisfies the
infinitesimal Torelli condition (C1’). Note that
$H^{j}(F,K_{F})=H^{j}(V,K_{V})$. and $H^{j+\dim V}(V,\mathbb{C})\subset
H^{j+\dim F}(F,\mathbb{C})$ is a direct summand, we have the commutative
diagram:

Thus the composition of maps

$T_{K,t}\rightarrow
H^{1}(V,T_{V})\rightarrow\mathrm{Hom}(H^{j}(V,K_{V}),H^{j+1}(V,\Omega_{V}^{\dim
V-1}))$

is also zero.

By assumption that $V$ satisfies the infinitesimal Torelli condition, the
Kodaira-Spencer map $T_{K,t}\rightarrow H^{1}(V,T_{V})$ induced by the family
${\widetilde{f}}_{b}$ is thus zero. Hence ${\widetilde{f}}_{b}$ is locally
isotrivial around $t\in K$ and thus the morphism $f_{b}:Y\rightarrow K$ is
birationally isotrivial.

In the following, we shall apply the condition (C2’) to show that for a
birational isotrivial morphism $f_{b}:Y\rightarrow K$, $\dim
V^{0}(R^{j}f_{b*}\omega_{Y})=0$ is absurd.

We assume without loss of generalities that $f_{b}:Y\rightarrow K$ is
primitive, i.e. $f_{b}$ does not factor through étale covers of $K$. We
consider the Stein factorization

$f_{b}:Y\xrightarrow{h_{1}}N\xrightarrow{h_{2}}K$

of $f_{b}$, where after birational modifications we may assume that $N$ is
smooth, $h_{1}$ is a fibration and $h_{2}$ is generically finite and
surjective. Since $K$ is a simple abelian variety and $f_{b}$ is primitive,
either $h_{2}$ is birational or $\deg h_{2}\geq 2$ and $N$ is of general type.

Note that $h_{1}$ is a birationally isotrivial fibration. Since $F$ is of
general type, its birational automorphism group is finite, we then take a
Galois cover $\rho:M\rightarrow N$ such that the flat base change
$M\times_{N}Y$ is birational to $M\times F$. If $\rho$ factors through an
abelian étale cover of $N$ induced by an isogeny $\widetilde{K}\rightarrow K$,
we replace $f_{b}:Y\rightarrow K$ by
$\widetilde{f_{b}}:\widetilde{Y}:=Y\times_{K}\widetilde{K}\rightarrow\widetilde{K}$.
Note that $V^{0}(\widetilde{f_{b}}_{*}\omega_{\widetilde{Y}})$ still consists
of finitely many points. Therefore, without loss of generalities, we assume
that $\rho:M\rightarrow N$ does not factor through abelian étale covers of $N$
induced by isogenies of $K$.

Let $G$ be the Galois group of the cover $\rho$. Taking an $G$-equivariant
resolution of singularities, we may assume that $M$ is smooth. We then
consider the canonical decomposition

$\rho_{*}\omega_{M}=\omega_{N}\bigoplus_{1_{G}\neq\chi\in\mathrm{Irr}(G)}\mathcal{V}_{\chi}$

with respect to the $G$-action, where $\mathrm{Irr}(G)$ denotes the set of
irreducible representations of $G$.

We claim that $h_{2*}\mathcal{V}_{\chi}$ is M-regular on $K$ for any
$1_{G}\neq\chi\in\mathrm{Irr}(G)$. If not, by the Chen-Jiang decomposition
theorem (see [PPS]), $h_{2*}\mathcal{V}_{\chi}$ has a numerically trivial line
bundle $P$ as a direct summand. Since $h^{\dim
M}(M,h_{2*}\rho_{*}\omega_{M})=h^{\dim M}(K,h_{2*}\omega_{N})=1$, $P$ is a
non-trivial torsion line bundle on $K$. Since $f_{b}$ is primitive,
$h_{2}^{*}P$ is also a non-trivial torsion line bundle on $N$. But $h^{\dim
M}(M,\omega_{M}\otimes\rho^{*}h_{2}^{*}P^{-1})\neq 0$, thus
$\rho^{*}h_{2}^{*}P=\mathcal{O}_{M}$, which implies that $\rho$ factors
through an abelian étale cover of $N$ induced by an isogeny of $K$, which is a
contradiction.

We are now ready to deduce a contradiction. We observe that $Y$ is birational
to the diagonal quotient $(M\times F)/G$, where $G$ acts on $F$ via an
injective homomorphism $G\rightarrow\mathrm{Bir}(F)$.

Since $G$ acts naturally on $H^{j}(F,K_{F})$, we then consider the
representation decomposition

$H^{j}(F,K_{F})=\bigoplus_{\chi\in\mathrm{Irr}(G)}V_{\chi}.$

By the condition (C2), $G\subset\mathrm{Bir}(F)$ acts faithfully on
$H^{j}(F,K_{F})$. Hence for some $\chi_{0}\neq 1_{G}$, $V_{\chi_{0}}$ is non-
zero.

From the commutative diagram:

we see that

$\displaystyle
R^{j}f_{b*}\omega_{Y}=h_{2*}R^{j}h_{1*}\omega_{Y}=h_{2*}R^{j}h_{1*}\omega_{(M\times
F)/G}$ $\displaystyle=h_{2*}(\rho_{*}\omega_{M}\otimes
H^{j}(F,K_{F}))^{G}=\bigoplus_{\chi,\chi^{\prime}\in\mathrm{Irr}(G)}(\mathcal{V}_{\chi}\otimes_{\mathbb{C}}V_{\chi^{\prime}})^{G}.$

It is known from the character theory of representation of finite groups that
for $\chi_{0}\neq 1_{G}$, there exists $\chi_{1}\neq
1_{G}\neq\in\mathrm{Irr}(G)$ such that
$(\mathcal{V}_{\chi_{1}}\otimes_{\mathbb{C}}V_{\chi_{0}})^{G}$ is a non-
trivial direct summand of
$\mathcal{V}_{\chi_{1}}\otimes_{\mathbb{C}}V_{\chi_{0}}$. Since
$\mathcal{V}_{\chi_{1}}$ is M-regular,
$(\mathcal{V}_{\chi_{1}}\otimes_{\mathbb{C}}V_{\chi_{0}})^{G}$ is also
M-regular on $K$. Thus $R^{j}f_{b*}\omega_{Y}$ has an M-regular sheaf as a
direct summand. This contradicts the assumption that $\dim
V^{0}(R^{j}f_{b*}\omega_{Y})=0$. ∎

######  Remark 2.2.

Perhaps the most useful case of Theorem 2.1 is when $j=0$, because the
conditions (C1) and (C2) have been extensively studied.

All smooth complete intersections of general type in projective spaces satisfy
$\mathrm{(C1)}$ (see [Pe] or [U]). Moreover, there is a fairly general
criterion for varieties of general type satisfying $\mathrm{(C1)}$ proved in
[Ki]: assume that $K_{F}=L^{\otimes m}$ for some $m\geq 1$, the base locus of
$|L|$ is of codimension $\geq 2$, and $h^{0}(F,\Omega_{F}^{\dim F-1}\otimes
L)\leq h^{0}(L)-2$, then

$H^{1}(F,T_{F})\rightarrow\mathrm{Hom}(H^{0}(F,K_{F}),H^{1}(F,\Omega_{F}^{\dim
F-1}))$

is injective.

For $\mathrm{(C2)}$, assuming that a finite subgroup $G\subset\mathrm{Bir}(F)$
acting trivially on $H^{0}(F,K_{F})$, then the canonical map of $F$ factors
through the quotient map by $G$. This is of course impossible if the canonical
system $|K_{F}|$ defines a birational map of $F$. Moreover, if $|K_{F}|$
defines a generically finite map of $F$, then the condition that the canonical
map of $F$ factors through the quotient map by $G$ implies that
$p_{g}(F)=p_{g}(F/G)$ (see [B, Théorème 3.1]), which is very restrictive.

In conclusion, we conclude that the assumptions (C1) and (C2) are satisfied by
a large number of varieties of general type, including all smooth complete
intersections of general type in projective spaces.

The proof of Theorem 2.1 provides more information.

######  Corollary 2.3.

Under the assumption of Theorem 2.1. Assume the sub-Hodge structure of
$H^{j+\dim F}(F,\mathbb{C})$ containing $H^{j}(F,K_{F})$ is simple and nonzero
and $f$ is a fibration onto its image, $R^{j}f_{*}\omega_{X}$ is M-regular.

###### Proof.

We have seen by Theorem 2.1 that $V^{0}(R^{j}f_{*}\omega_{X})$ generates
$\operatorname{Pic}^{0}(A)$. If $R^{j}f_{*}\omega_{X}$ is not M-regular, the
Chen-Jiang decomposition for $R^{j}f_{*}\omega_{X}$ has at least $2$ direct
summands. By Subsection 2.1, we know that the direct summands of
$R^{j}f_{*}\omega_{X}$ comes from the first grading pieces of the
decompositions of the Hodge module $M$. Restricting to a general fiber of $f$,
this implies that the sub-Hodge structure of $H^{j+\dim F}(F,\mathbb{C})$
containing $H^{j}(F,K_{F})$ is not simple, which is a contradiction. ∎

##  3\. Tricanonical maps of $X$ with $K_{F}$ sufficiently positive

######  Theorem 3.1.

Let $f:X\rightarrow A$ be a morphism from a projective variety to an abelian
variety. Let $F$ be an irreducible component of a general fiber of $f$. Assume
that $V^{0}(f_{*}\omega_{X})$ generates $\operatorname{Pic}^{0}(A)$ and
$|K_{F}|$ induces a birational map of $F$, then

  * (1)

the linear system $|3K_{X}+f^{*}P|$ induces a birational map of $X$ for
$P\in\operatorname{Pic}^{0}(A)$ general;

  * (2)

the linear system $|mK_{X}+f^{*}P|$ induces a birational map for $m\geq 4$ and
all $P\in\operatorname{Pic}^{0}(A)$.

###### Proof.

We first note that $(2)$ is easy. By the assumption that
$V^{0}(f_{*}\omega_{X})$ generates $\operatorname{Pic}^{0}(A)$ and Tirabassi’s
argument in [T], we know that $f_{*}(\omega_{X}^{2}\otimes\mathcal{I}_{x})$ is
M-regular for a general point $x\in X$. It follows immediately from standard
arguments that $|mK_{X}+f^{*}P|$ induces a birational map of $X$ for all
$m\geq 4$ and $P\in\operatorname{Pic}^{0}(X)$.

The proof of $(1)$ consists of several steps. Without loss of generalities, we
will assume that $f$ is primitive and $f(X)$ generates $A$.

First of all, by the decomposition theorem [PPS], we write

(4) $\displaystyle f_{*}\omega_{X}=\bigoplus_{p_{B_{i}}:A\rightarrow
B_{i}}\bigoplus_{j}p_{B_{i}}^{*}\mathcal{F}_{B_{i},j}\otimes
P_{B_{i},j}^{-1},$

where $p_{B_{i}}$ are surjective morphisms with connected fibers between
abelian varieties, $\mathcal{F}_{B_{i},j}$ are M-regular sheaves on $B_{i}$,
$P_{B_{i},j}$ are torsion line bundles. Then

$V^{0}(\omega_{X})=\bigcup_{i,j}\big{(}P_{B_{i},j}+\operatorname{Pic}^{0}(B_{i})\big{)},$

where $p_{B_{i}}$ and $j$ runs through the direct summands in the
decomposition formula (4).

Let $p:=\prod_{B_{i}}p_{B_{i}}:A\rightarrow\prod_{i}B_{i}$. By the assumption
that $V^{0}(f_{*}\omega_{X})$ generates $\operatorname{Pic}^{0}(A)$, we know
that $p$ is finite onto its image. Let $K$ be the image of $p$. Then
$p:A\rightarrow K\hookrightarrow\prod_{i}B_{i}$ is an isogeny between $A$ and
$K$.

Since $M$-regular sheaves are continuously globally generated, we conclude
that the twisted evaluation map

(5) $\displaystyle\bigoplus_{U_{ij}}\bigoplus_{Q\in
U_{ij}}H^{0}(A,f_{*}\omega_{X}\otimes Q)\otimes Q^{-1}\rightarrow
f_{*}\omega_{X}$

is surjective, where $U_{ij}\subset P_{B_{i},j}+\operatorname{Pic}^{0}(B_{i})$
are open subsets.

Step 1. $|3K_{X}+f^{*}P|$ for any $P\in\operatorname{Pic}^{0}(A)$, separates
two general points on the same general fiber of $f$.

Let $x,y\in X$ be two general points in the same general fiber $X_{a}$ of $f$
for $a\in A$. Since $V^{0}(f_{*}\omega_{X})$ contains positive dimensional
components, $h^{0}(X,2K_{X})=h^{0}(X,2K_{X}+f^{*}Q)\geq 2$ for any
$Q\in\operatorname{Pic}^{0}(A)$.

Fix $P\in\operatorname{Pic}^{0}(X)$.

If for some $B_{i}$ and $P_{B_{i},j}$ as in (4) and $Q\in
P_{B_{i},j}+{\widehat{B}}_{i}$ general, the linear system

$|2K_{X}+f^{*}P-f^{*}Q|$

separates $x$ and $y$, then for $Q\in P_{B_{i},j}+{\widehat{B}}_{i}$ general,
there exists $D_{1}\in|2K_{X}+f^{*}P-f^{*}Q|$ such that $x\in D$ and $y\notin
D$. Since $Q\in P_{B_{i},j}+{\widehat{B}}_{i}$ general, there exists
$D_{2}\in|K_{X}+f^{*}Q|$ containing neither $x$ nor $y$, then
$D_{1}+D_{2}\in|3K_{X}+f^{*}P|$ separating $x$ and $y$.

We then assume that for all $B_{i}$ and $P_{B_{i},j}$ and $Q\in
P_{B_{i},j}+\operatorname{Pic}^{0}(B_{i})$, the corresponding linear system

$|2K_{X}+f^{*}P-f^{*}Q|$

cannot separate $x$ and $y$. Since $|K_{F}|$ induces a birational map of $F$,
we conclude from (5) that there exist positive integers $N_{ij}$, $Q_{ijk}\in
P_{B_{i},j}+\operatorname{Pic}^{0}(B_{i})$ general for $1\leq k\leq N_{ij}$,
sections $s_{ijk}\in H^{0}(X,K_{X}+f^{*}Q_{ijk})$ such that the section

$s:=\sum_{ij}\sum_{1\leq k\leq N_{ij}}s_{ijk}\mid_{X_{a}}\in
H^{0}(X_{a},K_{X_{a}})$

separating $x$ and $y$, i.e. $x\in D_{s}$ and $y\in D_{s}$, where $D_{s}$ is
the corresponding divisor of $s$ on $X_{a}$. We choose $s^{\prime}_{ijk}\in
H^{0}(X,2K_{X}+f^{*}P-f^{*}Q_{ijk})$ which take the same nonzero value at both
$x$ and $y$ in an affine charts of $X_{a}$ containing both $x$ and $y$. Note
that $s_{ilk}^{\prime}$ exists since $\varphi_{ijk}(x)=\varphi_{ijk}(y)$,
where $\varphi_{ijk}$ is the rational map induced by
$|2K_{X}+f^{*}P-f^{*}Q_{ijk}|$.

Then

$\sum_{ij}\sum_{1\leq k\leq N_{ij}}s_{ijk}\cdot s^{\prime}_{ijk}\in
H^{0}(X,3K_{X}+f^{*}P)$

separating $x$ and $y$. Hence the twisted tricanonical map induced by
$|3K_{X}+f^{*}P|$ separates two general points on the same general fiber of
$f$.

Step 2. $|3K_{X}+f^{*}P|$ induces a generically finite map from $X$ onto its
image for all $P\in\operatorname{Pic}^{0}(A)$.

Assume the contrary, there exists a curve $C$ through a general point of $X$
contracting by the map $|3K_{X}+f^{*}P|$. Then the rank of the restriction map

$H^{0}(X,3K_{X}+f^{*}P)\rightarrow H^{0}(C,(3K_{X}+f^{*}P)|_{C})$

is $1$. By Step 1., $f|_{C}:C\rightarrow A$ is generically finite onto its
image. By the assumption, there exists $p_{B}$ appearing in the decomposition
formula (4) such that $p_{B}\circ f|_{C}:C\rightarrow B$ is generically finite
onto its image. Let $T=P+\operatorname{Pic}^{0}(B)$ be a subset of
$V^{0}(f_{*}\omega_{X})$. Then for $Q\in T$ general, the image $V_{Q}$ of the
restriction map

$H^{0}(X,K_{X}+f^{*}Q)\rightarrow H^{0}(C,(K_{X}+f^{*}Q)|_{C})$

is not zero. We also denote by $V_{Q}^{\prime}$ the image of the restriction
map

$H^{0}(X,2K_{X}+f^{*}P-f^{*}Q)\rightarrow
H^{0}(C,(2K_{X}+f^{*}P-f^{*}Q)|_{C}).$

We observe that the image of
$\operatorname{Pic}^{0}(B)\rightarrow\operatorname{Pic}^{0}(C)$ is positive
dimensional.

The image of the restriction map

$H^{0}(X,3K_{X}+f^{*}P)\rightarrow H^{0}(C,(3K_{X}+f^{*}P)|_{C})$

contains the map of natural multiplication map

$\bigcup_{Q\in T}V_{Q}\otimes V_{-Q}^{\prime}\rightarrow
H^{0}(C,(3K_{X}+f^{*}P)|_{C}).$

Thus the rank of the restriction map

$H^{0}(X,3K_{X}+f^{*}P)\rightarrow H^{0}(C,(3K_{X}+f^{*}P)|_{C})$

is bigger than $1$, which is a contradiction.

Step 3. For $P\in\operatorname{Pic}^{0}(A)$ general, let $x,y\in X$ be two
general points such that $p\circ f(x)\neq p\circ f(y)\in K$. Then the map
$|3K_{X}+f^{*}P|$ separates $x$ and $y$.

For a general point $x\in X$, for each component
$P_{B_{i},j}+{\widehat{B}}_{i}$ appearing in (4),
$x\notin\mathrm{Bs}(|K_{X}+f^{*}Q|)$ for $Q\in P_{B_{i},j}+{\widehat{B}}_{i}$
general. Then $x\notin\mathrm{Bs}(|2K_{X}+f^{*}Q|)$ for all
$Q\in{\widehat{T}}_{i,j}:=2P_{B_{i},j}+{\widehat{B}}_{i}$. Considering
$g_{i}:=p_{B_{i}}\circ f:X\rightarrow B_{i},$ we know that
$g_{i*}(\omega_{X}^{2}\otimes f^{*}Q)$ is IT0 for all
$Q\in\operatorname{Pic}^{0}(A)$ (see [LPS]). Hence from the short exact
sequence

$0\rightarrow g_{i*}(\omega_{X}^{2}\otimes
f^{*}Q\otimes\mathcal{I}_{x})\rightarrow g_{i*}(\omega_{X}^{2}\otimes
f^{*}Q)\rightarrow\mathbb{C}_{x}\rightarrow 0,$

we conclude that $g_{i*}(\omega_{X}^{2}\otimes f^{*}Q\otimes\mathcal{I}_{x})$
is again IT0 on $B_{i}$, for all $Q\in{\widehat{T}}_{ij}$. Note that there
exists an open subset $(x,Q)\in\mathcal{U}_{i}\subset
X\times\operatorname{Pic}^{0}(X)$ parametrizing the flat family of sheaves
$g_{i*}(\omega_{X}^{2}\otimes f^{*}Q\otimes\mathcal{I}_{x})$ on $B_{i}$. Being
IT0 is an open condition, hence there exists a non empty open subset
$\mathcal{U}_{i}^{\prime}$ of $\mathcal{U}_{i}$ parametrizing IT0 sheaves. Let
$\mathcal{V}$ be the intersection of all $\mathcal{U}_{i}^{\prime}$ for all
$p_{B_{i}}$ appearing in the decomposition formula. Then $\mathcal{V}$ is
again a non-empty open subset of $X\times\operatorname{Pic}^{0}(A)$. Then for
a general $P\in\operatorname{Pic}^{0}(A)$, there exists an open subset $W$ of
$X$ such that $g_{i*}(\omega_{X}^{2}\otimes f^{*}P\otimes\mathcal{I}_{x})$ is
IT0 for all $g_{i}$ and all $x\in W$. Hence for
$P\in\operatorname{Pic}^{0}(A)$ general, there exists an open subset
$W^{\prime}$ of $X$ such that

$g_{i*}(\omega_{X}^{2}\otimes f^{*}P\otimes
f^{*}P_{B_{i},j}^{-1}\otimes\mathcal{I}_{x})$

is IT0 for all $i,j$ and all $x\in W^{\prime}$.

We now take $W^{\prime\prime}$ to be the open subset of $W^{\prime}$ such that
for any $x\in W^{\prime\prime}$, $x\notin\mathrm{Bs}(|K_{X}+f^{*}P|)$ for any
$i,j$ and $P\in P_{B_{i},j}+{\widehat{B}}_{i}$ general. Then for $x,y\in
W^{\prime\prime}$ and $p(x)\neq p(y)$, there exists $p_{B_{i}}:A\rightarrow
B_{i}$ such that $g_{i}(x)\neq g_{i}(y)$. Since $g_{i*}(\omega_{X}^{2}\otimes
f^{*}P\otimes f^{*}P_{B_{i},j}^{-1}\otimes\mathcal{I}_{x})$ is IT0, for
$Q\in\operatorname{Pic}^{0}(B_{i})$ general, there exists a divisor
$D\in|2K_{X}+f^{*}P-f^{*}P_{B_{i},j}+f^{*}Q|$ separating $x$ and $y$ and hence
the linear system $|3K_{X}+f^{*}P|$ also separates $x$ and $y$.

Step 4. Conclusion.

We now argue that for $P\in\operatorname{Pic}^{0}(A)$ general,
$|3K_{X}+f^{*}P|$ induces a birational map of $X$. Assume the contrary, let
$\varphi:X\dashrightarrow Y\subset\mathbb{P}^{N}$ be the map induced by
$|3K_{X}+f^{*}P|$, where $Y$ is the image of $\varphi$. Note that there exists
a map $f_{Y}:Y\dashrightarrow K$ such that we have the commutative diagram:

Indeed, for $x\in Y$ general, $\varphi^{-1}(y)$ consists of finitely many
points and by Step 1 and Step 2, $p\circ f(\varphi^{-1}(y))$ is a single point
of $K$. Hence we can define the above map $Y\dashrightarrow K$.

After birational modifications of $X$ and $Y$, we may assume that $Y$ is
smooth, $\varphi$ is an morphism and $Y\rightarrow K$ is also a morphism.
After replacing $K$ by a suitable étale cover of it, we may alao assume that
$Y\rightarrow K$ is primitive. Since $\varphi$ maps a general fiber of $f$
birationally onto its image, we conclude that $\varphi:X\rightarrow Y$ is
birationally equivalent to the étale morphism $Y\times_{K}A\rightarrow Y$
induced by base change of $p:A\rightarrow K$. But then

$\varphi_{*}\omega_{X}^{3}=\bigoplus_{Q\in\ker p^{*}}\omega_{Y}^{3}\otimes Q.$

We then take $P^{\prime}\in\operatorname{Pic}^{0}(K)$ such that
$p^{*}P^{\prime}=P$. Then

$H^{0}(X,3K_{X}+f^{*}P)=\bigoplus_{Q\in\ker
p^{*}}H^{0}(Y,3K_{Y}+f_{Y}^{*}P^{\prime}+f_{Y}^{*}Q).$

Note that $H^{0}(Y,3K_{Y}+f_{Y}^{*}P^{\prime}+f_{Y}^{*}Q)\neq 0$ for all $Q$.
Hence it is absurd that $\varphi$ is birationally equivalent to the base
change of $p$.

∎

Combining Theorem 1.3 and Theorem 3.1, we have

######  Corollary 3.2.

Let $f:X\rightarrow A$ be a morphism from a smooth projective variety to an
abelian variety. Let $F$ be a connected component of a general fiber of $f$.
If $F$ satisfies the birationally infinitesimal Torelli condition
$\mathrm{(C1)}$ and $|K_{F}|$ induces a birational map of $F$, then

  * (1)

the linear system $|3K_{X}+f^{*}P|$ induces a birational map of $X$ for
$P\in\operatorname{Pic}^{0}(A)$ general;

  * (2)

the linear system $|mK_{X}+f^{*}P|$ induces a birational map for $m\geq 4$ and
all $P\in\operatorname{Pic}^{0}(A)$.

##  4\. $K_{F}$ is not so positive

The goal of this section is to prove Theorem 1.4, which partially generalize
the main result of [CCCJ].

######  Theorem 4.1.

Let $f:X\rightarrow A$ be the Albanese morphism of a smooth projective variety
of general type $X$ to its Albanese variety $A$. Let $F$ be a connected
component of a general fiber of $f$. Assume that $F$ is a surface with
$p_{g}(F)>0$ and $q(F)=0$, $|5K_{X}+P|$ induces a birational map of $X$ for
all $P\in\operatorname{Pic}^{0}(A)$.

###  4.1. Set-up

In this subsection we denote by $f:X\rightarrow A$ the Albanese morphism of
$X$ and let $F$ be a connected component of a general fiber of $f$ over its
image. For a line bundle $L$ on $X$, we define
$V^{0}(L):=\\{Q\in\operatorname{Pic}^{0}(A)\mid h^{0}(X,L\otimes Q)\neq 0\\}$.
For $Q\in\operatorname{Pic}^{0}(A)$, we can regard it as a line bundle on $X$
or on $A$, or just a point of $\operatorname{Pic}^{0}(A)$ when there is no
confusion.

Let $m_{2}\geq m_{1}\geq 2$ be integers such that

$|m_{1}K_{F}|\neq\emptyset$

and $|m_{2}K_{F}|$ induces a birational map of $F$. We explore some general
criteria of the birationality of $|(m_{1}+m_{2})K_{X}|$.

######  Remark 4.2.

If $F$ is a surface of general type, then we can take $m_{1}=2$ and $m_{2}=3$
(see [BPV]), unless its minimal model $F_{0}$ satisfies either
$(K_{F_{0}}^{2},p_{g}(F_{0}))=(1,2)$ or $(2,3)$. In the latter cases, the
tricanonical map of $F$ is of degree $2$.

We recall that for $m\geq 2$ and $h^{0}(F,mK_{F})\neq 0$, by the Theroem of
Lombardi-Popa-Schnell [LPS], $f_{*}\omega_{X}^{m}$ is IT0.

The starting point is the following lemma, which is essentially [CCCJ,
Propositiom 3.2 and Proposition 3.3].

######  Proposition 4.3.

Let $x\in X$ be a general point and $\mathcal{I}_{x}\subset\mathcal{O}_{X}$
the ideal sheaf of this point. Assume that either
$f_{*}(\omega_{X}^{m_{1}}\otimes\mathcal{I}_{x})$ or
$f_{*}(\omega_{X}^{m_{2}}\otimes\mathcal{I}_{x})$ is M-regular, then
$|(m_{1}+m_{2})K_{X}+P|$ induces a birational map of $X$ for all
$P\in\operatorname{Pic}^{0}(A)$.

###### Proof.

The argument is the same as in [CCCJ, Propositiom 3.2 and Proposition 3.3].
For reader’s convenience we repeat the argument here. We need to show that
$|(m_{1}+m_{2})K_{X}+P|$ separates two general points $x$ and $y$.

We first assume that $f_{*}(\omega_{X}^{m_{2}}\otimes\mathcal{I}_{x})$ is
M-regular and let $F$ be the component of $a_{X}^{-1}a_{X}(x)$ containing $x$.
Then by [PP], M-regular sheaves are continuously globally generated, which
implies that for any open subset $U$ of $\operatorname{Pic}^{0}(A)$, the
evaluation map

$\bigoplus_{Q\in U}H^{0}(X,\mathcal{I}_{x}(m_{2}K_{X})\otimes Q)\rightarrow
H^{0}(F,\mathcal{I}_{x}(m_{2}K_{F}))$

is surjective. Since $f_{*}\omega_{X}^{m_{1}}$ is IT0, $|m_{1}K_{X}+P-Q|$ is
nonzero for any $Q\in\operatorname{Pic}^{0}(A_{X})$. Hence
$|(m_{1}+m_{2})K_{X}+P|$ separates $x$ and $y$ if $y\notin F$. By the
assumption that $|m_{2}K_{F}|$ induces a birational map of $F$, we conclude as
well that $|(m_{1}+m_{2})K_{X}+P|$ separates $x$ and $y$ if $y\in F$.

We then assume that $f_{*}(\omega_{X}^{m_{1}}\otimes\mathcal{I}_{x})$ is
M-regular. The same argument shows that $|(m_{1}+m_{2})K_{X}+P|$ separates $x$
and $y$ either when $y\notin F$ or when $y\in F$ and $x,y\in F$ can be
separated by $|m_{1}K_{F}|$. If $x,y\in F$ cannot be separated by
$|m_{1}K_{F}|$, then take a section $f\in H^{0}(F,m_{2}K_{F})$ such that its
corresponding divisor contains $x$ and does not contain $y$. Since
$f_{*}\omega_{X}^{m_{2}}$ is IT0, for $U\subset\operatorname{Pic}^{0}(A_{X})$
open, there exists $P_{i}\in U$, $1\leq i\leq N$ and $f_{i}\in
H^{0}(X,m_{2}K_{X}+P+P_{i})$ such that

$\sum_{i}f_{i}\mid_{F}=f.$

We can pick $g_{i}\in H^{0}(X,m_{1}K_{X}-P_{i})$ which take the same nonzero
value at both $x$ and $y$ in a suitable affine open neighborhood of $x$ and
$y$. Then the divisor of $F:=\sum_{i}f_{i}g_{i}\in
H^{0}(X,(m_{1}+m_{2})K_{X}+P)$ separates $x$ and $y$. ∎

######  Corollary 4.4.

Assume that $V^{0}(K_{X})$ generates $\operatorname{Pic}^{0}(X)$. Then
$|(m_{2}+2)K_{X}+P|$ induces a birational map of $X$ for all
$P\in\operatorname{Pic}^{0}(A)$.

###### Proof.

Since $V^{0}(K_{X})$ generates $\operatorname{Pic}^{0}(X)$, we know that
$f_{*}(\omega_{X}^{2}\otimes\mathcal{I}_{x})$ is M-regular by [T]. We then
conclude by Proposition 4.3. ∎

###  4.2. The Abel-Jacobi maps induced by fixed components of pluricanonical
systems.

We assume in this subsection that for some $m\geq 2$ and $h^{0}(F,mK_{F})>0$,
$f_{*}(\omega_{X}^{m}\otimes\mathcal{I}_{x})$ is not M-regular for $x\in X$
general. Since [BLNP], we understand that the fixed component of $|mK_{X}+P|$
moves along with $P$ and the induced Abel-Jacobi map is quite interesting. We
extend some related results of [BLNP] and [CCCJ] in this setting.

Following the same argument in [CCCJ, Section 4], we consider the closed
subscheme of $X\times\operatorname{Pic}^{0}(A)$:

$B_{m}:=\\{(x,P)\mid x\in\mathrm{Bs}|mK_{X}+P|\\}\subset
X\times\operatorname{Pic}^{0}(A).$

Note that the defining ideal of $B_{m}$ is given by the relative evaluation
map of $\omega_{X}^{m}\otimes(f\times Id_{A})^{*}\mathcal{P}_{A}$, where
$\mathcal{P}_{A}$ is the Poincaré bundle on
$A\times\operatorname{Pic}^{0}(A)$.

By the assumption that $f_{*}(\omega_{X}^{m}\otimes\mathcal{I}_{x})$ is not
M-regular, $\dim B_{m}=\dim X+\dim\operatorname{Pic}^{0}(A)-1$ and $B_{m}$
dominates $X$ via the first projection (see for instance [CCCJ, Lemma 4.2]).
Let $\mathcal{D}_{m}$ be the union of the components of $B_{m}$ which are
divisors of $X\times\operatorname{Pic}^{0}(A)$ and dominate $X$ via the first
projection.

For $P\in\operatorname{Pic}^{0}(A)$ general, we have

(7) $\displaystyle|mK_{X}+P|=|M_{P}|+D_{P}+E_{m},$

where $D_{P}$ is the fiber of
$\mathcal{D}_{m}\rightarrow\operatorname{Pic}^{0}(A)$ over the point $P$, $E$
is the common fixed divisor, and $|M_{P}|$ has no fixed divisor.

We note that $\dim|M_{P}|=p_{m}(X)-1\geq 1$ if $p_{m}(X)\geq 2$ or $M_{P}=0$
when $p_{m}(X)=1$.

Since $f:X\rightarrow A$ is the Albanese morphism of $X$, we denote by
$\Phi_{m}:\operatorname{Pic}^{0}(A)\rightarrow\operatorname{Pic}^{0}(A)$ the
Abel-Jacobi map induced by $\mathcal{D}$, namely we have

$\mathcal{O}_{X}(D_{P_{1}}-D_{P_{2}})=\Phi_{m}(P_{1})-\Phi_{m}(P_{2})$

for $P_{1}$, $P_{2}\in\operatorname{Pic}^{0}(A)$ general and $\Phi_{m}(0)=0$.
We denote by ${\widehat{B}}_{m}\subset\operatorname{Pic}^{0}(A)$ the image of
$\Phi$ and ${\widehat{C}}_{m}\subset\operatorname{Pic}^{0}(A)$ the kernel of
$\Phi_{m}$.

######  Lemma 4.5.

[BLNP, Lemma 5.1] We have $\Phi_{m}\circ\Phi_{m}=\Phi_{m}$ and hence
$\operatorname{Pic}^{0}(A)={\widehat{B}}_{m}+{\widehat{C}}_{m}\simeq{\widehat{B}}_{m}\times{\widehat{C}}_{m}$.

###### Proof.

Let $P_{1},P_{2}\in\operatorname{Pic}^{0}(A)$ be general, then

$M_{P_{1}}+D_{P_{2}}+E_{m}\sim mK_{X}+P_{1}+\Phi_{m}(P_{2}-P_{1}).$

Since $f_{*}\omega_{X}^{m}$ is IT0, $h^{0}(X,mK_{X}+Q)=p_{m}(X)$ for any
$Q\in\operatorname{Pic}^{0}(A)$. Thus $D_{P_{2}}+E_{m}$ is the fixed part of
$|mK_{X}+P_{1}+\Phi_{m}(P_{2}-P_{1})|$. Thus
$\Phi_{m}\circ\Phi_{m}(P_{2}-P_{1})=\Phi_{m}(P_{2}-P_{1})$ and we have
$\Phi_{m}\circ\Phi_{m}=\Phi_{m}$. Hence
$\operatorname{Pic}^{0}(A)=\mathrm{Im}(\Phi_{m})+\mathrm{Im}(\mathrm{Id}_{A}-\Phi_{m})=\mathrm{Im}(\Phi_{m})+\mathrm{ker}(\Phi_{m})={\widehat{B}}_{m}+{\widehat{C}}_{m}\simeq{\widehat{B}}_{m}\times{\widehat{C}}_{m}$.
∎

Since $\mathcal{D}_{m}$ dominates $\operatorname{Pic}^{0}(A)$,
${\widehat{B}}_{m}$ is a positive dimensional abelian subvariety of
$\operatorname{Pic}^{0}(A)$. Given $P\in\operatorname{Pic}^{0}(A)$, we write
$P_{B}=\Phi_{m}(P)\in{\widehat{B}}_{m}$ and
$P_{C}=P-P_{B}\in{\widehat{C}}_{m}$.

Let $D=D_{P}-\Phi_{m}(P)$ for $P\in\operatorname{Pic}^{0}(A)$ general. Note
that $D$ is independent of the choice of general $P$ by the definition of
$\Phi_{m}$. Consider the composition of morphisms

$X\times\operatorname{Pic}^{0}(A)\xrightarrow{f\times\mathrm{Id}_{\operatorname{Pic}^{0}(A)}}A\times\operatorname{Pic}^{0}(A)\xrightarrow{\mathrm{Id}_{A}\times\Phi_{m}}A\times\operatorname{Pic}^{0}(A),$

we let $\mathcal{P}_{B}$ on $X\times\operatorname{Pic}^{0}(A)$ be the pull-
back of the Poincaré bundle $A\times\operatorname{Pic}^{0}(A)$ via this
composition of morphisms. Then by the see-saw principle, there exists a
divisor $\Theta$ on $\operatorname{Pic}^{0}(A)$ such that we have an
isomorphism of line bundles

$\mathcal{O}_{X\times\operatorname{Pic}^{0}(A)}(\mathcal{D}_{m})\simeq
p_{1}^{*}\mathcal{O}_{X}(D)\otimes
p_{2}^{*}\mathcal{O}_{\operatorname{Pic}^{0}(A)}(\Theta)\otimes\mathcal{P}_{B},$

where $p_{1}$ and $p_{2}$ are respectively the projections from
$X\times\operatorname{Pic}^{0}(A)$ to $X$ and $\operatorname{Pic}^{0}(A)$.

For $P\in\operatorname{Pic}^{0}(A)$ general, $D_{P}$ is linear equivalent to
$D+P_{B}$. Let $M=mK_{X}-D-E_{m}$. Then $M_{P}$ is linear equivalent to
$M+P_{C}$ for $P\in\operatorname{Pic}^{0}(A)$ general by (7).

######  Lemma 4.6.

We have $V^{0}(M+E_{m})=V^{0}(M)={\widehat{C}}_{m}$ and
$h^{0}(X,M+P_{C}+E_{m})=h^{0}(X,M+P_{C})=p_{m}(X)$ for any
$P_{C}\in{\widehat{C}}_{m}$. Similarly,
$V^{0}(D+E_{m})=V^{0}(D)={\widehat{B}}_{m}$ and
$h^{0}(X,D+P_{B}+E_{m})=h^{0}(X,D+P_{B})=1$ for any
$P_{B}\in{\widehat{B}}_{m}$.

###### Proof.

We know that for $P\in\operatorname{Pic}^{0}(A)$,
$h^{0}(X,\mathcal{O}_{X}(mK_{X}+P))=p_{m}(X)$ is constant. For $P$ general, we
have

$|mK_{X}+P|=|M_{P}|+D_{P}+E_{m},$

and $M_{P}\sim M+P_{C}$ and $D_{P}\sim D+P_{B}$. Thus $V^{0}(M+E_{m})\supset
V^{0}(M)\supset{\widehat{C}}_{m}$ and $V^{0}(D+E_{m})\supset
V^{0}(D)\supset{\widehat{B}}_{m}$. By semicontinuity and the fact that
$h^{0}(X,\mathcal{O}_{X}(mK_{X}+P))$ is constant,
$h^{0}(X,M+P_{C}+E_{m})=h^{0}(X,M+P_{C})=p_{m}(X)$ for any
$P_{C}\in{\widehat{C}}_{m}$ and $h^{0}(X,D+P_{B}+E_{m})=h^{0}(X,D+P_{B})=1$
for any $P_{B}\in{\widehat{B}}_{m}$.

In particular, $\mathcal{D}$ is flat over $\operatorname{Pic}^{0}(A)$ and
$D_{P}$ is the unique effective divisor linearly equivalent to $D+P_{B}$ for
any $P\in\operatorname{Pic}^{0}(A)$. Then, for any
$P\in\operatorname{Pic}^{0}(A)$,

$|mK_{X}+P|=|M+P_{C}|+D_{P_{B}}+E_{m}.$

It suffices to show that $V^{0}(M+E_{m})={\widehat{C}}_{m}$ and
$V^{0}(D+E_{m})={\widehat{B}}_{m}$. We argue by contradiction.

Assume that there exists $Q=Q_{B}+Q_{C}\in\operatorname{Pic}^{0}(A)$ with
$Q_{B}$ non-trivial such that $h^{0}(X,M+Q+E_{m})>0$. Then for
$P_{B}\in{\widehat{B}}_{m}$, we have

$|M+Q+E_{m}|+D_{P_{B}}\subset|mK_{X}+Q+P_{B}|=|M+Q_{C}|+D_{Q_{B}+P_{B}}+E_{m}.$

Thus $D_{Q_{B}+P_{B}}\preceq\mathrm{Fix}|M+Q+E_{m}|+D_{P_{B}}$. Let
$P_{B}\in{\widehat{B}}_{m}$ be general, we then have $D_{Q_{B}+P_{B}}\preceq
D_{P_{B}}$, which is absurd.

Assume that there exists $Q=Q_{B}+Q_{C}\in\operatorname{Pic}^{0}(A)$ with
$Q_{C}$ non-trivial such that $h^{0}(X,D+Q+E_{m})>0$. Let
$\widetilde{D}_{Q}\in|D+Q+E_{m}|$. Then for $P_{C}\in{\widehat{C}}_{m}$, we
have

$\widetilde{D}_{Q}+|M+P_{C}|\subset|mK_{X}+Q+P_{C}|.$

By dimension reason, the two linear systems are equal. Thus

$\widetilde{D}_{Q}+|M+P_{C}|=|mK_{X}+Q+P_{C}|=|M+P_{C}+Q_{C}|+D_{Q_{B}}+E_{m}.$

For $P_{C}\in{\widehat{C}}_{m}$ general, both $|M+P_{C}|$ and
$|M+P_{C}+Q_{C}|$ have no fixed part. Thus $\widetilde{D}_{Q}=D_{Q_{B}}+E_{m}$
is the fixed part of $|mK_{X}+Q+P_{C}|$, which is absurd. ∎

######  Proposition 4.7.

The natural morphism

$\displaystyle f_{*}\mathcal{O}_{X}(M)\otimes
f_{*}\mathcal{O}_{X}(D)\xrightarrow{\cdot E_{m}}f_{*}\mathcal{O}_{X}(mK_{X})$

is surjective. Hence restricting to $F$, we have the surjective map

$\displaystyle H^{0}(F,M|_{F})\otimes H^{0}(F,D|_{F})\xrightarrow{\cdot
E_{m}|_{F}}H^{0}(F,mK_{F})$

###### Proof.

We know that $f_{*}\omega_{X}^{m}$ is IT0 and hence is M-regular. Thus by
[PP], the evaluation map

$\bigoplus_{Q\in\operatorname{Pic}^{0}(A)}H^{0}(A,f_{*}\mathcal{O}_{X}(mK_{X})\otimes
Q)\otimes Q^{-1}\rightarrow f_{*}\mathcal{O}_{X}(mK_{X})$

is surjective.

For $Q\in\operatorname{Pic}^{0}(X)$, we write $Q=Q_{B}+Q_{C}$ with
$Q_{B}\in{\widehat{B}}_{m}$ and $Q_{C}\in{\widehat{C}}_{m}$. By Lemma 4.6, we
have

$|mK_{X}+Q|=|M+Q_{C}|+D_{Q_{B}}+E_{m}.$

Hence

$H^{0}(f_{*}\mathcal{O}_{X}(M)\otimes Q_{C})\otimes
H^{0}(f_{*}\mathcal{O}_{X}(D)\otimes Q_{B})\xrightarrow{\cdot
E_{m}}H^{0}(f_{*}\mathcal{O}_{X}(mK_{X})\otimes Q)$

is an isomorphism.

The composed surjective map

$\displaystyle\bigoplus_{Q\in\operatorname{Pic}^{0}(A)}\big{(}H^{0}(f_{*}\mathcal{O}_{X}(M)\otimes
Q_{C})\otimes H^{0}(f_{*}\mathcal{O}_{X}(D)\otimes Q_{B})\big{)}\otimes
Q^{-1}\rightarrow$ $\displaystyle f_{*}\mathcal{O}_{X}(mK_{X})$

factors through the map

(8) $\displaystyle f_{*}\mathcal{O}_{X}(M)\otimes
f_{*}\mathcal{O}_{X}(D)\rightarrow f_{*}\mathcal{O}_{X}(mK_{X}).$

Hence (8) is also surjective. Thus considering the above map restricted on
$F$, we have the surjective map

$H^{0}(F,M|_{F})\otimes H^{0}(F,D|_{F})\rightarrow H^{0}(F,mK_{F}).$

∎

We also need to consider the irreducible components of $\mathcal{D}_{m}$.
Write $\mathcal{D}_{m}=\sum_{j=1}^{s}a_{j}\widetilde{\mathcal{D}}_{j}$ as the
sum of irreducible components and let

$\Phi_{mj}:\operatorname{Pic}^{0}(A)\rightarrow\operatorname{Pic}^{0}(A)$

be the Abel-Jacobi map induced by $\widetilde{\mathcal{D}}_{j}$, namely
$\Phi_{mj}(P)=\mathcal{O}_{X}(\widetilde{D}_{jP}-\widetilde{D}_{j})$, where
$\widetilde{D}_{jP}$ is the fiber of $\widetilde{\mathcal{D}}_{j}$ over
$P\in\operatorname{Pic}^{0}(A)$ and $\widetilde{D}_{j}$ is the fiber over the
neutral point of $\operatorname{Pic}^{0}(A)$. Let ${\widehat{B}}_{mj}$ be the
image of $\Phi_{mj}$. Since $\widetilde{\mathcal{D}}_{j}$ dominates
$\operatorname{Pic}^{0}(A)$, $\dim{\widehat{B}}_{mj}>0$ for each $1\leq j\leq
s$.

######  Proposition 4.8.

We have

  * (1)

$\mathcal{D}$ is reduced, i.e. $a_{j}=1$ for $1\leq j\leq s$ and
$\mathcal{O}_{X}(\widetilde{D}_{j_{1}}-\widetilde{D}_{j_{2}})\notin\operatorname{Pic}^{0}(A)=\operatorname{Pic}^{0}(X)$
for $1\leq j_{1}\neq j_{2}\leq s$;

  * (2)

$\Phi_{mj}\circ\Phi_{mj}=\Phi_{mj}$ and $\Phi_{mj_{1}}\circ\Phi_{mj_{2}}=0$
for $j_{1}\neq j_{2}$ and
${\widehat{B}}_{i}\simeq{\widehat{B}}_{m1}\times\cdots\times{\widehat{B}}_{ms}$;

###### Proof.

Note that for any $P\in{\widehat{B}}_{mj}$, $|\widetilde{D}_{j}+P|$ is non-
empty by the definition of $\Phi_{mj}$. Thus if $a_{j}>1$, we have
$\dim|a_{j}\widetilde{D}_{j}+P|>1$, which contradicts the fact that
$h^{0}(X,D+P)=1$ for any $P\in\operatorname{Pic}^{0}(A)$. Thus $\mathcal{D}$
is reduced.

Similarly, if $\widetilde{D}_{j_{1}}\sim\widetilde{D}_{j_{2}}+Q$ for some
$Q\in\operatorname{Pic}^{0}(A)$, then
$h^{0}(X,\widetilde{D}_{j_{1}}+\widetilde{D}_{j_{2}}+Q)>1$, which is absurd.

We now consider $(2)$. For $P_{1},P_{2}\in\operatorname{Pic}^{0}(A)$ general,

$|mK_{X}+P_{1}+\Phi_{mj}(P_{2}-P_{1})|\supset|M_{P_{1}}|+\sum_{k\neq
j}\widetilde{D}_{kP_{1}}+\widetilde{D}_{jP_{2}}+E_{m}.$

By the dimension reason, the two linear systems are the same. Since
$\mathcal{O}_{X}(\widetilde{D}_{j_{1}}-\widetilde{D}_{j_{2}})\notin\operatorname{Pic}^{0}(A)$
for $j_{1}\neq j_{2}$, we conclude that

$\widetilde{D}_{k(P_{1}+\Phi_{mj}(P_{2}-P_{1}))}=\widetilde{D}_{kP_{1}}$

for $k\neq j$ and

$\widetilde{D}_{j(P_{1}+\Phi_{mj}(P_{2}-P_{1}))}=\widetilde{D}_{jP_{2}}.$

Therefore $\Phi_{mj}\circ\Phi_{mj}=\Phi_{mj}$ and
$\Phi_{mj_{1}}\circ\Phi_{mj_{2}}=0$ for $j_{1}\neq j_{2}$. Hence
${\widehat{B}}_{i}\simeq{\widehat{B}}_{m1}\times\cdots\times{\widehat{B}}_{ms}$.

∎

###  4.3. $F$ is a surface with $p_{g}(F)>0$ and $q(F)=0$

We now assume that $F$ is a surface of general type and this subsection is
devoted to the proof of Theorem 1.5.

By assumption $p_{g}(F)>0$, hence $f_{*}\omega_{X}$ is a nonzero sheaf on $A$.
By [PPS], we know that $f_{*}\omega_{X}$ admits the Chen-Jiang decomposition.
We denote by $\pi:X\rightarrow X_{0}$ the contraction from $X$ to its
canonical model $X_{0}$. Since the fibers of $\pi$ are rationally chain
connected (see [HM, Corollary 1.5]), we have the commutative diagram:

Note that a connected component of a general fiber of $f_{0}$ is then the
canonical model of the corresponding component of the fiber of $f$. Moreover,
for any divisor $E$ which is $\pi$ exceptional, we have

(10) $\displaystyle H^{0}(X,mK_{X}+P)=H^{0}(X,mK_{X}+E+P),$

for any $m\geq 0$ and $P\in\operatorname{Pic}^{0}(A)$.

We have a small variant of Proposition 4.3.

######  Lemma 4.9.

Assume that $f_{*}(\omega_{X}^{2}\otimes\mathcal{I}_{x})$ or
$f_{*}(\omega_{X}^{3}\otimes\mathcal{I}_{x})$ is M-regular for $x\in X$
general, then $|5K_{X}+P|$ induces a birational map of $X$ for any
$P\in\operatorname{Pic}^{0}(A)$.

###### Proof.

We know that $p_{2}(F)>1$ for any surfaces $F$ of general type. Moreover, if
$F$ is not birational to a $(1,2)$ surface or $(2,3)$ surface, then $|3K_{F}|$
induces a birational map of $F$ (see [BPV, Section 7]). We then conclude by
Proposition 4.3.

If $F$ is birational to a $(1,2)$ surface or $(2,3)$ surface, then $|3K_{F}|$
induces a degree $2$ map. We can conclude by Proposition 4.3 and the same
argument in [CCCJ, Proposition 3.1 and Proposition 3.2]. ∎

From now on, we will assume that both

$f_{*}(\omega_{X}^{2}\otimes\mathcal{I}_{x})$

and

$f_{*}(\omega_{X}^{3}\otimes\mathcal{I}_{x})$

are not M-regular. We now apply the results in Subsection 4.2.

For $m=2$ and $3$, we consider (7) and write respectively for
$P\in\operatorname{Pic}^{0}(A)$

$\displaystyle|2K_{X}+P|=|M_{P}|+D_{P}+E_{2};$
$\displaystyle|3K_{X}+P|=|L_{P}|+V_{P}+E_{3},$

where $|M_{P}|$ and $|L_{P}|$ have no fixed components when $P$ is general and
$D_{P}$ and $V_{P}$ are respectively the fibers of $\mathcal{D}_{2}$ and
$\mathcal{D}_{3}$ over $P$.

There exist divisors $M$ and $D$ on $X$ such that $D_{P}\sim D+P_{B}$ and
$|M_{P}|=|M+P_{C}|$, where $P=P_{B}+P_{C}$ and
$P_{B}=\Phi_{2}(P)\in{\widehat{B}}_{2}$ and $P_{C}\in{\widehat{C}}_{2}$.

Similarly, there exist divisors $L$ and $V$ on $X$ such that $V_{Q}\sim
V+Q_{B}$ and $|L_{Q}|=|L+Q_{C}|$, where $Q=Q_{B}+Q_{C}$ with
$Q_{B}=\Phi_{3}(Q)\in{\widehat{B}}_{3}$ and $Q_{C}\in{\widehat{C}}_{3}$.

We can study the relation between $\mathcal{D}_{2}$ and $\mathcal{D}_{3}$ with
the help of $|K_{X}+Q_{0}|$ for $Q_{0}\in V^{0}(K_{X})$. Note that

$|K_{X}+Q_{0}|+|2K_{X}+P|\subset|3K_{X}+P+Q_{0}|.$

Hence

$|K_{X}+Q_{0}|+|M_{P}|+D_{P}+E_{2}\subset|L_{P+Q_{0}}|+V_{P+Q_{0}}+E_{3}.$

For $P\in\operatorname{Pic}^{0}(A)$ general, comparing the fixed parts of the
above linear systems, we have

(11) $\displaystyle V_{P+Q_{0}}+E_{3}\preceq
D_{P}+\mathrm{Fix}|K_{X}+Q_{0}|+E_{2}.$

Since each component of $\mathcal{D}_{2}$ and $\mathcal{D}_{3}$ dominates
$\operatorname{Pic}^{0}(A)$, we hav3

(12) $\displaystyle V_{P+Q_{0}}\preceq D_{P}$

for $P\in\operatorname{Pic}^{0}(A)$ general and hence for all
$P\in\operatorname{Pic}^{0}(A)$. Hence $(\mathrm{Id}_{X}\times
t_{Q_{0}})_{*}\mathcal{D}_{3}$ is a sub-divisor of $\mathcal{D}_{2}$, where
$t_{Q_{0}}$ is the translation by $Q_{0}$ on $\operatorname{Pic}^{0}(A)$. Let
$\mathcal{D}_{2}=\mathcal{S}+(\mathrm{Id}_{X}\times
t_{Q_{0}})_{*}\mathcal{D}_{3}$ and let $S_{P}$ be the fiber of $\mathcal{S}$
over $P\in\operatorname{Pic}^{0}(A)$.

We then have $D_{P}=V_{P+Q_{0}}+S_{P}$.

By Lemma 4.8, we know that the image ${\widehat{B}}_{3}$ of $\Phi_{3}$ is an
abelian subvariety of the image ${\widehat{B}}_{2}$ of $\Phi_{2}$. Moreover,
there exists a abelian subvariety of ${\widehat{K}}$, which is the image of
the map $\Phi_{2}-\Phi_{3}$, such that
${\widehat{B}}_{2}={\widehat{B}}_{3}+{\widehat{K}}\simeq{\widehat{B}}_{3}\times{\widehat{K}}$
and
${\widehat{C}}_{3}={\widehat{C}}_{2}+{\widehat{K}}\simeq{\widehat{C}}_{2}\times{\widehat{K}}$.

######  Lemma 4.10.

Any translate through the origin of an irreducible component of $V^{0}(K_{X})$
is contained in ${\widehat{C}}_{3}$.

###### Proof.

For $Q_{0}\in V^{0}(K_{X})$ and $P\in\operatorname{Pic}^{0}(A)$, we have
$V_{P+Q_{0}}\preceq D_{P}$. Fix $P$, we see that any translate through the
origin of an irreducible component of $V^{0}(K_{X})$ is contained in
$\Phi_{3}={\widehat{C}}_{3}$.

∎

######  Lemma 4.11.

We then have

  * (1)

Any translate through the origin of an irreducible component of
$V^{0}(K_{X}+M+S)$ is contained in ${\widehat{C}}_{3}$.

  * (2)

$V^{0}(K_{X}+V)\subset V^{0}(K_{X})+{\widehat{B}}_{3}$.

###### Proof.

By (11) and (12),
$|K_{X}+Q_{0}|+|M_{P}|+S_{P}+E_{2}\subset|L_{P+Q_{0}}|+E_{3}$ for $Q_{0}\in
V^{0}(K_{X})$ and $P\in\operatorname{Pic}^{0}(A)$. Because
$V^{0}(L+E_{3})={\widehat{C}}_{3}$ by Lemma 4.6, $V^{0}(K_{X}+M+S)$ is
contained in $Q_{0}+{\widehat{C}}_{3}$. We conclude the proof of (1) by Lemma
4.10.

Fix $P\in V^{0}(K_{X}+V)$ and let $Q\in\operatorname{Pic}^{0}(A)$ be general.
Let $T$ be a general member of $|M_{Q}|+S_{Q}$. We see that
$(K_{X}+V+P)+T+E_{2}$ is linearly equivalent to $3K_{X}+P+Q$. Hence

$|K_{X}+V+P|+T+E_{2}\subset|3K_{X}+P+Q|=|L_{P+Q}|+V_{P+Q}+E_{3}.$

Since $Q$ general, no irreducible components of $T$ is a sub-divisor of
$V_{P+Q}+E_{3}$, thus $T\preceq|L_{P+Q}|$ and

$V_{P+Q}\preceq|K_{X}+V+P|+E_{2}.$

Thus $P+V-V_{P+Q}=P-\Phi_{3}(P+Q)\in V^{0}(K_{X}+E_{2})$. By the following
Lemma 4.12 and (10), $V^{0}(K_{X}+E_{2})=V^{0}(K_{X})$ and hence

$V^{0}(K_{X}+V)\subset V^{0}(K_{X})+{\widehat{B}}_{3}.$

∎

######  Lemma 4.12.

$E_{2}$ is $\pi$-exceptional and $E_{2}|_{F}$ is $\pi_{F}$-exceptional, where
$\pi_{F}:F\rightarrow F_{0}$ is the contraction from $F$ to its canonical
model $F_{0}$.

###### Proof.

Since $f_{*}\omega_{X}^{2}$ is M-regular, thus

$\bigoplus_{P\in\operatorname{Pic}^{0}(A)}H^{0}(f_{*}\omega_{X}^{2}\otimes
P)\otimes P^{-1}\rightarrow f_{*}\omega_{X}^{2}$

is surjective. Let $F$ be a connected component of a general fiber of $f$ and
$F_{0}$ its canonical model. Hence
$\bigoplus_{P\in\operatorname{Pic}^{0}(A)}H^{0}(X,2K_{X}+P)\rightarrow
H^{0}(F,2K_{F})$ is surjective. Since $|2K_{X}+P|=|M_{P}|+D_{P}+E_{2}$, we
conclude that $E_{2}\mid_{F}$ is contained in the fixed component of
$|2K_{F}|$. On the other hand, since $p_{g}(F)>0$, the bicanonical system of
the minimal model of $F$ is base point free and hence $E_{2}|_{F}$ is an
exceptional divisor for the contraction $F\rightarrow F_{0}$. Hence $E$ is
$\pi$-exceptional. ∎

######  Lemma 4.13.

Assume moreover that $q(F)=0$, then for $P\in\operatorname{Pic}^{0}(A)$
general,

  * (1)

$M_{P}|_{F}$ is trivial and hence $(D_{P}+E_{2})|_{F}$ is linearly equivalent
to $2K_{F}$;

  * (2)

We have ${\widehat{B}}_{3}={\widehat{B}}_{2}$ and
${\widehat{C}}_{3}={\widehat{C}}_{2}$ and hence
$\mathcal{D}_{2}=(\mathrm{Id}_{X}\times t_{Q_{0}})_{*}\mathcal{D}_{3}$ for
$Q_{0}\in V^{0}(K_{X})$;

  * (3)

$V^{0}(K_{X})\subset{\widehat{C}}_{2}$.

###### Proof.

Let $Y$ be an irreducible component of a general member of $|M_{P}|$ or an
irreducible component of $S_{P}$ for $P$ general. Then considering the short
exact sequence:

$0\rightarrow K_{X}\rightarrow K_{X}+Y\rightarrow K_{Y}\rightarrow 0,$

we see that $V^{0}(K_{Y})\subset V^{0}(K_{X}+Y)\cup V^{1}(K_{X})$. By the
assumption that $q(F)=0$, we know $R^{1}f_{*}\omega_{X}=0$ and hence by
generic Vanishing (see [H])

$V^{1}(K_{X})=V^{1}(f_{*}\omega_{X})\subset
V^{0}(f_{*}\omega_{X})=V^{0}(K_{X}).$

Hence $V^{0}(K_{Y})\subset V^{0}(K_{X}+Y)$. By Lemma 4.11, we conclude that
any translate through the origin of an irreducible component of $V^{0}(K_{Y})$
is contained in ${\widehat{C}}_{3}$. Then for any resolution of singularities
$\rho:\widetilde{Y}\rightarrow Y$, we have
$V^{0}(\rho_{*}\omega_{\widetilde{Y}})\subset
V^{0}(K_{Y})\subset{\widehat{C}}_{3}$ 111Since $Y$ is Gorenstein,
$\rho_{*}\omega_{\widetilde{Y}}$ is sub-sheaf of $K_{Y}$.. Since a deformation
of $\widetilde{Y}$ covers $X$, $\widetilde{Y}$ is of general type, by the main
Theorem of [JS], a general fiber of $f|_{Y}\circ\rho:\widetilde{Y}\rightarrow
A$ is not of dimension $1$. We conclude that both $M_{P}|_{F}$ and
$S_{P}|_{F}$ are trivial. Hence $(D_{P}+E_{2})|_{F}$ is linearly equivalent to
$2K_{F}$. Since $D_{P}=V_{P+Q_{0}}+S_{P}$ for $Q_{0}\in V^{0}(K_{X})$, $E_{2}$
is $\pi$-exceptional by Lemma 4.12, and $S_{P}|_{F}$ are trivial, there exists
an irreducible component $N$ of $V_{P+Q_{0}}$ whose general fiber over $f(N)$
is a curve. We then consider

$0\rightarrow K_{X}\rightarrow K_{X}+N\rightarrow K_{N}\rightarrow 0$

and apply the same argument as above. By Lemma 4.11, we know that
$V^{0}(K_{N})\subset V^{0}(K_{X})+{\widehat{B}}_{3}$. Note that the family of
$N$ covers $X$. Hence any desingularization $\widetilde{N}$ of $N$ is of
general type and a general fiber of the morphism $\widetilde{N}\rightarrow
f(N)\subset A$ is a curve. Hence by the main result of [JS], $V^{0}(K_{N})$
generates $\operatorname{Pic}^{0}(A)$.

Because each translate through the origin of an irreducible component of
$V^{0}(K_{X})$ is contained in ${\widehat{C}}_{3}$ and
$\operatorname{Pic}^{0}(A)={\widehat{B}}_{3}+{\widehat{C}}_{3}$, the
translates through origin of irreducible components of $V^{0}(K_{X})$
generates ${\widehat{C}}_{3}$.

Assume that $V^{0}(K_{X})=\bigcup_{i}(Q_{i}+{\widehat{T}}_{i})$, where $Q_{i}$
is a torsion point and ${\widehat{T}}_{i}$ is an abelian subvariety of
${\widehat{C}}_{3}$. For any positive dimensional component
$Q_{i}+{\widehat{T}}_{i}$ and $P\in{\widehat{T}}_{i}$ general, we write

$|K_{X}+Q_{i}+P|=|M_{iP}|+N_{iP}+E,$

where $E$ is the common fixed divisor and $|M_{iP}|$ has no fixed part. Hence
for $P\in{\widehat{T}}_{i}$ general,

$|M_{iP}|+|M_{i(-P)}|+N_{iP}+N_{i(-P)}+2E\subset|2K_{X}+2Q_{i}|=|M_{2Q_{i}}|+D_{2Q_{i}}+E_{2},$

we see that $M_{iP}+M_{i(-P)}+N_{iP}+N_{i(-P)}\preceq M_{2Q_{i}}$. Thus
${\widehat{T}}_{i}\subset{\widehat{C}}_{2}$. Since these ${\widehat{T}}_{i}$
generates ${\widehat{C}}_{3}$, we have ${\widehat{C}}_{2}={\widehat{C}}_{3}$
and ${\widehat{B}}_{2}={\widehat{B}}_{3}$. Therefore by Lemma 4.8,
$\mathcal{S}=0$ and $\mathcal{D}_{2}=(\mathrm{Id}_{X}\times
t_{Q_{0}})_{*}\mathcal{D}_{3}$ for $Q_{0}\in V^{0}(K_{X})$.

Finally, recall that $V_{P+Q_{0}}\preceq D_{P}$ for any $Q_{0}\in
V^{0}(K_{X})$ and $P\in\operatorname{Pic}^{0}(A)$. We now know that $V$ and
$D$ are algebraically equivalent. Thus fix $P$, $V_{P+Q_{0}}=D_{P}$ is the
same divisor for any $Q\in V^{0}(K_{X})$. Thus
$V^{0}(K_{X})\subset{\widehat{C}}_{2}$.

∎

######  Theorem 4.14.

Assume that $p_{g}(F)>0$ and $q(F)=0$, $|5K_{X}|$ induces a birational map of
$X$.

###### Proof.

Note that by the Chen-Jiang decomposition theorem, the natural twisted
evaluation map

$\bigoplus_{Q\in V^{0}(K_{X})}H^{0}(A,f_{*}\omega_{X}\otimes Q)\otimes
Q^{-1}\rightarrow f_{*}\omega_{X}$

is surjective, which implies the surjectivity of the restriction map

(13) $\displaystyle\bigoplus_{Q\in V^{0}(K_{X})}H^{0}(X,K_{X}+Q)\rightarrow
H^{0}(F,K_{F}).$

Note that for any $Q\in V^{0}(K_{X})$, we have

$|K_{X}+Q|+|K_{X}+Q|\subset|2K_{X}+2Q|=|M_{2Q}|+D_{2Q}+E_{2}.$

We know by Lemma 4.10 and Lemma 4.13 that
$V^{0}(K_{X})\subset{\widehat{C}}_{2}$. Note that
${\widehat{C}}_{2}=\mathrm{ker}(\Phi_{2})$. Hence $D_{2Q}=D_{2P_{0}}$ is fixed
for all $Q\in V^{0}(K_{X})$. Moreover, $M_{2Q}|_{F}$ is trivial by Lemma 4.13.
Hence the image of sum of evaluation map (13) is $1$-dimensional. We have
$h^{0}(F,K_{F})=1$.

On the other hand, by Proposition 4.7, the map

$H^{0}(F,L|_{F})\otimes H^{0}(F,V|_{F})\xrightarrow{\cdot
E_{3}|_{F}}H^{0}(F,3K_{F})$

is surjective. Recall that $D=V_{Q_{0}}$ and $(D+E_{2})|_{F}$ is linear
equivalent to $2K_{F}$, where $E_{2}|_{F}$ is $\pi_{F}$-exceptional. Thus
$L|_{F}\sim K_{F}+E_{2}|_{F}-E_{3}|_{F}$. We have $h^{0}(F,V|_{F})=p_{2}(F)$
and $h^{0}(F,L|_{F})\leq h^{0}(F,K_{F})=1$. This implies that $p_{3}(F)\leq
p_{2}(F)$, which is absurd. ∎

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