# Effective characterization of quasi-abelian surfaces

Margarida Mendes Lopes  CAMGSD/Departamento de Matemática, Instituto Superior
Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
<EMAIL_ADDRESS>,  Rita Pardini  Dipartimento di Matematica,
Università degli studi di Pisa, Largo Pontecorvo 5, 56127 Pisa (PI), Italy
<EMAIL_ADDRESS>and  Sofia Tirabassi  Department of Mathematics,
Stockholm University, Kräftriket, Stockholm, Sweden<EMAIL_ADDRESS>
###### Abstract.

Let $V$ be a smooth quasi-projective complex surface such that the first three
logarithmic plurigenera $\overline{P}_{1}(V)$, $\overline{P}_{2}(V)$ and
$\overline{P}_{3}(V)$ are equal to 1 and the logarithmic irregularity
$\overline{q}(V)$ is equal to $2$. We prove that the quasi-Albanese morphism
$a_{V}\colon V\to A(V)$ is birational and there exists a finite set $S$ such
that $a_{V}$ is proper over $A(V)\setminus S$, thus giving a sharp effective
version of a classical result of Iitaka ([Ii79]).

###### Key words and phrases:

Quasi-abelian varieties, Quasi-abelian surfaces, Semi-abelian varieties, Semi-
abelian surfaces, Open surfaces, Logarithmic plurigenus, Quasi-Albanese
morphism, birational geometry of log-varieties, WWPB-maps, affine varieties

###### 2020 Mathematics Subject Classification:

Primary 14E05, Secondary 14J10, 14K99, 14L20, 14L40, 14R05

###### Contents

  1. 1 Preliminaries
    1. 1.1 Log varieties
    2. 1.2 Quasi-abelian varieties and the quasi-Albanese map
      1. 1.2.1 The quasi-Albanese map
    3. 1.3 Logarithmic ramification formula
    4. 1.4 Generic vanishing
    5. 1.5 Curves on smooth surfaces
  2. 2 Results in arbitrary dimension via generic vanishing
  3. 3 The geometry of the quasi-Albanese morphism
    1. 3.1 The quasi-Albanese map when $q(V)\geq 1$
    2. 3.2 The quasi-Albanese map when $q(V)=0$
  4. 4 Proof of Theorem A
    1. 4.1 Preliminary steps
    2. 4.2 Conclusion

## Introduction

Let $V$ be a smooth complex quasi-projective variety. By Hironaka’s theorem on
the resolution of singularities we can write $V=X\backslash D$ where $X$ is a
smooth projective variety and $D$ is a reduced divisor on $X$ with simple
normal crossings support (in fact in the case of surfaces, our main interest
here, $V$ is automatically quasi-projective since it is smooth).

Denoting by $K_{X}$ the canonical divisor of $X$, one defines the following
invariants of $V$:

  * -

for $m$ a positive integer, the $m$-th _log-plurigenus_ of $V$ is
$\overline{P}_{m}(V):=h^{0}(X,m(K_{X}+D))$;

  * -

the _log-Kodaira dimension_ of $V$ is
$\overline{\kappa}(V):=\kappa(X,K_{X}+D);$

  * -

the _log-irregularity_ of $V$ is $\overline{q}(V):=h^{0}(X,\Omega^{1}_{X}(\log
D))$.

In addition, we say that the _irregularity_ $q(V)$ of $V$ is the irregularity
of $X$, that is

$q(V):=h^{0}(X,\Omega^{1}_{X}).$

It easy to see that these invariants do not depend on the choice of the
compactification $X$.

Similarly to what happens for projective varieties, to $V$ we can associate a
quasi-abelian variety (i.e., an algebraic group which does not contain
$\mathbb{G}_{a}$) $A(V)$, called the _quasi-Albanese variety_ of $V$. This
comes equipped with a morphism $a_{V}\colon V\rightarrow A(V)$ which is called
the _quasi-Albanese morphism_. Iitaka in [Ii79] characterizes quasi-abelian
surfaces as surfaces of log-Kodaira dimension 0 and log-irregularity 2. More
precisely he proves the following.

######  Theorem (Iitaka, [Ii79]).

Let $V$ be a smooth complex algebraic surface. Then $\overline{\kappa}(V)=0$
and $\overline{q}(V)=2$ if, and only if, the quasi-Albanese morphism
$a_{V}\colon V\rightarrow A(V)$ is birational and there are an open subset
$V^{0}\subseteq V$, and finitely many points $\\{p_{1},\ldots
p_{t}\\}\subseteq A(V)$, such that the restriction $a_{V|_{V^{0}}}\colon
V^{0}\rightarrow A(V)\backslash\\{p_{1},\ldots p_{t}\\}$ is proper.

In this paper, we give a characterization of quasi-abelian surfaces using the
three first logarithmic plurigenera instead of the logarithmic Kodaira
dimension. Our main result is the following Theorem:

######  Theorem A.

Let $V$ be a smooth complex algebraic surface with $\overline{q}(V)=2$. Assume
that:

  * (a)

$\overline{P}_{1}(V)=\overline{P}_{2}(V)=1$ and $q(V)>0$; or

  * (b)

$\overline{P}_{1}(V)=\overline{P}_{3}(V)=1$ and $q(V)=0$.

Then the quasi-Albanese morphism $a_{V}\colon V\rightarrow A(V)$ is
birational. In addition, there are an open subset $V^{0}\subseteq V$ and
finitely many points $\\{p_{1},\ldots p_{t}\\}\subseteq A(V)$, such that the
restriction $a_{V|_{V^{0}}}\colon V^{0}\rightarrow
A(V)\backslash\\{p_{1},\ldots p_{t}\\}$ is proper.

Using the language of WWPB equivalences introduced by Iitaka in [Ii77],
Theorem A above can be rephrased in the following manner:

######  Theorem A*.

Let $V$ be a smooth algebraic surface. Then $V$ is WWPB equivalent to a quasi-
abelian variety if, and only if either

  * (a)

$\overline{q}(V)=2$, $\overline{P}_{1}(V)=\overline{P}_{2}(V)=1$ and $q(V)>0$;
or

  * (b)

$\overline{q}(V)=2$ , $\overline{P}_{1}(V)=\overline{P}_{3}(V)=1$ and
$q(V)=0$.

As WWPB-maps between normal affine varieties are actually isomorphisms (see
[Ii77, Corollary on p. 498]), we have:

######  Corollary B.

A smooth complex affine surface $V$ is isomorphic to $\mathbb{G}_{m}^{2}$ if,
and only if, it has $\overline{P}_{1}(V)=\overline{P}_{3}(V)=1$ and
$\overline{q}(V)=2$.

We remark that Kawamata, in his celebrated work [Ka81], proved in any
dimension a weaker form of Iitaka’s Theorem, showing that the quasi-Albanese
morphism of an algebraic variety of log-Kodaira dimension 0 and log-
irregularity equal to the dimension is birational. In the compact case,
effective versions of this result have been given in [CH01] and [PPS17]: in
[CH01], it is proven that the Albanese map of a projective variety $X$ is
surjective and birational iff $\dim X=q(X)$ and $P_{1}(X)=P_{2}(X)=1$; in
[PPS17], the analogous statement is proven for compact Kähler manifolds.

So we are led to formulate the following conjecture:

######  Conjecture.

Let $V$ a smooth complex quasi-projective variety with $\overline{q}(V)=\dim
V$, then there exists a positive integer $k$, independent of the dimension of
$V$, such that $\overline{P}_{1}(V)=\overline{P}_{k}(V)=1$ implies that the
quasi-Albanese morphism of $V$ is birational (and there exist an open set
$V^{0}\subseteq V$ and a closed set $W\subseteq A(V)$ of codimension $>1$ such
that $a_{V|_{V^{0}}}\colon V^{0}\rightarrow A(V)\backslash W$ is proper).

In view of the results recalled above, one might hope that $k=2$ is the right
bound also in the open setting, but in fact our Theorem A is sharp because
there is a surface $V$ with $\overline{P}_{1}(V)=\overline{P}_{2}(V)=1$,
$\overline{q}(V)=2$ and quasi-Albanese map not dominant (see Example 3.18).

Our argument is completely independent of Iitaka’s Theorem and its proof. The
first step consists in showing that the assumptions on the logarithmic
irregularity and on the logarithmic plurigenera imply that the quasi-Albanese
map of $V$ is dominant. When $q(V)=2$, one obtains as an immediate consequence
of the work of Chen and Hacon ([CH01]) that the quasi-Albanese map of $V$ is
birational. When $q(V)=0$, the lengthy proof uses a fine analysis of the
boundary divisor and classical arguments from the theory of surfaces. However,
when $q(V)=1$, we get a considerably shorter proof leveraging on the fact that
the Albanese map of a compactification of $V$ is nontrivial, using techniques
coming from Green–Lazarsfeld generic vanishing theorems ([GL1987]) and their
more recent extensions to pairs (see [PS14] and [Sh16]).

Once we know that the quasi-Albanese map of $V$ is dominant, then we get that
its extension to the compactifications of $V$ and $A(V)$ respectively is
generically finite. Then we use Iitaka’s logarithmic ramification formula (see
1.3 for more details) together with some geometric considerations to conclude
that the quasi-Albanese map is birational.

The last step of our argument consists in proving that the quasi-Albanese map
is proper outside a finite set of points. This is done by showing that the
boundary divisor $D$ gets contracted by the quasi-Albanese map. Again, the key
tool here is the logarithmic ramification formula together with some more
classical arguments exploiting the geometry of the divisor $D$.

The paper is organized as follows. In §1 we recall the necessary
prerequisites; §2 contains some results that hold in arbitrary dimension for
$q(V)>0$ (see in particular Corollary 2.2). Then we focus on surfaces. Section
3 is devoted to proving that the quasi-Albanese map is dominant: here the
results of §2 are crucial in case $q(V)>0$. In §4 we complete the proof of
Theorem A.

Notation. We work over the complex numbers. If $X$ is a smooth projective
variety, we denote by $K_{X}$ the canonical class, by
$q(X):=h^{0}(X,\Omega^{1}_{X})=h^{1}(X,\mathcal{O}_{X})$ the _irregularity_
and by $p_{g}(X):=h^{0}(X,K_{X})$ the geometric genus.

We identify invertible sheaves and Cartier divisors and we use the additive
and multiplicative notation interchangeably. Linear equivalence is denoted by
$\sim$. Given two divisors $D_{1}$ and $D_{2}$ on $X$, we write $D_{1}\succeq
D_{2}$ (respectively $D_{1}\succ D_{2}$) if the divisor $D_{1}-D_{2}$ is
(strictly) effective.

Finally note that throughout this paper, we use the term _(-1)-curve_ to
indicate the total transform $E$ of a point $q\in Y$ via a birational morphism
$g\colon X\to Y$ of smooth projective surfaces such $g^{-1}$ is not defined at
$q$; so one has $E^{2}=K_{X}E=-1$ but $E$ may be reducible and/or non reduced.

Acknowledgments. We wish to thank the referee for the extremely careful
reading of the paper.

M. Mendes Lopes was partially supported by FCT/Portugal through Centro de
Análise Matemática, Geometria e Sistemas Dinâmicos (CAMGSD), IST-ID, projects
UIDB/04459/2020 and UIDP/04459/2020. R. Pardini was partially supported by
project PRIN 2017SSNZAW$\\_$004 “Moduli Theory and Birational Classification"
of Italian MIUR and is a member of GNSAGA of INDAM. S.Tirabassi was partially
supported by grant 261756 of the Research Council of Norway and grant KAW
2019.0493 of Knut and Alice Wallenberg Foundation.

##  1\. Preliminaries

Our proof of Theorem A combines different arguments and techniques; in this
section we recall briefly the necessary prerequisites and set the notation.

###  1.1. Log varieties

Let $X$ be a smooth complex projective variety of dimension $n$. A reduced
effective divisor $D=\sum D_{i}$ on $X$ is said to have _simple normal
crossings_ (in short we say $D$ is snc) if all the $D_{i}$ are smooth and for
every $p\in\operatorname{Supp}D$ there are local coordinates
$(x_{1},\ldots,x_{n})$ around $p$ such that $D$ is cut out by the equation
$x_{1}\cdots x_{r}=0$, for some $r\leq n$.

Given a smooth projective $n$-dimensional variety $X$ together with a snc
divisor $D$, we can define the sheaf of logarithmic 1-forms along $D$ by
setting

$\Omega^{1}_{X}(\log
D)_{p}:=\sum_{i=1}^{r}\mathcal{O}_{X,p}\frac{dx_{i}}{x_{i}}+\sum_{i=r+1}^{n}\mathcal{O}_{X,p}dx_{i}\subset(\Omega^{1}_{X}\otimes
k(X))_{p}$

where $(x_{1},\ldots,x_{n})$ are local coordinates around $p$ such that
$D=\\{x_{1}\cdots x_{r}=0\\}$. It is a locally free sheaf of rank equal to
$n$. The sheaf of logarithmic $m$-forms is defined as

$\Omega_{X}^{m}(\log D):=\bigwedge^{m}\Omega_{X}(\log D),$

and, in particular, we have that

$\Omega_{X}^{n}(\log D)\simeq\mathcal{O}_{X}(K_{X}+D).$

We recall the following condition for the existence of logarithmic 1-forms
with prescribed poles.

######  Proposition 1.1.

Let $D$ be a simple normal crossing divisor on a smooth projective variety $X$
and let $D_{1},\dots D_{k}$ be a subset of the irreducible components of $D$.

There exists $\sigma\in H^{0}(X,\Omega^{1}_{X}(\log D))$ with poles precisely
on $D_{1}\dots D_{k}$ if and only if in $H^{2}(X,\mathbb{Z})$ there is a
relation $\sum_{i}a_{i}[D_{i}]=0$ between the classes $[D_{1}],\dots[D_{k}]$
such that $a_{i}\neq 0$ for $i=1,\dots k$.

###### Proof.

Let $D_{1},\dots D_{t}$ be the irreducible components of $D$; consider the
residue sequence

$0\to\Omega^{1}_{X}\to\Omega^{1}_{X}(\log
D)\to\oplus_{i=1}^{t}{\mathcal{O}}_{D_{i}}\to 0.$

The claim follows from the fact that the associated coboundary map
$\delta\colon\oplus_{i=1}^{t}H^{0}(X,{\mathcal{O}}_{D_{i}})\to
H^{1}(X,\Omega^{1}_{X})$ sends $1\in H^{0}(X,{\mathcal{O}}_{D_{i}})$ to the
class $[D_{i}]\in H^{1}(X,\Omega^{1}_{X})$. ∎

Given a smooth quasi-projective variety $V$, by Hironaka’s resolution of
singularities we can embed it in a smooth projective variety $X$ such that
$X\backslash V$ is a snc divisor $D$, called the _boundary of $V$_. Then we
can use the sheaves of logarithmic forms to define logarithmic (in short
“log”) invariants on $V$, as explained in the introduction. In the case when
$V$ is a curve, we call $\overline{P}_{1}(V)=\overline{q}(V)$ the logarithmic
genus of $V$.

While the usual plurigenera and irregularity are birational invariants, this
is not the case for logarithmic invariants. For example, an abelian variety
$X$ and $X\backslash D$, where $D>0$ is a smooth ample divisor, are birational
but they have different logarithmic Kodaira dimensions. For this reason Iitaka
in [Ii77] introduced the notion of WPB-equivalence (weakly proper birational
equivalence). It is the equivalence relation generated by the proper
birational morphisms and by the open inclusions $V\subset V^{\prime}$ such
that $V^{\prime}\setminus V$ has codimension at least 2.

WPB-equivalent varieties have the same plurigenera and log irregularity: this
is obvious for open immersions as above and it is proven for proper birational
morphisms in [Ii77b, Prop. 1]. Therefore, it would seem that WPB-equivalence
is the right notion of equivalence when studying the birational geometry of
open varieties. However, the set of WPB-maps is not _saturated_ , namely it is
possible, for instance, to have rational maps $g\colon U\dashrightarrow V$ and
$f\colon V\dashrightarrow W$ such that $f\circ g$ is WPB, but $f$ or $g$ is
not. In order to get around this kind of difficulty, Iitaka in [Ii77] defines
the notion of WWPB-equivalence, proving that WWPB-equivalent varieties have
the same logarithmic plurigenera and irregularity.

###  1.2. Quasi-abelian varieties and the quasi-Albanese map

Here we introduce one of the main characters of our story: quasi-abelian
varieties, which for many aspects can be thought of as a non-projective
analogue of abelian varieties. We recall briefly the facts that we need.

######  Definition 1.2.

A _quasi-abelian variety_ \- in some sources also called a _semiabelian
variety_ \- is a connected algebraic group $G$ that is an extension of an
abelian variety $A$ by an algebraic torus. More precisely, $G$ sits in the
middle of an exact sequence of the form

(1) $1\rightarrow\mathbb{G}_{m}^{r}\longrightarrow G\longrightarrow
A\rightarrow 0.$

We call $A$ the compact part and $\mathbb{G}_{m}^{r}$ the linear part of $G$.

Over the complex numbers, $G\simeq\mathbb{C}^{\dim G}/\pi_{1}(G)$. Observe
that $\pi_{1}(G)$ is a finitely generated free abelian group. When the rank of
$\pi_{1}(G)$ is equal to $2\dim(G)$, then $G$ is an abelian variety.

For later use we recall the following from [III, §10]:

######  Proposition 1.3.

Let $G$ be a quasi abelian variety, let $A$ be its compact part and let
$r:=\dim G-\dim A$. Then there exists a compactification $G\subset Z$ such
that:

  * (a)

$Z$ is a $\mathbb{P}^{r}$-bundle over $A$;

  * (b)

$\Delta:=Z\setminus G$ is a simple normal crossing divisor and
$\Omega^{1}_{Z}(\log\Delta)$ is a trivial bundle of rank equal to $\dim G$.

In particular, $\overline{q}(G)=\dim G$ and $\overline{P}_{m}(G)=1$ for all
$m>0$.

We are especially interested in the following particular case of Proposition
1.3:

######  Corollary 1.4.

Let $G$ be a quasi-abelian variety of dimension 2 with compact part $A$ of
dimension 1. Then there is a compactification $G\subset Z$, where
$Z=\mathbb{P}({\mathcal{O}}_{A}\oplus L)$ with $L\in\operatorname{Pic}^{0}(A)$
and the boundary $\Delta$ is the disjoint union of two sections $\Delta_{1}$
and $\Delta_{2}$ of $Z\to A$.

###### Proof.

By Proposition 1.3 we may find a compactification $Z$ which is a
$\mathbb{P}^{1}$-bundle over $A$ and such that $\Delta$ is a normal crossing
divisor. The divisor $\Delta$ meets every fibre of $Z\to A$ in exactly two
points, so it is a smooth bisection. By Proposition 1.3, there is a non
regular logarithmic 1-form of $Z$ with poles contained in $\Delta$, hence by
Proposition 1.1 $\Delta$ is reducible and its components satisfy a non trivial
relation in cohomology. Since $\Delta$ is a smooth bisection of $Z\to A$, we
have $\Delta=\Delta_{1}+\Delta_{2}$ with $\Delta_{i}$ disjoint sections. So
$Z={\mathbf{P}}({\mathcal{O}}_{A}\oplus L)$ for some
$L\in\operatorname{Pic}(A)$. Since in $\operatorname{Pic}(Z)$ we have
$\Delta_{2}=\Delta_{1}+p^{*}L$, where $p\colon Z\to A$ is the natural
projection, $\Delta_{1}$ and $\Delta_{2}$ are independent in
$H^{2}(Z,\mathbb{Z})$ unless $p^{*}L=0$ in $H^{2}(Z,\mathbb{Z})$, namely
unless $p^{*}L\in\operatorname{Pic}^{0}(Z)$. So Proposition 1.1 implies that
$L$ is an element of $\operatorname{Pic}^{0}(A)$.∎

Finally we recall some fundamental properties of abelian varieties that extend
verbatim to the quasi-abelian case:

######  Proposition 1.5.

Let $G$ be a quasi-abelian variety. Then:

  1. (i).

if $G^{\prime}$ is a quasi-abelian variety and $\phi\colon G^{\prime}\to G$ is
a morphism with $\phi(0)=0$, then $\phi$ is a homomorphism;

  2. (ii).

if $G^{\prime}\to G$ is a finite étale cover, then $G^{\prime}$ is a quasi-
abelian variety;

  3. (iii).

if $H\subset G$ is closed and $\overline{\kappa}(H)=0$ then $H$ is a quasi-
abelian variety.

###### Proof.

Item (i) is [Nevanlinna, Thm. 5.1.37], (ii) is [Fu14, Thm. 4.2], and (iii) is
[III, Thm. 4]. ∎

####  1.2.1. The quasi-Albanese map

The classical construction of the Albanese variety of a projective variety can
be extended to the non projective case, by replacing regular 1-forms by
logarithmic ones and abelian varieties by quasi-abelian ones. The key fact is
that by Deligne [De71] logarithmic 1-forms are closed (for the details of the
construction see [III], [Fu14, Section 3]).

######  Theorem 1.6.

Let $V$ be a smooth algebraic variety. Then there exists a quasi-abelian
variety $A(V)$ and a morphism $a_{V}\colon V\to A(V)$ such that:

  1. (i).

if $h\colon V\rightarrow G$ is a morphism to a quasi-abelian variety, then $h$
factors through $a_{V}$ in a unique way;

  2. (ii).

if $X$ is a compactification of $V$ with snc boundary $D$, then we have the
following exact sequence:

(2) $1\rightarrow\mathbb{G}_{m}^{r}\longrightarrow A(V)\longrightarrow
A(X)\rightarrow 0,$

where $r=\overline{q}(V)-q(V)$.

###### Proof.

Item (i) is proven in [III], in the discussion immmediately after Proposition
4. Item (ii) is [Fu14, p. 13] ∎

The variety $A(V)$ is called the quasi-Albanese variety of $V$ and $a_{V}$ the
quasi-Albanese map. Note that the compact part of $A(V)$ is $A(X)$. We note
the following logarithmic version of Abel’s Theorem:

######  Proposition 1.7.

Let $C$ be a smooth curve with $\overline{P}_{1}(C)>0$. Then the quasi-
Albanese map $a_{C}\colon C\to A(C)$ is an embedding.

###### Proof.

Denote by $\overline{C}$ the compactification of $C$. If $g(\overline{C})>0$,
then the Abel-Jacobi map $\overline{C}\to A(\overline{C})$ factorizes through
$a_{C}$, so the claim follows by Abel’s Theorem. If $g(\overline{C})=0$, we
have $C:=\overline{C}\setminus\\{p_{0},\dots p_{k}\\}$, where
$k:=\overline{P}_{1}(C)$. By the universal property, the inclusion
$C\to\overline{C}\setminus\\{p_{0},p_{1}\\}\simeq\mathbb{G}_{m}$ factorizes
through $a_{C}$, which therefore is an isomorphism. ∎

###  1.3. Logarithmic ramification formula

Let $V$, $W$ be smooth varieties of dimension $n$ and let $h\colon V\to W$ be
a dominant morphism. Let $g\colon X\to Z$ be a morphism extending $h$, where
$X$, $Z$ are smooth compactifications of $V$, respectively $W$, such that
$D:=X\setminus V$ and $\Delta:=Z\setminus W$ are snc divisors. Then the pull
back of a logarithmic $n$-form on $Z$ is a logarithmic $n$-form on $X$, and a
local computation shows that there is an effective divisor $\overline{R}_{g}$
of $X$ \- the logarithmic ramification divisor \- such that the following
linear equivalence holds:

(3) $K_{X}+D\sim g^{*}(K_{Z}+\Delta)+\overline{R}_{g}.$

Equation (3) is called the logarithmic ramification formula (cf. [Ii82,
§11.4]).

We note the following for later use:

######  Lemma 1.8.

In the above set-up, denote by $R_{g}$ the (usual) ramification divisor of
$g$. Let $\Gamma$ be an irreducible divisor such that
$g(\Gamma)\not\subseteq\Delta$. Then $\Gamma$ is a component of
$\overline{R}_{g}$ if and only if $\Gamma$ is a component of $D+R_{g}$.

###### Proof.

Let $x\in\Gamma$ be a general point, so that $y=g(x)$ does not lie on
$\Delta$. Then $K_{Z}+\Delta$ is generated locally near $y$ by a nowhere
vanishing regular $n$-form, while $K_{X}+D$ is generated locally near $x$
either by an $n$-form with a logarithmic pole on $\Gamma$ or by a regular
$n$-form, according to whether $\Gamma$ is a component of $D$ or not. In the
former case $\Gamma$ is always a component of $\overline{R}_{g}$, in the
latter case it is a component of $\overline{R}_{g}$ if and only if $g$ is
ramified along $\Gamma$. ∎

###  1.4. Generic vanishing

The theory of generic vanishing was introduced by Green–Lazarsfeld in
[GL1987]. It has since developed in a powerful tool to study the geometry of
projective varieties via their Albanese morphism (see, for example, [Pa12] for
a nice survey). We are going to see in Section 2 that these techniques can be
useful instruments also in the quasi-projective setting.

Let $X$ be a smooth projective variety. Given a coherent sheaf $\mathcal{F}$
on $X$, the _cohomological support loci_ of $\mathcal{F}$ are the subsets

$V^{i}(X,\mathcal{F}):=\\{\alpha\in{\rm Pic}^{0}(X)\ |\
h^{i}(X,\mathcal{F}\otimes\alpha)\neq 0\\}\subseteq{\rm Pic}^{0}(X).$

We say that $\mathcal{F}$ is a _$GV$ -sheaf_ if, for every $i>0$, we have that

$\mathrm{codim}_{{\rm Pic}^{0}(X)}V^{i}(X,\mathcal{F})\geq i.$

When all the $V^{i}(X,\mathcal{F})$ are empty for $i>0$, one says, following
the original terminology of Mukai ([Mu1981]), that $\mathcal{F}$ is a
_IT(0)-sheaf_.

We recall the following well known useful observation:

######  Lemma 1.9.

Let $X$ be a smooth projective variety and $L$ a line bundle of $X$.

If $V^{0}(X,L)\cap(-V^{0}(X,L))$ has positive dimension, then $h^{0}(X,2L)\geq
2$.

###### Proof.

For $\alpha\in V^{0}(X,L)\cap(-V^{0}(X,L))$ consider the multiplication map

$H^{0}(X,L\otimes\alpha)\otimes H^{0}(X,L\otimes\alpha^{-1})\longrightarrow
H^{0}(X,2L).$

As $\alpha$ varies, the image of the map must vary, since a divisor can be
written as the sum of two effective divisors only in finitely many ways. As a
consequence $h^{0}(X,2L)\geq 2$. ∎

###  1.5. Curves on smooth surfaces

Let $D\succ 0$ be a divisor (a “curve”) on a smooth projective surface $X$.
The _arithmetic genus_ of $D$ is $p_{a}(D):=1-\chi({\mathcal{O}}_{D})$ and can
be computed by means of the adjunction formula

(4) $p_{a}(D)-1=\frac{1}{2}D(K_{X}+D).$

By Serre duality, one also has
$p_{a}(D)-1=h^{0}(D,\omega_{D})-h^{0}(D,{\mathcal{O}}_{D})$, hence
$h^{0}(D,\omega_{D})\geq p_{a}(D)$.

For $m\in\mathbb{N}$, we say that $D$ is _$m$ -connected_ if for every
decomposition $D=D_{1}+D_{2}$, with $D_{1}$ and $D_{2}$ effective, one has
$D_{1}D_{2}\geq m$.

We recall some well known facts.

######  Lemma 1.10.

Let $D$ be a $1$-connected divisor, then:

  1. (i).

$h^{0}(D,{\mathcal{O}}_{D})=1$ (so in particular,
$p_{a}(D)=h^{0}(D,\omega_{D})\geq 0$);

  2. (ii).

if $L$ is a line bundle on $D$ that has degree 0 on every component of $D$,
then $h^{0}(D,L)\leq 1$, and $h^{0}(D,L)=1$ if and only if
$L={\mathcal{O}}_{D}$.

###### Proof.

See [BHPV, Lem. II 12.2]. ∎

######  Lemma 1.11.

Let $D_{1}$, $D_{2}$ be two effective non zero divisors on a smooth projective
surface $X$. Then

  1. (i).

$p_{a}(D_{1}+D_{2})=p_{a}(D_{1})+p_{a}(D_{2})+D_{1}D_{2}-1$;

  2. (ii).

if $D_{1}\prec D_{2}$, then $h^{0}(D_{1},\omega_{D_{1}})\leq
h^{0}(D_{2},\omega_{D_{2}})$.

###### Proof.

(i) Use the adjunction formula (4).

(ii) Write $D_{2}=D_{1}+A$ and consider the decomposition sequence:

$0\to{\mathcal{O}}_{D_{1}}(-A)\to{\mathcal{O}}_{D_{2}}\to{\mathcal{O}}_{A}\to
0.$

Twisting by $K_{X}+D_{2}$ and taking cohomology gives the result. ∎

######  Lemma 1.12.

Let $D$ be an effective non zero divisor on a smooth projective surface $X$.
Then we have the following formulae:

  1. (i).

$h^{0}(X,K_{X}+D)=p_{a}(D)+p_{g}(X)-q(X)+h^{1}(X,-D)$;

  2. (ii).

$h^{0}(X,K_{X}+D)=h^{0}(D,\omega_{D})+p_{g}(X)-q(X)+d$,

where $d$ is the dimension of the kernel of the restriction map
$H^{1}(X,{\mathcal{O}}_{X})\to H^{1}(D,{\mathcal{O}}_{D})$.

###### Proof.

(i) follows from the Riemann-Roch theorem, the adjunction formula (4) and
Serre duality.

Taking cohomology in the restriction sequence

$0\to{\mathcal{O}}_{X}(-D)\to{\mathcal{O}}_{X}\to{\mathcal{O}}_{D}\to 0,$

one obtains $h^{1}(X,-D)=h^{0}(D,{\mathcal{O}}_{D})-1+d$. Plugging this
relation and the equality
$p_{a}(D)-1=h^{0}(D,\omega_{D})-h^{0}(D,{\mathcal{O}}_{D})$ in (i), one
obtains (ii). ∎

######  Remark 1.13.

Note that by Serre duality $d$ is exactly the codimension of the image of the
map $t\colon H^{0}(D,\omega_{D})\to H^{1}(X,K_{X})$.

##  2\. Results in arbitrary dimension via generic vanishing

Let $V$ be a smooth quasi-projective variety and let $X$ be a smooth
compactification of $V$ with snc boundary. In this section we investigate the
geometric constraints imposed on the Albanese map $a_{X}$ of $X$, and on the
image of $V$ via $a_{X}$, by the fact the logarithmic plurigenera of $V$ are
small. Our main tool will be the theory of generic vanishing (§1.4), and its
recent extensions to the “log-canonical” setting by Popa–Schnell ([PS14])and
Shibata ([Sh16]).

######  Proposition 2.1.

Let $V$ be a smooth quasi-projective variety and denote by $X$ a
compactification of $V$ with simple normal crossing boundary divisor $D$.

If $\overline{P}_{1}(V)=\overline{P}_{2}(V)=1$, then the Albanese morphism
$a_{X}\colon X\to A(X)$ is surjective.

###### Proof.

Since $V^{0}(X,\mathcal{O}_{X}(K_{X}+D))$ is a union of translates of subtori
of $A(X)$ (cf. [Sh16]*Theorem 1.3), Lemma 1.9 implies that ${\mathcal{O}}_{X}$
is an isolated point of $V^{0}(X,{\mathcal{O}}_{X}(K_{X}+D))$. Thus by the
projection formula ${\mathcal{O}}_{A(X)}$ is an isolated point of
$V^{0}(A(X),a_{X,*}{\mathcal{O}}_{X}(K_{X}+D))$. Thanks to [PS14]*Variant 5.5,
we know that $a_{X,*}{\mathcal{O}}_{X}(K_{X}+D)$ is a $GV$-sheaf on $A(X)$,
and therefore by [Pa12]*Lemma 1.8 we know that ${\mathcal{O}}_{A(X)}$ is a
component of $V^{q(X)}(A(X),a_{X,*}{\mathcal{O}}_{X}(K_{X}+D))$. In particular
$h^{q(X)}(A(X),a_{X,*}{\mathcal{O}}_{X}(K_{X}+D))\neq 0$ and we deduce that
the dimension of the Albanese image of $X$ is equal the dimension of $A(X)$.
We conclude because $X$ and $A(X)$ are projective. ∎

As an immediate consequence we have:

######  Corollary 2.2.

Let $V$ an $n$-dimensional smooth quasi-projective variety, $X$ a
compactification of $V$ such that $D:=X\setminus V$ is a snc divisor. If
$\overline{P}_{1}(V)=\overline{P}_{2}(V)=1$ and $\overline{q}(V)=q(X)=n$, then
the quasi-Albanese morphism $a_{V}\colon V\rightarrow A(V)$ is birational.

###### Proof.

By Proposition 2.1, we know that $a_{X}$ is generically finite, and so $X$ is
of maximal Albanese dimension. In particular, we have that

$0<h^{0}(X,K_{X})\leq h^{0}(X,K_{X}+D)=1.$

We conclude that $h^{0}(X,K_{X})=1$. Similarly, we see that
$h^{0}(X,2K_{X})=1$ and so by the characterization Theorem of Chen–Hacon
([CH01]) we conclude that $a_{X}$ is a birational morphism. In addition, by
Theorem 1.6 there is a commutative diagram

$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{V}}$$\textstyle{A(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{X}}$$\textstyle{A(X)}$

and so also $a_{V}$ is birational.

∎

The next result refines Proposition 2.1:

######  Proposition 2.3.

Let $V$ be a smooth quasi-projective variety, $X$ a compactification of $V$
such that $D:=X\setminus V$ is a snc divisor. Let $a_{X}\colon X\to A(X)$ be
the Albanese morphism and let $E>0$ be a divisor on $A(X)$.

If $\overline{P}_{1}(V)>0$ and $D$ contains the support of $a_{X}^{*}E$, then
$\overline{P}_{2}(V)\geq 2$.

###### Proof.

The divisor $E$ is of the form $\pi^{*}H$, where $\pi\colon A\to B$ is a
morphism onto a positive dimensional abelian variety and $H$ is an ample
divisor on $B$. Set $f:=\pi\circ a_{X}$; by assumption there is a positive
integer $N$ such that $ND\succeq f^{*}H$. Set $\Delta:=D-\frac{1}{N}f^{*}H$:
by assumption the $\mathbb{Q}$-divisor $\Delta$ has snc support and
$\Delta=\sum_{i}d_{i}\Delta_{i}$, with $\Delta_{i}$ irreducible divisors and
$0\leq d_{i}\leq 1$. Given $\alpha\in\operatorname{Pic}^{0}(B)$, the divisor
$K_{X}+D+f^{*}(\alpha)$ is $\mathbb{Q}$-linearly equivalent to
$K_{X}+\Delta+f^{*}(\frac{1}{N}H+\alpha)$. Since $\frac{1}{N}H+\alpha$ is
ample if $H$ is, the assumptions of [Fu11, Thm. 6.3] or [Ambro03, Thm. 3.2]
are satisfied and we have

$H^{i}(B,f_{*}\mathcal{O}_{X}(K_{X}+D)\otimes\alpha)=0,\quad\text{for all
}i>0,\quad\text{and all }\alpha\in\operatorname{Pic}^{0}(B).$

So $f_{*}(K_{X}+D)$ is an IT(0)-sheaf, and therefore for all
$\alpha\in\operatorname{Pic}^{0}(B)$ we have
$h^{0}(B,f_{*}\mathcal{O}_{X}(K_{X}+D)\otimes\alpha)=h^{0}(B,f_{*}(K_{X}+D))=\overline{P}_{1}(V)>0$.
So $V^{0}(X,K_{X}+D)$ contains the positive dimensional abelian subvariety
$f^{*}\operatorname{Pic}^{0}(B)=\pi^{*}\operatorname{Pic}^{0}(B)$ and
$\overline{P}_{2}(V)\geq 2$ by Lemma 1.9. ∎

When $q(V)$ is equal to 1, Proposition 2.3 gives:

######  Corollary 2.4.

Let $V$ be a smooth quasi-projective variety, $X$ a compactification of $V$
such that $D:=X\setminus V$ is a snc divisor. Let $a_{X}\colon X\to A(X)$ be
the Albanese morphism.

If $q(V)=\overline{P}_{1}(V)=\overline{P}_{2}(V)=1$, then $a_{X}(V)=A(X)$.

######  Remark 2.5.

Observe that, by Proposition 2.1, we can deduce that if
$\overline{P}_{1}(V)=\overline{P}_{2}(V)=1$ then the map $a_{X}|_{V}$ is
dominant. In addition, as a consequence of Proposition 2.3, we know that the
complement of $a_{X}(V)$ in $A(X)$ does not contain a divisor. Notice that, if
$q(X)>1$, this does not mean that the complement of $a_{X}(V)$ has codimension
>1, since the image of $V$ is not necessarily open in $A(X)$.

##  3\. The geometry of the quasi-Albanese morphism

The aim of this section is to establish the following fundamental step in the
proof of Theorem A:

######  Proposition 3.1.

Let $V$ be a smooth complex algebraic surface with $\overline{q}(V)=2$. Assume
that:

  * (a)

$\overline{P}_{1}(V)=\overline{P}_{2}(V)=1$ and $q(V)>0$; or

  * (b)

$\overline{P}_{1}(V)=\overline{P}_{3}(V)=1$ and $q(V)=0$.

Then the quasi-Albanese morphism $a_{V}\colon V\rightarrow A(V)$ is dominant.

We use different approaches for the case $q(V)=0$ and $q(V)>0$, so we treat
them separately. In fact, Proposition 3.1 is just the combination of
Propositions 3.2 and 3.3 below.

###  3.1. The quasi-Albanese map when $q(V)\geq 1$

######  Proposition 3.2.

Let $V$ be a smooth algebraic surface such that
$\overline{P}_{1}(V)=\overline{P}_{2}(V)=1$, $\overline{q}(V)=2$, and
$q(V)\geq 1$. Then the quasi-Albanese morphism of $V$ is dominant, and hence
generically finite.

###### Proof.

When $q(V)=\overline{q}(V)=2$ the map $a_{V}$ is dominant by Corollary 2.2.

Assume now that $q(V)=1$ and, by contradiction, that the image of $a_{V}$ is a
curve $C$. Denote by $\overline{C}$ the smooth projective model of $C$; since
$\overline{C}$ dominates the elliptic curve $A(X)$, the curve $\overline{C}$
has positive genus and $a_{V}$ extends to a morphism $f\colon
X\to\overline{C}$ such that the Albanese morphism factorizes through
$\overline{C}$. By the universal property of the Albanese map, the map
$\overline{C}\to A(X)$ is an isomorphism. Now Corollary 2.4 implies that
$C=\overline{C}$. In particular, by Proposition 1.5 up to translation $C$ is
an algebraic subgroup of $A(V)$. On the other hand, $C$ must generate $A(V)$
as an algebraic group, so it follows $C=A(V)$, a contradiction. ∎

###  3.2. The quasi-Albanese map when $q(V)=0$

In this section we prove the following

######  Proposition 3.3.

Let $V$ be a smooth algebraic surface such that
$\overline{P}_{1}(V)=\overline{P}_{3}(V)=1$, $\overline{q}(V)=2$, and
$q(V)=0$. Then the quasi-Albanese morphism of $V$ is dominant, and hence
generically finite.

The proof is based on numerical arguments and is quite intricate, so we break
it into several steps. We start by proving two results on curves on smooth
projective surfaces.

######  Lemma 3.4.

Let $X$ be a non-singular projective surface and $A$ a 1-connected effective
divisor with $p_{a}(A)=1$.

Then $A$ contains a 2-connected divisor $B$ such that $p_{a}(B)=1$.

###### Proof.

If $A$ is 2-connected there is of course nothing to prove. Otherwise, take any
decomposition $A=A_{1}+A_{2}$ with $A_{1}A_{2}=1$ and $A_{1}$ minimal with
respect to $A_{1}A_{2}=1$. $A_{2}$ is 1-connected, $A_{1}$ is 2-connected and
by Lemma 1.11 $p_{a}(A_{1})+p_{a}(A_{2})=1$, $p_{a}(A_{1})\geq 0$,
$p_{a}(A_{2})\geq 0$. If $p_{a}(A_{1})=1$ we have proved the statement, if not
we repeat the argument on $A_{2}$. ∎

######  Lemma 3.5.

Let $X$ be a non-singular projective surface and $B$ a 2-connected effective
divisor with $p_{a}(B)=1$. Then:

  1. (i).

if $B$ is not irreducible then every component $\Gamma$ of $B$ is smooth
rational and satisfies $(K_{X}+B)\Gamma=0$;

  2. (ii).

$\omega_{B}=\mathcal{O}_{B}$;

  3. (iii).

if $\Gamma$ is an irreducible component of $B$ and $B-\Gamma\succ 0$, then
$B-\Gamma$ is 1-connected.

###### Proof.

(i) Since $p_{a}(B)=1$, $(K_{X}+B)B=0$. On the other hand for every
irreducible component $\Gamma$ of $B$ one has
$(K_{X}+B)\Gamma=(K_{X}+\Gamma)\Gamma+(B-\Gamma)\Gamma\geq 0$ because
$(K_{X}+\Gamma)\Gamma\geq-2$ by adjunction and $(B-\Gamma)\Gamma\geq 2$ since
$B$ is 2-connected and reducible. So necessarily $(K_{X}+B)\Gamma=0$, and
$\Gamma$ is smooth rational.

(ii) Since, by (i) $\omega_{B}$ has degree $0$ on every component of $B$ and
$h^{0}(B,\omega_{B})=1$, by Lemma 1.10 $\omega_{B}={\mathcal{O}}_{B}$.

(iii) By the proof of (i) one has $\Gamma(B-\Gamma)=2$. Let
$B-\Gamma=A_{1}+A_{2}$ with $A_{1}>0,A_{2}>0$. Then because $B$ is 2-connected
$A_{i}(A_{j}+\Gamma)\geq 2$ for $\\{i,j\\}=\\{1,2\\}$. Since
$(A_{1}+A_{2})\Gamma=2$ necessarily $A_{1}A_{2}\geq 1$ and so $B-\Gamma$ is
1-connected. ∎

######  Lemma 3.6.

Let $X$ be a non-singular projective surface with $q(X)=0$ and let $B$ be an
effective 2-connected divisor satisfying $p_{a}(B)=1$. Then one of the
following occurs:

  1. (a)

$h^{0}(X,2K_{X}+2B)\geq 2$;

  2. (b)

$X$ is rational, and there is a blow-down morphism $\rho\colon X\to T$ with
exceptional divisor $\sum_{j=1}^{n}E_{j}$ (where the $E_{j}$ are $-1$-curves)
such that $K_{X}+B\sim\sum_{j=1}^{n}E_{j}$ and $B$ is disjoint from
$\sum_{j=1}^{n}E_{j}$.

###### Proof.

From Lemma 1.12 we obtain $h^{0}(X,K_{X}+B)=p_{g}(X)+1\geq 1$.

Assume that $K_{X}+B$ is nef. Then since $(K_{X}+B)B=0$, we have
$K_{X}(K_{X}+B)=(K_{X}+B)^{2}\geq 0$ and
$(2K_{X}+B)(K_{X}+B)=2K_{X}(K_{X}+B)\geq 0$. So we have two possibilities:
either $K_{X}+B\sim 0$ (and $p_{g}(X)=0$) or there is a non zero effective
divisor $B_{1}$ in $|K_{X}+B|$, and $p_{a}(B_{1})\geq 1$ implying by Lemma
1.12 that $h^{0}(X,K_{X}+B_{1})\geq 1+p_{g}$, i.e. $h^{0}(X,2K_{X}+B)\geq 1$.

In the second case the restriction map $H^{0}(X,K_{X}+B)\to
H^{0}(B,\omega_{B})$ is surjective because $q(X)=0$. Since
$\omega_{B}=\mathcal{O}_{B}$, the map $H^{0}(X,2K_{X}+2B)\to
H^{0}(B,\omega_{B}^{\otimes 2})$ is also nonzero. So the exact sequence

$0\to H^{0}(X,2K_{X}+B)\to H^{0}(X,2K_{X}+2B)\to H^{0}(B,\omega_{B}^{\otimes
2})$

gives $h^{0}(X,2K_{X}+2B)\geq 2$.

Assume now that $B_{1}$ is not nef and let $\theta$ be an irreducible curve
with $B_{1}\theta<0$. Since $B_{1}$ is effective, $\theta$ is a component of
$B_{1}$ with $\theta^{2}<0$. In addition, for every component $\Gamma$ of $B$,
we have, by Lemma 3.5, $B_{1}\Gamma=0$, so $\theta$ is not a component of $B$,
and thus $B\theta\geq 0$. Then the only possibility is that $\theta$ is an
irreducible $-1$-curve disjoint from $B$. We contract $\theta$ and replace $B$
by its image under the contraction; repeating this process we eventually end
up with a birational morphism of smooth surfaces $\rho\colon X\to T$ such that
$K_{T}+\rho(B)$ is nef, and $\rho(B)$ is still a 2-connected divisor with
$p_{a}=1$.

By the discussion in the previous case, we see that either
$h^{0}(T,2K_{T}+2\rho(B))\geq 2$ or $K_{T}+\rho(B)\sim 0$. In the former case
$h^{0}(X,2K_{X}+2B)\geq h^{0}(T,2K_{T}+2\rho(B))\geq 2$. In the latter case,
taking pull-backs we get $\rho^{*}(K_{T})+B=K_{X}-\sum_{j=1}^{n}E_{j}+B\sim
0$. So, if $L$ is an ample divisor on $T$, then $\rho^{*}L$ is nef and big and
it satisfies $K_{X}\rho^{*}L=-B\rho^{*}L<0$, so $\kappa(X)=-\infty$. Since
$q(X)=0$, the surface $X$ is rational. ∎

For the rest of the section, we refer to the following situation:

######  Setting 3.7.

We let $V$ be a smooth open algebraic surface with $\overline{q}(V)=2$,
$q(V)=0$, and $\overline{P}_{1}(V)=\overline{P}_{3}(V)=1$. We consider the
standard compactification $Z=\mathbb{P}^{1}\times\mathbb{P}^{1}$ of
$A(V)=\mathbb{G}_{m}^{2}$, and we denote by $\Delta:=Z\setminus A(V)$ the
boundary. Finally, we fix a compactification $X$ of $V$ with snc boundary $D$
such that the quasi-Albanese map $a_{V}\colon V\to A(V)$ extends to a morphism
$g\colon X\to Z$.

We exploit the previous results to gain more information on the pair $(X,D)$.

######  Lemma 3.8.

There is a blow down morphism $\rho\colon X\to T$ with exceptional divisor
$\sum_{j=1}^{n}E_{j}$ (where the $E_{j}$ are $-1$-curves), such that one of
the following holds:

  1. (a)

$p_{g}(X)=0$, $h^{0}(D,\omega_{D})=1$, $X$ is a rational surface and there is
a 2-connected divisor $B\preceq D$ such that $K_{X}+B\sim\sum_{j=1}^{n}E_{j}$
and $B$ is disjoint from $\sum_{j=1}^{n}E_{j}$;

  2. (b)

$p_{g}(X)=1$, $h^{0}(D,\omega_{D})=0$, and $T$ is a K3 surface.

In either case there is a divisor $B\succeq 0$ such that
$K_{X}+B\sim\sum_{j=1}^{n}E_{j}$.

###### Proof.

The assumption $\overline{P}_{1}(V)=1$ implies that either $p_{g}(X)=0$ or
$p_{g}(X)=1$; in either case the value of $h^{0}(D,\omega_{D})$ can be
computed by Lemma 1.12.

Assume first $p_{g}(X)=0$. Then, by Lemma 1.12, $q(X)=0$ implies that
$h^{0}(D,\omega_{D})=1$. So there is a connected component $A$ of $D$ such
that $p_{a}(A)=h^{0}(A,\omega_{A})=1$ and by Lemma 3.4 there is a 2-connected
divisor $B\preceq A$ with $p_{a}(B)=1$. By Lemma 3.6 and the hypothesis
$\overline{P}_{2}(V)=1$, $X$ is rational and there is a blow down morphism
$\rho\colon X\to T$ as in (a).

Assume now $p_{g}(X)=1$. In this case $X$ has non-negative Kodaira dimension
and we take $\rho\colon X\to T$ to be the morphism to the minimal model. If
$X$ is of general type, $K_{T}^{2}>0$ and
$h^{0}(X,2K_{X})=h^{0}(T,2K_{T})=K_{T}^{2}+\chi(\mathcal{O}_{X})\geq 2$,
contradicting $\overline{P}_{2}(V)=1$.

If $X$ is properly elliptic then, because $p_{g}(T)=1$ and
$K_{T}\neq\mathcal{O}_{T}$, $h^{0}(T,-K_{T})=0$, and so by duality
$h^{2}(T,2K_{T})=0$. Since $K_{T}^{2}=0$, by the Riemann-Roch theorem we
obtain $h^{0}(T,2K_{T})=h^{1}(T,2K_{T})+\chi(\mathcal{O}_{X})\geq 2$.

So $\kappa(X)=0$ and by the classification of projective surfaces we conclude
that $T$ is a $K3$-surface. We have $K_{X}=\sum_{j=1}^{n}E_{j}$, so in this
case the last claim holds with $B=0$. ∎

######  Lemma 3.9.

One has:

  1. (i).

$h^{0}(X,2K_{X}+D)=h^{0}(X,3K_{X}+2D)=p_{g}(X)$;

  2. (ii).

if $D_{i}$ is the unique divisor in $|i(K_{X}+D)|$, $i=1,2$, then
$h^{0}(D_{i},\omega_{D_{i}})=0$.

###### Proof.

(i) Since $D\succ 0$ and $\overline{P}_{i}(V)=1$ for $i=1,2,3$ by assumption,
we have $p_{g}(X)\leq h^{0}(X,2K_{X}+D)\leq h^{0}(X,3K_{X}+2D)\leq 1$. So if
$p_{g}(X)=1$ the assertion is trivial.

For $p_{g}(X)=0$ and $m\geq 1$ consider the exact sequence:

(5) $0\to H^{0}(mK_{X}+(m-1)D)\to H^{0}(X,m(K_{X}+D))\to
H^{0}(D,\omega_{D}^{\otimes m})$

The second map in (5) is an isomorphism for $m=1$, since $q(X)=0$, so it is
nonzero for all $m\geq 1$, since $D$ is a reduced divisor. Then
$\overline{P}_{2}(V)=1$ and $\overline{P}_{3}(V)=1$ imply that
$h^{0}(X,2K_{X}+D)=h^{0}(X,3K_{X}+2D)=0$.

(ii) is an immediate consequence of $q(X)=0$ and of (i) (see Lemma 1.12).

∎

We now turn to the study of the quasi-Albanese map.

######  Lemma 3.10.

If the image of $a_{V}$ is a curve, then it is isomorphic to
$\mathbb{P}^{1}\setminus\\{0,1,\infty\\}$ and the general fiber of $a_{V}$ is
connected.

###### Proof.

Let $g$ as in Setting 3.7. By assumption, the image $\Gamma$ of $g$ is a
curve. Let $X\to\widetilde{\Gamma}\to\Gamma$ be the Stein factorization of $g$
and let $\Gamma_{0}\subset\widetilde{\Gamma}$ be the image of $V$. Then by the
universal property of the quasi-Albanese map, $A(V)$ is isomorphic to
$A(\Gamma_{0})$. In addition, the image of $a_{V}$ is isomorphic to
$\Gamma_{0}$ by Proposition 1.7. It follows that $g$ has connected fibers,
hence the general fiber of $a_{V}$ is also connected. Finally $\Gamma_{0}$ is
rational, since $q(X)=0$, and has logarithmic genus 2, hence it is isomorphic
to $\mathbb{P}^{1}\setminus\\{0,1,\infty\\}$.∎

We get immediately:

######  Corollary 3.11.

If the image of $a_{V}$ is a curve, then there is a fibration $f\colon
X\to\mathbb{P}^{1}$ such that $D$ contains the supports $F_{1}^{s}$,
$F_{2}^{s}$ and $F_{3}^{s}$ of three distinct fibers $F_{1}$, $F_{2}$ and
$F_{3}$ of $f$.

######  Lemma 3.12.

Let $E_{j}$, $j=1,\dots n$, be the $-1$-curves contracted by the blow down
morphism $\rho\colon X\to T$ of Lemma 3.8.

Then $E_{j}F_{i}^{s}\geq 0$ for all $j,i$.

###### Proof.

Assume by contradiction that $E_{j}F_{1}^{s}<0$. Then $E_{j}$ and $F_{1}^{s}$
have common components. Write $E_{j}=A+C_{1}$ and $F_{1}^{s}=A+C_{2}$ where
$A\succ 0$ and $C_{1},C_{2}\succeq 0$ have no common components. Note that
$C_{2}\succ 0$ otherwise blowing down $E_{j}$ we would contract the whole
fiber. Now $E_{j}F_{1}^{s}<0$ yields $A^{2}+AC_{1}+AC_{2}+C_{1}C_{2}<0$. But,
because $E_{j}$ is a $(-1)$-curve, $E_{j}A=A^{2}+AC_{1}\geq-1$ (see [ADE,
Prop. 3.2]) and because $F_{1}^{s}$ is 1-connected $AC_{2}\geq 1$. Since
$C_{1}C_{2}\geq 0$ we have a contradiction. ∎

######  Lemma 3.13.

Let $E_{j}$, $j=1,\dots n$, be the $-1$-curves contracted by the blow down
morphism $\rho\colon X\to T$ of Lemma 3.8.

Then $\sum_{j=1}^{n}E_{j}+D$ has at most one component transversal to $f$.

###### Proof.

Let $M_{1},M_{2}$ be two distinct components of $\sum_{j=1}^{n}E_{j}+D$
tranversal to $f$. Then because $F_{i}^{s}$ is the support of a full fiber
$M_{k}(\sum_{i=1}^{3}F_{i}^{s})\geq 3$ for $k=1,2$ yielding
$p_{a}(M_{1}+M_{2}+\sum_{i=1}^{3}F_{i}^{s})\geq 2$ and so, by Lemma 1.12,
$h^{0}(X,K_{X}+M_{1}+M_{2}+\sum_{i=1}^{3}F_{i}^{s})\geq 2$.

By Lemma 3.8 there is a divisor $0\preceq B\preceq D$ such that
$K_{X}+B\sim\sum_{j}E_{j}$. Since
$M_{1}+M_{2}+\sum_{i=1}^{3}F_{i}^{s}\leq\sum_{j=1}^{n}E_{j}+D\in|K_{X}+B+D|$
we obtain $h^{0}(X,2K_{X}+B+D)\geq 2$, which contradicts
$\overline{P}_{2}(V)=1$.

∎

######  Corollary 3.14.

Assume $p_{g}(X)=0$, and let $0\prec B\preceq D$ be the divisor such that
$K_{X}+B\sim\sum_{j}E_{j}$ (cf. Lemma 3.8). If $B$ has components in common
with $\sum_{i=1}^{3}F_{i}^{s}$, then one of the following happens:

  1. (i).

$B\leq F_{i}^{s}$ for some $i\in\\{1,2,3\\}$, or ;

  2. (ii).

$B$ has a unique component $H$ transversal to $f$ and $B-H$ is contained in,
say, $F_{1}^{s}$. Furthermore $HF_{1}^{s}=2$ and $HF_{i}^{s}=1$ for $i=2,3$.

###### Proof.

Suppose no component of $B$ is transversal to $f$. Then because $B$ is
connected and we are assuming that $B$ has common components with $\sum
F_{i}^{s}$ we have statement (i).

If there is a component $H$ of $B$ transversal to $f$ the assumption that $B$
has common components with $\sum_{i=1}^{3}F_{i}^{s}$ implies $B-H\neq 0$.
Recall that by Lemma 3.8, the divisor $B$ is $2$-connected and $p_{a}(B)=1$,
so, by Lemma 3.5, $H\simeq\mathbb{P}^{1}$ and $B-H$ is connected. From Lemma
3.13 $B-H$ is contained in fibers of $f$ and since $B-H$ is connected we
obtain $B-H$ contained in $F_{i}^{s}$ for one of $i=1,2,3$, say $F_{1}^{s}$.

In this case $HF_{1}^{s}\geq 2$ (since $B$ is 2-connected) and $HF_{i}^{s}\geq
1$ for $i=2,3$, because $F_{i}^{s}$ is the support of a full fiber. Now
$p_{a}(H+\sum_{i=1}^{3}F_{i}^{s})\geq
p_{a}(H)+\sum_{i=1}^{3}p_{a}(F_{i}^{s})+H(\sum_{i=1}^{3}F_{i}^{s})-3$. Since
$h^{0}(D,\omega_{D})=1$ (cf. Lemma 3.8) $D$ cannot contain an effective
divisor with $p_{a}\geq 2$ and so necessarily $HF_{1}^{s}=2$ and
$HF_{i}^{s}=1$ for $i=2,3$. ∎

######  Lemma 3.15.

Let $E_{j}$, $j=1,\dots n$, be the $-1$-curves contracted by the blow down
morphism $\rho\colon X\to T$ and let $B\preceq D$ be the divisor such that
$K_{X}+B\sim\sum_{j}E_{j}$ (cf. Lemma 3.8):

  1. (i).

if $F_{i}^{s}\leq D-B$ then $E_{j}F_{i}^{s}\leq 1$ for all $E_{j}$;

  2. (ii).

the general fiber $F$ of $f$ satisfies $F\sum_{j=1}^{n}E_{j}=0$.

###### Proof.

(i) Assume that $E_{j}F_{i}^{s}\geq 2$. Then
$h^{0}(E_{j}+F_{i}^{s},\omega_{E_{j}+F_{i}^{s}})=p_{a}(E_{j}+F_{i}^{s})\geq
p_{a}(F_{i}^{s})+1\geq 1$ and so since by the assumption $F_{i}^{s}\leq D-B$,
we have $E_{j}+F_{i}^{s}\leq D_{1}\in|K_{X}+D|$, contradicting Lemma 3.9 (cf.
Lemma 1.11).

(ii) Suppose otherwise, i.e., that there is an irreducible curve
$M\preceq\sum_{j=1}^{n}E_{j}$ transversal to the fibration $f$. Since $B$ and
$\sum_{j=1}^{n}E_{j}$ are disjoint (cf. Lemma 3.8), by Lemma 3.13 the divisor
$B$ must be contained in a fiber of $f$.

Since each $F_{i}^{s}$ is the support of a whole fiber we have $MF_{i}^{s}\geq
1$. Since $M\preceq\sum_{j=1}^{n}E_{j}$, we have $M^{2}=-l<0$. Then among the
$E_{j}$’s there are $E_{p_{1}},...,E_{p_{l-1}}$ such that
$E_{m}=M+\sum_{k=1}^{l-1}E_{p_{k}}$ is one of the $E_{j}$. This is true by
[ADE, Lemma 3.2] if $l=1$, by [ADE, Prop. 4.1] if $l=2$ and by [ADE, Prop.
4.2] if $l>2$.

Since $E_{p_{k}}F_{i}^{s}\geq 0$ for all $k=1,\dots l-1$ and $i=1,2,3$ by
Lemma 3.12, we obtain $E_{m}(\sum F_{i}^{s})\geq 3$. Since
$E_{m}(E_{m}+\sum_{i=1}^{3}F_{i}^{s})\geq 2$, one obtains

  * (a)

$p_{a}(2E_{m}+\sum_{i=1}^{3}F_{i}^{s})\geq 2$ if $p_{g}(X)=0$ and $B\neq 0$ is
contained in one of the $F_{i}^{s}$.

  * (b)

$p_{a}(2E_{m}+\sum_{i=1}^{3}F_{i}^{s})\geq 1$, if $p_{g}(X)=1$ or $p_{g}(X)=0$
and $B$ is not contained in $\sum_{i=1}^{3}F_{i}^{s}$;

Since $E_{m}\leq K_{X}+B$, we have $2E_{m}+\sum F_{i}^{s}\leq 2K_{X}+2B+D$,
which in case (a) yields $h^{0}(X,3K_{X}+2B+D)\geq 2$, contradicting
$\overline{P}_{3}(V)=1$. In case (b) we have $2E_{m}+\sum F_{i}^{s}\leq
2(K_{X}+B)+(D-B)=2K_{X}+B+D$, yielding $h^{0}(X,3K_{X}+B+D)\geq p_{g}(X)+1$
and contradicting Lemma 3.9. ∎

######  Corollary 3.16.

The fibration $f\colon X\to\mathbb{P}^{1}$ descends to a fibration
$\bar{f}\colon T\to\mathbb{P}^{1}$.

######  Lemma 3.17.

The general fiber $F$ of $\bar{f}$ has genus $0$.

###### Proof.

Clearly $f$ and $\bar{f}$ have the same general fiber by construction. Since
$K_{T}+\rho(B)=0$ and $F$ is nef, we have $FK_{T}=-F\rho(B)\leq 0$, so either
$FK_{T}=0$ and $F$ has genus 1, or $K_{T}F=-2$ and $F$ is smooth rational.

Assume by contradiction that $F$ has genus $1$. In case $p_{g}(X)=0$ all the
components of $B$ are contracted by $f$ since $F\rho(B)=0$, so $B$, being
connected, is contained in a fiber of $f$ and we may assume that $B$ and
$F_{2}^{s}+F_{3}^{s}$ are disjoint. The divisor $D_{0}=B+F_{2}^{s}+F_{3}^{s}$
satisfies
$h^{0}(D_{0},\omega_{D_{0}})=p_{a}(B)+p_{a}(F_{2}^{s})+p_{a}(F_{3}^{s})=1+p_{a}(F_{2}^{s})+p_{a}(F_{3}^{s})$.
Since $h^{0}(D_{0},\omega_{D_{0}})\leq
h^{0}(D,\omega_{D})=\overline{P}_{1}(V)=1$ (cf. Lemma 1.12), we conclude that
$p_{a}(F_{2}^{s})=p_{a}(F_{3}^{s})=0$. If $p_{g}(X)=1$, applying the same
argument to $D_{0}:=F_{1}^{s}+F_{2}^{s}+F_{3}^{s}$ we obtain
$p_{a}(F_{i}^{s})=0$ for $i=1,2,3$.

Denote by $\overline{F}_{i}$ the full fiber of $\bar{f}$ corresponding to
$F_{i}^{s}$ for $i=1,2,3$. If $p_{g}(X)=1$ then $T$ is minimal by Lemma 3.8,
hence $\overline{F}_{i}$ does not contain $-1$-curves for $i=1,2,3$. If
$p_{g}(X)=0$ and $E$ is an irreducible $-1$ curve of $T$, then
$\rho(B)E=-K_{T}E=1$, namely $E$ meets $\rho(B)$. So in this case there is no
$-1$-curve contained in $\overline{F}_{2}+\overline{F}_{3}$.

By Lemmas 3.12 and 3.15, we know that $0\leq F_{i}^{s}E_{j}\leq 1$ for $i=2,3$
and also for $i=1$ if $p_{g}(X)=1$. Therefore $\rho^{*}(\rho(F_{i}^{s}))\leq
F_{i}^{s}+\sum_{j=1}^{n}E_{j}$. So the reduced divisor $\rho(F_{i}^{s})$ still
has $p_{a}=0$ for $i=1,2,3$ in the case $p_{g}(X)=1$ and for $i=2,3$ in the
case $p_{g}(X)=0$.

For $i=1,2,3$ in the case $p_{g}(X)=1$ and for $i=2,3$ in the case
$p_{g}(X)=0$, the elliptic fibers $\overline{F}_{i}$, since they do not
contain $-1$-curves and have support with $p_{a}=0$, must be of type ∗ (see
[BHPV, Chp.V, §7]). Note that, in particular, the $\overline{F}_{i}$ cannot be
multiple fibers of $\bar{f}$ (see [BHPV, Chp.V, §7]). So if $\overline{F}$ is
a fiber of type ∗ with support $F_{0}$, we have $2F_{0}\geq\overline{F}$ if
$\overline{F}$ is of type $I_{b}^{*}$ ($=\tilde{D}_{4+b}$) and
$3F_{0}\geq\overline{F}$ if $\overline{F}$ is of type $IV^{*}$
($=\tilde{E}_{6}$).

On the other hand, if $p_{g}(X)=0$, note that the sum of the Euler numbers
$e(\overline{F}_{2})+e(\overline{F}_{3})$ cannot exceed 12, since a relatively
minimal elliptic fibration on a rational surface has $c_{2}=12$. Since the
Euler numbers of fibers of type ∗ are always bigger than $6$ except for type
$I_{0}^{*}$, $\overline{F}_{2}$ and $\overline{F}_{3}$ must be of type
$I_{0}^{*}$ and so, for instance, $2(F_{2}^{s}+\sum_{j=1}^{n}E_{j})$ contains
$\rho^{*}(\overline{F}_{2})=F_{2}$. Since $h^{0}(X,F_{2})=2$ we obtain
$h^{0}(X,2K_{X}+2B+2F_{2}^{s})\geq 2$, contradicting $h^{0}(X,2K_{X}+2D)=1$.

Similarly, if $p_{g}(X)=1$ then the sum of the Euler numbers
$e(\overline{F}_{1})+e(\overline{F}_{2})+e(\overline{F}_{3})$ cannot exceed
24. If one of the $\overline{F}_{i}$ is of type $I_{b}^{*}$ we have a
contradiction as above and if one of the $\overline{F}_{i}$ is of type
$IV^{*}$ ($=\tilde{E}_{6}$), then $F_{i}\leq
3F_{i}^{s}+3\sum_{j=1}^{n}E_{j}\leq 3K_{X}+3D$, contradicting
$h^{0}(X,3K_{X}+3D)=1$. On the other hand if all the $\overline{F}_{i}$ are of
type $II^{*}$ ($=\tilde{E}_{8}$) or type $III^{*}$ ($=\tilde{E}_{7}$) the sum
of the Euler numbers is larger than 24. ∎

###### Proof of Proposition 3.3.

By Lemma 3.17 the general fiber $F$ of $f$ has genus $0$, and in particular we
have $p_{g}(X)=0$. Since $K_{X}F=-2$, we have $BF=2$ and so $B$ has a
component $H$ transversal to $f$. If $B$ has no common component with
$F_{i}^{s}$, $i=1,2,3$, then we set $D_{0}=B+F_{1}^{s}+F_{2}^{s}+F_{3}^{s}$
and we compute

$p_{a}(D_{0})=p_{a}(B)+p_{a}(F_{1}^{s})+p_{a}(F_{2}^{s})+p_{a}(F_{3}^{s})+B(F_{1}^{s}+F_{2}^{s}+F_{3}^{s})-3.$

Since $D_{0}$ is reduced and connected, we have
$p_{a}(D_{0})=h^{0}(D_{0},\omega_{D_{0}})\leq
h^{0}(D,\omega_{D})=\overline{P}_{1}(V)=1$ (cf. Lemma 1.11 and 1.12) and we
conclude that $p_{a}(F_{i}^{s})=0$, $BF_{i}^{s}=1$ for $i=1,2,3$. On the other
hand, if $B$ and $F_{1}^{s}+F_{2}^{s}+F_{3}^{s}$ have common components, then
by Corollary 3.14 at least two of the $F_{i}^{s}$, say $F_{2}^{s}$ and
$F_{3}^{s}$, have no common components with $B$ and satisfy $F_{i}^{s}B=1$ for
$i=2,3$.

So in any case $F_{2}^{s}$ contains a unique irreducible curve $\Gamma$ such
that $\Gamma B\neq 0$. Since $BF=2$ for a general fiber of $F$ and
$BF_{2}^{s}=1$, $\Gamma$ appears with multiplicity $2$ in the full fiber
$F_{2}$ containing $F_{2}^{s}$. The curve $\Gamma$ is not contracted by
$\rho$, since $B\Gamma=1$ and $B$ does not meet the $\rho$-exceptional curves.

Write $\rho^{*}(\rho(\Gamma))=\Gamma+Z$, with $Z$ an exceptional divisor.
Since $B=\rho^{*}(\rho(B))$ the projection formula gives

$1=B\Gamma=\rho^{*}(\rho(B))\Gamma=\rho^{*}(\rho(B))(\Gamma+Z)=\rho^{*}(\rho(B))\rho^{*}(\Gamma)=\rho(B)\rho(\Gamma).$

Since $K_{T}+\rho(B)=0$, we have $K_{T}\rho(\Gamma)=-1$. The curve
$\rho(\Gamma)$ is contained in a fiber of $\bar{f}$, so $\rho(\Gamma)^{2}\leq
0$ and $\rho(\Gamma)$ is a $-1$-curve by the adjunction formula. So
$\overline{F}_{2}=2\rho(\Gamma)+C$, where $C$ does not contain $\rho(\Gamma)$.
The components of $C$ do not meet $\rho(B)=-K_{T}$, hence they are all
$-2$-curves. From $\rho(\Gamma)\overline{F}_{2}=0$, we get $C\rho(\Gamma)=2$.
If there are two distinct components $N_{1}$ and $N_{2}$ of $C$ with
$\rho(\Gamma)N_{i}=1$, then $N_{1}$ and $N_{2}$ are disjoint, since the dual
graph of $\overline{F}_{2}$ is a tree. So $(2\rho(\Gamma)+N_{1}+N_{2})^{2}=0$
and therefore $\overline{F}_{2}=2\rho(\Gamma)+N_{1}+N_{2}$. If there is only a
component $N$ of $G$ with $N\rho(\Gamma)=1$, then $N$ appears in
$\overline{F}_{2}$ with multiplicity 2 and $M_{1}:=\rho(\Gamma)+N$ is a
(reducible and reduced ) $-1$-curve such that $\overline{F}_{2}=2M_{1}+C_{1}$
and $C_{1}$ and $M_{1}$ have no common component. So we may repeat the
previous argument and either write $\overline{F}_{2}=2M_{1}+N_{1}+N_{2}$,
where $N_{1},N_{2}$ are disjoint $-2$-curves contained in $C_{1}$ with
$N_{i}M_{1}=1$, or $\overline{F}_{2}=2(M_{1}+N)+C_{2}$, where $N$ is a
component of $C$ such that $M_{2}:=M_{1}+N$ is a $-1$-curve and $C_{2}$ and
$M_{2}$ have no common component. This process must of course terminate,
showing that $\overline{F}_{2}=2M_{0}+N_{1}+N_{2}$, where $M_{0}$ is a reduced
$-1$-curve, and $N_{1}$ and $N_{2}$ are disjoint $-2$-curves not contained in
$M_{0}$. In particular we have shown that $\overline{F}_{2}$ has no component
of multiplicity $>2$.

Since (as in the proof of Lemma 3.17) $\rho^{*}(\rho(F_{2}^{s}))\leq
F_{2}^{s}+\sum_{j=1}^{n}E_{j}$ we conclude that
$2F_{2}^{s}+2\sum_{j=1}^{n}E_{j}$ contains the full fiber $F_{2}$ of $f$ and
as in the in the proof of Lemma 3.17 we obtain
$h^{0}(X,2K_{X}+2B+2F_{2}^{s})\geq 2$, contradicting
$h^{0}(X,2K_{X}+2D)=\overline{P}_{2}(V)=1$.

So we have excluded all the possibilities for the genus of the general fiber
$F$ of the fibration induced by $a_{V}$ if $a_{V}$ is not dominant and the
proof is complete. ∎

######  Example 3.18.

The hypothesis $\overline{P}_{3}(V)=1$ in Proposition 3.3 is necessary. A K3
surface with an elliptic fibration with three singular fibers of type
$IV^{*}$(cf. the proof of Lemma 3.17) does exist. Take for instance the
elliptic curve $C$ with an automorphism $h$ of order 3, let $\mathbb{Z}/3$ act
on $C\times C$ by $(x,y)\mapsto(hx,h^{2}y)$ and denote by $X_{0}$ the quotient
surface. Then $X_{0}$ has 9 singular points of type $A_{2}$, and the first
projection $C\times C\to C$ descends to an isotrivial elliptic fibration
$X_{0}\to C/\mathbb{Z}_{3}\cong\mathbb{P}^{1}$ with three “triple” fibers,
each containing three of the 9 singular points. The minimal resolution $X$ of
$X_{0}$ is a K3 surface with an isotrivial elliptic fibration with three
fibers of type $IV^{*}$. Alternatively, $X$ can be constructed as the minimal
resolution of a simple $\mathbb{Z}_{3}$-cover of
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ branched over three fibers of one of the
fibrations plus three fibers of the other.

Consider the surface $V:=X\backslash\\{F_{1}^{s},\;F_{2}^{s},\;F_{3}^{s}\\}$,
where the $F_{i}^{s}$ denote the supports of the 3 fibers of type IV*. Then
$V$ has $\overline{P}_{1}(V)=\overline{P}_{2}(V)=1$ and $\overline{q}(V)=2$
(see Proposition 1.1), and its quasi-Albanese map is the restriction of the
elliptic fibration $X\to\mathbb{P}^{1}$ and so it is not dominant.

##  4\. Proof of Theorem A

Thanks to what we have proven in the previous section, we know that in the
assumptions of Theorem A the quasi-Albanese map of $V$ is dominant. Since $V$
and $\operatorname{Alb}(V)$ have the same dimension, we can conclude that
$a_{V}$ is generically finite. In particular there is a generically finite
morphism $g:X\rightarrow Z$, where $Z$ is the compactification of
$\operatorname{Alb}(V)$ described in Proposition 1.3. Now, our proof boils
down to the following two facts

  1. 1.

the morphism $g$ has degree 1;

  2. 2.

all the components of $D$ that are not mapped to the boundary $\Delta$ of
$\operatorname{Alb}(V)$ are contracted by $g$.

When $q(V)=2$, the first assertion is Corollary 2.2. The second statement can
be proven by contradiction (see §4.2).

The situation is more involved when $q(V)<2$. We start by proving a slightly
weaker version of assertion 2 (Lemma 4.5) and we use it to show that the
finite part of the Stein factorization of $g$ is étale over $A(V)$ (in case
$q(V)=1$ this requires also a topological argument). Now the universal
property of the quasi-Albanese map implies that $g$ has indeed degree 1.
Finally, we complete the proof of assertion 2 by means of a local computation.

###  4.1. Preliminary steps

######  Notation 4.1.

We let $V$ be a smooth open algebraic surface $V$ with $\overline{q}(V)=2$. We
assume $\overline{P}_{1}(V)=\overline{P}_{2}(V)=1$ if $q(V)\geq 1$ and
$\overline{P}_{1}(V)=\overline{P}_{3}(V)=1$ if $q(V)=0$.

If $q(V)\leq 1$ fix a compactification $Z$ with boundary $\Delta$ of the
quasi-Albanese variety $A(V)$ of $V$ as follows:

  * •

if $q(V)=1$, we take $Z$ a $\mathbb{P}^{1}$-bundle over the compact part $A$
of $A(V)$ as in Corollary 1.4, and we write $\Delta=\Delta_{1}+\Delta_{2}$,
where $\Delta_{i}$ are disjoint sections of $Z\to A$;

  * •

if $q(V)=0$ (and thus $A(V)=\mathbb{G}_{m}^{2}$) we take
$Z=\mathbb{P}^{1}\times\mathbb{P}^{1}$ with the obvious choice of the boundary
$\Delta$.

We fix a compactification $X$ of $V$ with snc boundary $D$ such that the
quasi-Albanese map $a_{V}\colon V\to A(V)$ extends to a morphism $g\colon X\to
Z$. In addition, we write $H:=g^{-1}(\Delta)$ (set-theoretic inverse image)
and in case $q(V)=1$ we also write $H_{i}=g^{-1}(\Delta_{i})$, $i=1,2$. Note
that $D\succeq H$ by construction. When $q(V)=1$ we denote by $a_{X}\colon
X\to A=A(X)$ the Albanese map of $X$.

The proof is quite involved so we break it into several smaller steps and we
examine the case $q(V)\leq 1$ since by Corollary 2.2, for $q(V)=2$ we already
know that $a_{V}$ is birational.

######  Lemma 4.2.

If $q(V)\leq 1$, then the divisor of poles of a generator of
$H^{0}(X,K_{X}+D)$ is a non zero subdivisor of $H$.
In particular, $p_{g}(X)=0$ and $h^{0}(X,K_{X}+H)=1$.

###### Proof.

The vector space $H^{0}(Z,\Omega^{1}_{Z}(\log\Delta))$ is generated by two
logarithmic $1$-forms $\tau_{1}$ and $\tau_{2}$ such that
$\omega:=\tau_{1}\wedge\tau_{2}$ vanishes nowhere on $A(V)$ (cf. Proposition
1.3) and has poles exactly on $\Delta$. More precisely, if $q(V)=1$ then we
can take $\tau_{1}$ to be the pull back of a nonzero regular 1-form on
$A=A(X)$ via the projection $Z\to A$ and $\tau_{2}$ a logarithmic $1$-form
with poles on $\Delta_{1}$ and $\Delta_{2}$ (cf. proof of Corollary 1.4); if
$q(V)=0$ then we can take $\tau_{1}$ and $\tau_{2}$ to be pullbacks of non
zero logarithmic forms via the two projections
$\mathbb{G}_{m}^{2}\to\mathbb{G}_{m}$.

Since $g$ is surjective by Propositions 3.2 and 3.3, $g^{*}\omega$ is a non-
zero logarithmic form on $X$ and a local computation shows that if $\Gamma$ is
an irreducible component of $H$ not contracted by $g$, then $g^{*}\omega$ has
a pole along $\Gamma$. On the other hand the poles of $\omega$ are contained
in $H$ by construction.

Since $\overline{P}_{1}(V)=h^{0}(X,K_{X}+D)=1$, it follows immediately that
$p_{g}(X)=0$ and $h^{0}(X,K_{X}+H)=1$. ∎

######  Lemma 4.3.

  1. (i).

If $q(V)=0$, then $H$ is connected and $h^{0}(\omega_{H})=1$;

  2. (ii).

If $q(V)=1$, then for $i=1,2$ the divisor $H_{i}$ is connected and
$h^{0}(\omega_{H_{i}})=1$

###### Proof.

(i) The divisor $H$ is connected since it is the support of the nef and big
divisor $g^{*}\Delta$. Lemma 1.12 gives $h^{0}(H,\omega_{H})=1$ since
$p_{g}(X)=0$ by Lemma 4.2.

(ii) Consider the exact sequence

$H^{1}(X,-H)\to H^{1}(X,{\mathcal{O}}_{X})\to H^{1}(H,{\mathcal{O}}_{H}).$

The second map in the sequence is non zero, since $H$ is mapped onto $A$ by
the Albanese map $a_{X}$, so in this case Lemma 1.12 gives
$h^{0}(H,\omega_{H})=2$. The divisor $H$ is the disjoint union of $H_{1}$ and
$H_{2}$, so
$2=h^{0}(H,\omega_{H})=h^{0}(H_{1},\omega_{H_{1}})+h^{0}(H_{2},\omega_{H_{2}})$.
For $i=1,2$ let $S_{i}$ be a component of $H_{i}$ such that
$g(S_{i})=\Delta_{i}$. Since $\Delta_{i}$ has geometric genus 1, by Lemma 1.11
we have $1\leq h^{0}(S_{i},\omega_{S_{i}})\leq h^{0}(H_{i},\omega_{H_{i}})$.
So we have $h^{0}(S_{i},\omega_{S_{i}})=h^{0}(H_{i},\omega_{H_{i}})=1$ for
$i=1,2$ and $S_{i}$ is the only component of $H$ with $p_{a}>0$. Assume now by
contradiction that $H_{i}=B_{i}+C_{i}$, where $B_{i}$ and $C_{i}$ are disjoint
non zero effective divisors and $S_{i}\preceq B_{i}$. Then all the components
of $C_{i}$ are rational and so their images via $g$ are contained in fibers of
$Z\to A$. Since $g(C_{i})\subset\Delta_{i}$, it follows that $g(C_{i})$ is a
finite set. So the intersection form on the set of components of $C_{i}$ is
negative definite, contradicting the fact that $B_{i}+C_{i}$ is the support of
the nef divisor $g^{*}\Delta_{i}$. ∎

Since $K_{Z}+\Delta=0$, the logarithmic ramification formula (3) gives
$K_{X}+D\sim\overline{R}_{g}$. We aim to show that the components of
$\overline{R}_{g}$ not contained in $H$ are contracted to points. We begin
with a simple observation:

######  Lemma 4.4.

If $\Gamma$ is an irreducible component of $\overline{R}_{g}$, then
$h^{0}(X,K_{X}+H+\Gamma)\leq 1$.

###### Proof.

We have $(K_{X}+H)+\Gamma\preceq(K_{X}+D)+\overline{R}_{g}=2(K_{X}+D)$. So
$h^{0}(X,K_{X}+H+\Gamma)\leq\overline{P}_{2}(V)=1$. ∎

######  Lemma 4.5.

Let $C$ be the union of all the components of $\overline{R}_{g}$ that are not
contained in $H$ and are not contracted by $g$. Then:

  1. (i).

if $q(V)=0$, then $C=0$;

  2. (ii).

if $q(V)=1$ and $C>0$, then $C$ is irreducible and $g(C)$ is a ruling of $Z$.

###### Proof.

(i) Let $\Gamma$ be an irreducible component of $C$. Since $g(\Gamma)$ is a
curve not contained in $\Delta$ and since $\Delta$ is the union of two fibers
of the first projection $\mathbb{G}^{2}_{m}\to\mathbb{G}_{m}$ and two fibers
of the second projection, $g(\Gamma)\cap\Delta$ contains at least two distinct
points. So $\Gamma\cap H$ also contains at least two distinct points, and
therefore $H\Gamma\geq 2$ and
$p_{a}(H+\Gamma)=p_{a}(H)+p_{a}(\Gamma)+H\Gamma-1\geq p_{a}(H)+1$. Since both
the reduced divisors $H$ and $H+\Gamma$ are connected by Lemma 4.3, we have
$p_{a}(H)=h^{0}(H,\omega_{H})$ and
$h^{0}(H+\Gamma,\omega_{H+\Gamma})=p_{a}(H+\Gamma)\geq
h^{0}(H,\omega_{H})+1=2$, where the last equality follows from Lemma 4.3. Now
Lemma 1.12 gives
$h^{0}(X,K_{X}+H+\Gamma)=h^{0}(H+\Gamma,\omega_{H+\Gamma})\geq 2$ (recall that
$p_{g}(X)=0$ by Lemma 4.2), contradicting Lemma 4.4.

(ii) Let again $\Gamma$ be a component of $C$ such that $g(\Gamma)$ is not a
ruling of $Z$. Then $\Gamma$ dominates $A$ and therefore $p_{a}(\Gamma)\geq
1$. If $\Gamma$ is disjoint from $H$, then
$h^{0}(H+\Gamma,\omega_{H+\Gamma})=h^{0}(H,\omega_{H})+h^{0}(\Gamma,\omega_{\Gamma})\geq
3$ by Lemma 4.3. If $\Gamma$ intersects $H$, then it intersects both $H_{1}$
and $H_{2}$, since they support the numerically equivalent divisors
$g^{*}\Delta_{1}$ and $g^{*}\Delta_{2}$. So $\Gamma+H$ is connected and
$h^{0}(\Gamma+H,\omega_{\Gamma+H})=p_{a}(\Gamma+H)\geq
p_{a}(\Gamma)+p_{a}(H)+1=3$. In either case, Lemma 1.12 gives
$h^{0}(X,K_{X}+H+\Gamma)\geq 2$, contradicting Lemma 4.4. So we conclude that
$g(\Gamma)$ is a ruling of $Z$. Assume that $C$ has at least two components
$\Gamma_{1}$ and $\Gamma_{2}$: then the same argument as above gives
$h^{0}(X,K_{X}+H+\Gamma_{1}+\Gamma_{2})\geq 2$, contradicting Lemma 4.4 again.
∎

Now we consider the Stein factorization
$X\overset{\nu}{\to}\overline{X}\overset{\overline{g}}{\to}Z$ of $g$.

######  Lemma 4.6.

The morphism $\overline{g}$ is étale over $A(V)$.

###### Proof.

Let $m:=\deg g$; if $m=1$ then $\overline{g}$ is an isomorphism and the claim
is of course true. So we may assume $m>1$.

The map $\overline{g}$ is finite by construction, so by purity of the branch
locus it is enough to show that there is no component of the (usual)
ramification divisor of $g$ that is not contained in $H$ and is not contracted
to a point. By Lemma 1.8 such a curve is a component of the logarithmic
ramification divisor $\overline{R}_{g}$. So if $q(V)=0$ the statement follows
directly from Lemma 4.5. Therefore we assume for the rest of the proof that
$q(V)=1$.

Again by Lemma 4.5, if $\overline{g}$ is not étale then there is exactly one
irreducible curve $\Gamma$ in $R_{g}$ such that $\Gamma$ is not contained in
$H$ and is not contracted by $g$, and the image of $\Gamma$ is a ruling $\Phi$
of $Z$. So $\overline{g}$ restricts to a connected étale cover
$q\colon\widetilde{W}\to W:=A(V)\setminus\Phi$. The preimage in $X$ of a
ruling of $Z$ is a fiber of the Albanese map $a_{X}\colon X\to A(V)$ and so it
is connected. So $q$ restricts to a connected cover of $\mathbb{G}_{m}$.

Since algebraically trivial line bundles are topologically trivial, Corollary
1.4 implies that $Z$ is homeomorphic to $A\times\mathbb{P}^{1}$, $A(V)$ is
homeomorphic to $A\times\mathbb{G}_{m}$ and $W$ is homeomorphic to
$(A\setminus\\{a\\})\times\mathbb{P}^{1}$, where $a\in A$ is a point. Fix base
points in $\widetilde{W}$ and $W$ and denote by $N$ the subgroup of index $m$
of $\pi_{1}(W)\simeq\pi_{1}(A\setminus\\{a\\})\times\pi_{1}(\mathbb{G}_{m})$
corresponding to the cover $q$. If $\gamma$ is a generator of
$\pi_{1}(\mathbb{G}_{m})$, we have
$N\cap\pi_{1}(\mathbb{G}_{m})=<\gamma^{m}>$. So the elements
$1,\gamma,\dots\gamma^{m-1}$ represent distinct left cosets of $\pi_{1}(W)$
modulo $N$. Since $\pi_{1}(\mathbb{G}_{m})$ is a central subgroup of
$\pi_{1}(W)$, it follows that left and right cosets modulo $N$ coincide,
namely $N$ is a normal subgroup of $\pi_{1}(W)$ and the quotient
$\pi_{1}(W)/N$ is cyclic of order $m$. Since the map
$\pi_{1}(W)\to\pi_{1}(A(V))$ is just abelianization, $N$ is the preimage of an
index $m$ subgroup $\overline{N}<\pi_{1}(A(V))$. This shows that $q$ extends
to an étale cover $q^{\prime}\colon W^{\prime}\to A(V)$. Since by [DG94, Thm.
3.4] both $q$ and $q^{\prime}$ extend uniquely to an analytically branched
covering of $Z$, it follows that $\overline{g}\colon\overline{X}\to Z$ extends
$q^{\prime}$, that is, $\overline{g}$ is étale over $A(V)$ as claimed. ∎

###  4.2. Conclusion

We are finally ready to complete the proof of Theorem A.

###### Proof of Theorem A.

Consider the case $q(V)=2$ first. By Corollary 2.2, we only need to show that
all components of $D$ are contracted by $a_{X}$. So assume for contradiction
that there is an irreducible component $\Gamma$ of $D$ that is not contracted
by $a_{X}$ and denote by $\overline{\Gamma}$ its image in $A(X)$; note that
the map $\Gamma\to\overline{\Gamma}$ is birational. The geometric genus of
$\overline{\Gamma}$ is positive and if it is equal to 1, then
$\overline{\Gamma}$ is a translate of an abelian subvariety of $A(X)$, so in
particular it is smooth.

Assume first that $p_{a}(\Gamma)=1$: then $\Gamma$ is smooth of genus 1 and
$\Gamma\to\overline{\Gamma}$ is an isomorphism. It follows that
$a_{X}^{*}\overline{\Gamma}\preceq K_{X}+\Gamma\preceq K_{X}+D$. Since
$h^{0}(A(X),2\overline{\Gamma})=2$, we have a contradiction to
$\overline{P}_{2}(V)=2$. If $p_{a}(\Gamma)\geq 2$, then by Serre duality and
Riemann–Roch for all $\alpha\in\operatorname{Pic}^{0}(X)$ we have
$h^{0}(X,K_{X}+\Gamma+\alpha)\geq\chi(K_{X}+\Gamma)=p_{a}(\Gamma)-1\geq 1$.
Lemma 1.9 gives $h^{0}(X,2(K_{X}+\Gamma))\geq 2$, contradicting again the
assumption $\overline{P}_{2}(V)=1$. So all the components of $D$ are
$a_{X}$-exceptional.

From now we assume $q(V)\leq 1$. By Lemma 4.6, the quasi-Albanese map
$a_{V}\colon V\to A(V)$ factors through an étale cover of degree $m:=\deg g$.
Since a finite étale cover of a quasi-abelian variety is also a quasi-abelian
variety (Proposition 1.5) by the universal property of $a_{V}$ (cf. Theorem
1.6) we have $m=1$, namely $g$ is birational. To complete the proof we need to
show that all the components of $D-H$ are contracted by $g$. Since
$D-H\preceq\overline{R}_{g}$ by Lemma 1.8, in case $q(V)=0$ the claim follows
by Lemma 4.5.

Assume $q(V)=1$. By Lemma 4.5 there is at most an irreducible curve
$\Gamma\preceq D-H$ such that $\Gamma$ is not contracted by $g$, and
$g(\Gamma)$ is a ruling $\Phi$ of $Z$. We are going to show that
$g^{*}\Phi\preceq\overline{R}_{g}$, hence $g^{*}(2\Phi)\preceq
2\overline{R}_{g}\sim 2(K_{X}+D)$. Since $h^{0}(Z,2\Phi)=2$, this contradicts
the assumption $\overline{P}_{2}(V)=1$.

Write $g^{*}\Phi=\Gamma+\sum_{i}\alpha_{i}C_{i}$, where the $C_{i}$ are
distinct irreducible curves contracted by $g$ and
$\alpha_{i}\in\mathbb{N}_{>0}$. Let $u,v$ be local coordinates on $Z$ centered
at the point $P_{i}:=g(C_{i})$ such that $u=0$ is the ruling $\Phi$ of $Z$; if
in addition $P_{i}\in\Delta$, we assume that $v$ is a local equation of
$\Delta$. At a general point of $C_{i}$ we have $u=x^{\alpha_{i}}a$,
$v=x^{\beta_{i}}b$, where $x$ is a local equation for $C_{i}$, $a,b$ are non
zero regular functions and $\beta_{i}>0$ is an integer. A simple computation
gives

(6) $g^{*}\frac{dv}{v}=\beta_{i}\frac{dx}{x}+\frac{db}{b},\qquad
g^{*}\left(\frac{du}{u}\wedge\frac{dv}{v}\right)=\frac{dx}{x}\wedge\left(\alpha_{i}\frac{db}{b}-\beta_{i}\frac{da}{a}\right).$

Note that $\alpha_{i}\frac{db}{b}-\beta_{i}\frac{da}{a}$ is a regular 1-form.
So if $P_{i}\in\Delta$, then $C_{i}\preceq D$, $K_{Z}+\Delta$ is locally
generated by $u\wedge\frac{dv}{v}$, $K_{X}+D$ is locally generated by
$\frac{dx}{x}\wedge dy$ and $C_{i}$ appears in $\overline{R}_{g}$ with
multiplicity $\geq\alpha_{i}$.

If $P_{i}\notin\Delta$, then $K_{Z}+\Delta$ is locally generated by $du\wedge
dv$, $K_{X}+D$ is locally generated by $\frac{dx}{x}\wedge dy$ or by $dx\wedge
dy$ according to whether $C_{i}\preceq D$ or not; in either case $C_{i}$
appears in $\overline{R}_{g}$ with multiplicity
$\geq\alpha_{i}+\beta_{i}-1\geq\alpha_{i}$. Summing up, we have shown
$g^{*}\Phi\preceq\overline{R}_{g}$, as promised. ∎

## References