1 Introduction
Let V be a smooth complex quasiprojective 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 quasiprojective 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 mth logplurigenus of V is $\overline {P}_m(V):=h^0(X,m(K_X+D))$ ;

• the logKodaira dimension of V is $\overline {\kappa }(V):=\kappa (X,K_X+D);$

• the logirregularity 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,
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 quasiabelian variety (i.e., an algebraic group which does not contain $\mathbb {G}_a$ ) $A(V)$ , called the quasiAlbanese variety of V. This comes equipped with a morphism $a_V\colon V\rightarrow A(V)$ which is called the quasiAlbanese morphism. Iitaka in [Reference Iitaka12] characterizes quasiabelian surfaces as surfaces of logKodaira dimension 0 and logirregularity 2. More precisely, he proves the following.
Theorem (Iitaka, [Reference Iitaka12]).
Let V be a smooth complex algebraic surface. Then $\overline {\kappa }(V)=0$ and $\overline {q}(V)=2$ if and only if the quasiAlbanese 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 quasiabelian 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 quasiAlbanese 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 weakly weak proper birational (WWPB) equivalences introduced by Iitaka in [Reference Iitaka10], 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 quasiabelian 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 WWPBmaps between normal affine varieties are actually isomorphisms (see [Reference Iitaka10, 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 [Reference Kawamata14], proved in any dimension a weaker form of Iitaka’s theorem, showing that the quasiAlbanese morphism of an algebraic variety of logKodaira dimension 0 and logirregularity equal to the dimension is birational. In the compact case, effective versions of this result have been given in [Reference Chen and Hacon3] and [Reference Pareschi, Popa and Schnell19]: in [Reference Chen and Hacon3], 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 [Reference Pareschi, Popa and Schnell19], the analogous statement is proven for compact Kähler manifolds.
So we are led to formulate the following conjecture:
Conjecture. Let V a smooth complex quasiprojective 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 quasiAlbanese 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 the quasiAlbanese map not dominant (see Example 4.18).
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 quasiAlbanese map of V is dominant. When $ q(V)=2$ , one obtains as an immediate consequence of the work of Chen and Hacon [Reference Chen and Hacon3] that the quasiAlbanese 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 [Reference Green and Lazarsfeld8] and their more recent extensions to pairs (see [Reference Popa and Schnell20] and [Reference Shibata21]).
Once we know that the quasiAlbanese 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 2.3 for more details) together with some geometric considerations to conclude that the quasiAlbanese map is birational.
The last step of our argument consists in proving that the quasiAlbanese map is proper outside a finite set of points. This is done by showing that the boundary divisor D gets contracted by the quasiAlbanese 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 §2, we recall the necessary prerequisites; §3 contains some results that hold in arbitrary dimension for $q(V)>0$ (see in particular Corollary 3.2). Then we focus on surfaces. Section 4 is devoted to proving that the quasiAlbanese map is dominant: here the results of §3 are crucial in case $q(V)>0$ . In §5 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 $D_1D_2$ 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 that $g^{1}$ is not defined at q; so one has $E^2=K_XE=1$ but E may be reducible and/or nonreduced.
2 Preliminaries
Our proof of Theorem A combines different arguments and techniques; in this section, we recall briefly the necessary prerequisites and set the notation.
2.1 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 ndimensional variety X together with a snc divisor D, we can define the sheaf of logarithmic 1forms along D by setting
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 mforms is defined as
and, in particular, we have that
We recall the following condition for the existence of logarithmic 1forms with prescribed poles.
Proposition 2.1. 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 $a_i\ne 0$ for $i=1,\dots k$ .
Proof. Let $D_1, \dots D_t$ be the irreducible components of D; consider the residue sequence
The claim follows from the fact that the associated coboundary map $\delta \colon \oplus _{i=1}^tH^0(X,{\mathcal O}_{D_i})\to H^1(X,\Omega ^1_X)$ 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 quasiprojective 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 $D\succ0$ is a smooth ample divisor, are birational but they have different logarithmic Kodaira dimensions. For this reason, Iitaka in [Reference Iitaka10] introduced the notion of WPBequivalence (weakly proper birational equivalence). It is the equivalence relation generated by the proper birational morphisms and by the open inclusions $V\subset V'$ such that $V'\setminus V$ has codimension at least 2.
WPBequivalent 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 [Reference Iitaka11, Prop. 1]. Therefore, it would seem that WPBequivalence is the right notion of equivalence when studying the birational geometry of open varieties. However, the set of WPBmaps 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 [Reference Iitaka10] defines the notion of WWPBequivalence, proving that WWPBequivalent varieties have the same logarithmic plurigenera and irregularity.
2.2 Quasiabelian varieties and the quasiAlbanese map
Here, we introduce one of the main characters of our story: quasiabelian varieties, which for many aspects can be thought of as a nonprojective analogue of abelian varieties. We recall briefly the facts that we need.
Definition 2.2. A quasiabelian 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
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 [Reference Iitaka9, §10]:
Proposition 2.3. 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 2.3:
Corollary 2.4. Let G be a quasiabelian 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 $L\in \operatorname {\mathrm {Pic}}^0(A)$ and the boundary $\Delta $ is the disjoint union of two sections $\Delta _1$ and $\Delta _2$ of $Z\to A$ .
Proof. By Proposition 2.3, 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 fiber of $Z\to A$ in exactly two points, so it is a smooth bisection. By Proposition 2.3, there is a nonregular logarithmic 1form of Z with poles contained in $\Delta $ , hence by Proposition 2.1 $\Delta $ is reducible and its components satisfy a nontrivial 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 $L\in \operatorname {\mathrm {Pic}}(A)$ . Since in $\operatorname {\mathrm {Pic}}(Z)$ , 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 $p^*L\in \operatorname {\mathrm {Pic}}^0(Z)$ . So Proposition 2.1 implies that L is an element of $\operatorname {\mathrm {Pic}}^0(A)$ .
Finally we recall some fundamental properties of abelian varieties that extend verbatim to the quasiabelian case:
Proposition 2.5. Let G be a quasiabelian variety. Then:

1. if $G'$ is a quasiabelian variety and $\phi \colon G'\to G$ is a morphism with $\phi (0)=0$ , then $\phi $ is a homomorphism;

2. if $G'\to G$ is a finite étale cover, then $G'$ is a quasiabelian variety;

3. if $H\subset G$ is closed and $\overline \kappa (H)=0$ , then H is a quasiabelian variety.
Proof. Item 1. is [Reference Noguchi and Winkelmann17, Thm. 5.1.37], 2. is [Reference Fujino7, Thm. 4.2] and 3. is [Reference Iitaka9, Thm. 4].
2.2.1 The quasiAlbanese map
The classical construction of the Albanese variety of a projective variety can be extended to the nonprojective case by replacing regular 1forms by logarithmic ones and abelian varieties by quasiabelian ones. The key fact is that by Deligne [Reference Deligne4] logarithmic 1forms are closed (for the details of the construction see [Reference Iitaka9], [Reference Fujino7, Section 3]).
Theorem 2.6. Let V be a smooth algebraic variety. Then there exists a quasiabelian variety $A(V)$ and a morphism $a_V\colon V\to A(V)$ such that:

1. if $h\colon V\rightarrow G$ is a morphism to a quasiabelian variety, then h factors through $a_V$ in a unique way;

2. if X is a compactification of V with snc boundary D, then we have the following exact sequence:
(2.2) $$ \begin{align}1\rightarrow \mathbb{G}_m^r\longrightarrow A(V)\longrightarrow A(X)\rightarrow 0, \end{align} $$where $r=\overline q(V)q(V)$ .
Proof. Item 1. is proven in [Reference Iitaka9] in the discussion immediately after Proposition 4. Item 2. is [Reference Fujino7, p. 13].
The variety $A(V)$ is called the quasiAlbanese variety of V and $a_V$ the quasiAlbanese map. Note that the compact part of $A(V)$ is $A(X)$ . We note the following logarithmic version of Abel’s Theorem:
Proposition 2.7. Let C be a smooth curve with $\overline P_1(C)>0$ . Then the quasiAlbanese 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.
2.3 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 pullback of a logarithmic nform on Z is a logarithmic nform 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:
Equation (2.3) is called the logarithmic ramification formula (cf. [Reference Iitaka13, §11.4]).
We note the following for later use:
Lemma 2.8. In the above setup, 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 nform, while $K_X+D$ is generated locally near x either by an nform with a logarithmic pole on $\Gamma $ or by a regular nform, 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 $ .
2.4 Generic vanishing
The theory of generic vanishing was introduced by Green–Lazarsfeld in [Reference Green and Lazarsfeld8]. It has since developed in a powerful tool to study the geometry of projective varieties via their Albanese morphism (see, for example, [Reference Pareschi18] for a nice survey). We are going to see in Section 3 that these techniques can be useful instruments also in the quasiprojective 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
We say that $\mathcal {F}$ is a $GV$ sheaf if, for every $i>0$ , we have that
When all the $V^i(X,\mathcal {F})$ are empty for $i>0$ , one says, following the original terminology of Mukai [Reference Mukai16], that $\mathcal {F}$ is a IT(0)sheaf.
We recall the following wellknown useful observation:
Lemma 2.9. 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 $h^0(X, 2L)\ge 2$ .
Proof. For $\alpha \in V^0(X,L)\cap (V^0(X,L))$ , consider the multiplication map
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, $h^0(X,2L)\ge 2$ .
2.5 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
By Serre duality, one also has $p_a(D)1= h^0(D,\omega _D)h^0(D, {\mathcal O}_D)$ , hence $h^0(D, \omega _D)\ge p_a(D)$ .
For $m\in \mathbb N$ , we say that D is mconnected if, for every decomposition $D=D_1+D_2$ , with $D_1$ and $D_2$ effective, one has $D_1D_2\ge m$ .
We recall some wellknown facts.
Lemma 2.10. Let D be a $1$ connected divisor, then:

1. $h^0(D,{\mathcal O}_D)=1$ (so in particular, $p_a(D)=h^0(D, \omega _D)\ge 0$ );

2. if L is a line bundle on D that has degree 0 on every component of D, then $h^0(D,L)\le 1$ , and $h^0(D,L)=1$ if and only if $L= {\mathcal O}_D$ .
Proof. See [Reference Barth, Hulek, Peters and Van de Ven2, Lem. II 12.2].
Lemma 2.11. Let $D_1$ , $D_2$ be two effective nonzero divisors on a smooth projective surface X. Then

1. $p_a(D_1+D_2)=p_a(D_1)+p_a(D_2)+D_1D_21$ ;

2. if $D_1\prec D_2$ , then $h^0(D_1,\omega _{D_1})\le h^0(D_2, \omega _{D_2})$ .
Proof. 1. Use the adjunction formula (2.4).
2. Write $D_2=D_1+A$ , and consider the decomposition sequence:
Twisting by $K_X+D_2$ and taking cohomology gives the result.
Lemma 2.12. Let D be an effective nonzero divisor on a smooth projective surface X. Then we have the following formulae:

1. $h^0(X, K_X+D)= p_a(D)+ p_g(X)q(X)+h^1(X,D)$ ;

2. $h^0(X, K_X+D)= h^0(D,\omega _D)+ p_g(X)q(X)+d$ ,
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. 1. follows from the Riemann–Roch theorem, the adjunction formula (2.4) and Serre duality.
Taking cohomology in the restriction sequence
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 1. one obtains 2.
Remark 2.13. Note that by Serre duality d is exactly the codimension of the image of the map $t\colon H^0(D, \omega _D)\to H^1(X, K_X)$ .
3 Results in arbitrary dimension via generic vanishing
Let V be a smooth quasiprojective 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 (§2.4), and its recent extensions to the ‘logcanonical’ setting by Popa–Schnell [Reference Popa and Schnell20] and Shibata [Reference Shibata21].
Proposition 3.1. Let V be a smooth quasiprojective 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. [Reference Shibata21, Theorem 1.3]), Lemma 2.9 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 [Reference Popa and Schnell20, Variant 5.5], we know that $a_{X,*}{\mathcal O}_X(K_X+D)$ is a $GV$ sheaf on $A(X)$ , and therefore by [Reference Pareschi18, 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 3.2. Let V an ndimensional smooth quasiprojective 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 quasiAlbanese morphism $a_V\colon V\rightarrow A(V)$ is birational.
Proof. By Proposition 3.1, we know that $a_X$ is generically finite, and so X is of maximal Albanese dimension. In particular, we have that
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 [Reference Chen and Hacon3] we conclude that $a_X$ is a birational morphism. In addition, by Theorem 2.6 there is a commutative diagram
and so also $a_V$ is birational.
The next result refines Proposition 3.1:
Proposition 3.3. Let V be a smooth quasiprojective 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 $E\succ0$ be a divisor on $A(X)$ .
If $\overline P_1(V)>0$ and D contains the support of $a_X^*E$ , then $\overline P_2(V)\ge 2$ .
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 $\Delta =\sum _id_i\Delta _i$ , with $\Delta _i$ irreducible divisors and $0\le d_i\le 1$ . Given $\alpha \in \operatorname {\mathrm {Pic}}^0(B)$ , 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 [Reference Fujino6, Thm. 6.3] or [Reference Ambro1, Thm. 3.2] are satisfied and we have
So $f_*(K_X+D)$ is an IT(0)sheaf, and therefore for all $\alpha \in \operatorname {\mathrm {Pic}}^0(B)$ 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 $f^*\operatorname {\mathrm {Pic}}^0(B)=\pi ^*\operatorname {\mathrm {Pic}}^0(B)$ and $\overline P_2(V)\ge 2$ by Lemma 2.9.
When $q(V)$ is equal to 1, Proposition 3.3 gives:
Corollary 3.4. Let V be a smooth quasiprojective 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 3.5. Observe that, by Proposition 3.1, 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 3.3, 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 $A(X)$ .
4 The geometry of the quasiAlbanese morphism
The aim of this section is to establish the following fundamental step in the proof of Theorem A:
Proposition 4.1. 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 quasiAlbanese 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 4.1 is just the combination of Propositions 4.2 and 4.3 below.
4.1 The quasiAlbanese map when $q(V)\ge 1$
Proposition 4.2. Let V be a smooth algebraic surface such that $\overline {P}_1(V)=\overline {P}_2(V)=1$ , $\overline {q}(V)=2$ and $q(V)\ge 1$ . Then the quasiAlbanese 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 3.2.
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 3.4 implies that $C=\overline C$ . In particular, by Proposition 2.5 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.
4.2 The quasiAlbanese map when $q(V)=0$
In this section, we prove the following
Proposition 4.3. 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 quasiAlbanese 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 4.4. Let X be a nonsingular projective surface and A a 1connected effective divisor with $p_a(A)=1$ .
Then A contains a 2connected divisor B such that $p_a(B)=1$ .
Proof. If A is 2connected, there is of course nothing to prove. Otherwise, take any decomposition $A=A_1+A_2$ with $A_1A_2=1$ and $A_1$ minimal with respect to $A_1A_2=1$ . $A_2$ is 1connected, $A_1$ is 2connected, and by Lemma 2.11 $p_a(A_1)+p_a(A_2)=1$ , $p_a(A_1)\geq 0$ , $p_a(A_2)\geq 0$ . If $p_a(A_1)=1$ , we have proved the statement; if not, we repeat the argument on $A_2$ .
Lemma 4.5. Let X be a nonsingular projective surface and B a 2connected effective divisor with $p_a(B)=1$ . Then:

1. if B is not irreducible, then every component $\Gamma $ of B is smooth rational and satisfies $(K_X+B)\Gamma =0$ ;

2. $\omega _B=\mathcal O_B$ ;

3. if $\Gamma $ is an irreducible component of B and $B\Gamma \succ 0$ , then $B\Gamma $ is 1connected.
Proof. 1. 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 2connected and reducible. So necessarily $(K_X+B)\Gamma =0$ , and $\Gamma $ is smooth rational.
2. Since, by 1. $\omega _B$ has degree $0$ on every component of B and $h^0(B, \omega _B)=1$ , by Lemma 2.10 $\omega _B={\mathcal O}_B$ .
3. By the proof of 1. one has $\Gamma (B\Gamma )=2$ . Let $B\Gamma =A_1+A_2$ with $A_1>0,A_2>0$ . Then because B is 2connected, $A_i(A_j+\Gamma )\geq 2$ for $\{i,j\}=\{1,2\}$ . Since $(A_1+A_2)\Gamma =2$ necessarily $A_1A_2\geq 1$ and so $B\Gamma $ is 1connected.
Lemma 4.6. Let X be a nonsingular projective surface with $q(X)=0$ , and let B be an effective 2connected divisor satisfying $p_a(B)=1$ . Then one of the following occurs:

(a) $h^0(X, 2K_X+2B)\geq 2$ ;

(b) X is rational, and there is a blowdown morphism $\rho \colon X\to T$ with exceptional divisor $\sum _{j=1}^nE_j$ (where the $E_j$ are $1$ curves) such that $K_X+B\sim \sum _{j=1}^nE_j$ and B is disjoint from $\sum _{j=1}^nE_j$ .
Proof. From Lemma 2.12, 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 nonzero effective divisor $B_1$ in $K_X+B$ , and $p_a(B_1)\geq 1 $ implying by Lemma 2.12 that $h^0(X, K_X+B_1)\geq 1+p_g$ , that is, $h^0(X, 2K_X+B)\geq 1$ .
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
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 4.5, $B_1\Gamma = 0$ , so $\theta $ is not a component of B, and thus $B\theta \ge 0$ . 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 2connected divisor with $p_a=1$ .
By the discussion in the previous case, we see that either $h^0(T, 2K_T+2\rho (B))\ge 2$ or $K_T+\rho (B)\sim 0$ . In the former case $h^0(X, 2K_X+2B)\ge h^0(T, 2K_T+2\rho (B))\ge 2$ . In the latter case, taking pullbacks we get $\rho ^*(K_T)+B=K_X\sum _{j=1}^nE_j+B \sim 0$ . 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 4.7. 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 quasiAlbanese 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 4.8. There is a blowdown 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:

(a) $p_g(X)=0$ , $h^0(D,\omega _D)=1$ , X is a rational surface, and there is a 2connected divisor $B\preceq D$ such that $K_X+B\sim \sum _{j=1}^n E_j$ and B is disjoint from $\sum _{j=1}^n E_j$ ;

(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 2.12.
Assume first $p_g(X)=0$ . Then, by Lemma 2.12, $q(X)=0$ 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 4.4 there is a 2connected divisor $B\preceq A$ with $p_a(B)=1$ . By Lemma 4.6 and the hypothesis $\overline P_2(V)=1$ , X is rational and there is a blowdown morphism $\rho \colon X\to T$ as in (a).
Assume now $p_g(X)=1$ . In this case, X has nonnegative 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 4.9. One has:

1. $h^0(X, 2K_X+D)=h^0(X, 3K_X+2D)=p_g(X)$ ;

2. 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. 1. Since $D\succ 0$ and $\overline P_i(V)=1$ for $i=1,2,3$ by assumption, we have $p_g(X)\le h^0(X, 2K_X+D) \le h^0(X, 3K_X+2D)\le 1$ . So if $p_g(X)=1$ the assertion is trivial.
For $p_g(X)=0$ and $m\ge 1$ , consider the exact sequence:
The second map in equation (4.1) is an isomorphism for $m=1$ since $q(X)=0$ , so it is nonzero for all $m\ge 1$ since D is a reduced divisor. Then $\overline P_2(V)=1$ and $\overline P_3(V)=1$ imply that $h^0(X, 2K_X+D)=h^0(X, 3K_X+2D)=0$ .
2. is an immediate consequence of $q(X)=0$ and of 1. (see Lemma 2.12).
We now turn to the study of the quasiAlbanese map.
Lemma 4.10. 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 4.7. 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 quasiAlbanese map, $A(V)$ is isomorphic to $A(\Gamma _0)$ . In addition, the image of $a_V$ is isomorphic to $\Gamma _0$ by Proposition 2.7. 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 4.11. 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 4.12. Let $E_j$ , $j=1,\dots n$ , be the $1$ curves contracted by the blowdown morphism $\rho \colon X\to T$ of Lemma 4.8.
Then $ E_jF_i^s\geq 0$ for all $j,i$ .
Proof. Assume by contradiction that $E_jF_1^s< 0$ . 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, $E_jF_1^s< 0$ yields $A^2+AC_1+AC_2+C_1C_2<0$ . But, because $E_j$ is a $(1)$ curve, $E_jA=A^2+AC_1\geq 1$ (see [Reference Lorenzo, Mendes Lopes and Pardini15, Prop. 3.2]) and because $F_1^s$ is 1connected $AC_2\geq 1$ . Since $C_1C_2\geq 0$ , we have a contradiction.
Lemma 4.13. Let $E_j$ , $j=1,\dots n$ , be the $1$ curves contracted by the blow downmorphism $\rho \colon X\to T$ of Lemma 4.8.
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 2.12, $h^0(X, K_X+M_1+M_2+\sum _{i=1}^3 F_i^s)\geq 2$ .
By Lemma 4.8, 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 4.14. 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 4.8). If B has components in common with $\sum _{i=1}^3 F_i^s$ , then one of the following happens:

(a) $B\leq F_i^s$ for some $i\in \{1,2,3\}$ , or

(b) B has a unique component H transversal to f and $BH$ 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 (a).
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 $BH\neq 0$ . Recall that, by Lemma 4.8, the divisor B is $2$ connected and $p_a(B)=1$ , so, by Lemma 4.5, $H\simeq \mathbb P^1$ and $BH$ is connected. From Lemma 4.13, $BH$ is contained in fibers of f, and since $BH$ is connected, we obtain $BH$ 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 2connected) and $HF_i^s\geq 1$ for $i=2,3$ because $F_i^s$ is the support of a full fiber. Now, $p_a(H+\sum _{i=1}^3 F_i^s)\geq p_a(H)+\sum _{i=1}^3 p_a(F_i^s) + H(\sum _{i=1}^3F_i^s)3$ . Since $h^0(D,\omega _D)=1$ (cf. Lemma 4.8), 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 4.15. Let $E_j$ , $j=1,\dots n$ , be the $1$ curves contracted by the blowdown 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 4.8):

1. if $F_i^s\leq DB$ , then $ E_jF_i^s\leq 1$ for all $E_j$ ;

2. the general fiber F of f satisfies $F\sum _{j=1}^nE_j=0$ .
Proof. 1. Assume that $E_jF_i^s\geq 2$ . 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 $F_i^s\leq DB$ , we have $E_j+F_i^s\leq D_1\in K_X+D$ , contradicting Lemma 4.9 (cf. Lemma 2.11).
2. Suppose otherwise, that is, 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 4.8), by Lemma 4.13 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 $E_{p_1}, \ldots ,E_{p_{l1}}$ such that $E_m=M+\sum _{k=1}^{l1} E_{p_k}$ is one of the $E_j$ . This is true by [Reference Lorenzo, Mendes Lopes and Pardini15, Lemma 3.2] if $l=1$ , by [Reference Lorenzo, Mendes Lopes and Pardini15, Prop. 4.1] if $l=2$ and by [Reference Lorenzo, Mendes Lopes and Pardini15, Prop. 4.2] if $l>2$ .
Since $E_{p_k}F_i^s\geq 0$ for all $k=1,\dots l1$ and $i=1,2,3$ by Lemma 4.12, we obtain $E_m(\sum F_i^s)\geq 3$ . Since $E_m(E_m+\sum _{i=1}^3F_i^s)\geq 2$ , one obtains

(a) $p_a(2E_m+\sum _{i=1}^3F_i^s)\geq 2$ if $p_g(X)=0$ and $B\neq 0$ is contained in one of the $F_i^s$ .

(b) $p_a(2E_m+\sum _{i=1}^3F_i^s)\geq 1$ , if $p_g(X)=1$ or $p_g(X)=0$ and B is not contained in $\sum _{i=1}^3F_i^s$ .
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 $2E_m+\sum F_i^s\leq 2(K_X+B)+ (DB)=2K_X+B+D$ , yielding $h^0(X, 3K_X+B+ D)\geq p_g(X)+1$ and contradicting Lemma 4.9.
Corollary 4.16. The fibration $f\colon X\to \mathbb P^1$ descends to a fibration $\bar f\colon T\to \mathbb P^1$ .
Lemma 4.17. 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 $FK_T=F\rho (B)\le 0$ , so either $FK_T=0$ and F has genus 1, or $K_TF=2$ 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 $h^0(D_0,\omega _{D_0})\le h^0(D,\omega _{D})=\overline P_1(V)=1$ (cf. Lemma 2.12), 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 4.8; 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 $\rho (B)E=K_TE=1$ ; namely, E meets $\rho (B)$ . So, in this case, there is no $1$ curve contained in $\overline F_2+\overline F_3$ .
By Lemmas 4.12 and 4.15, 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 [Reference Barth, Hulek, Peters and Van de Ven2, Chp.V, §7]). Note that, in particular, the $\overline F_i$ cannot be multiple fibers of $\bar f$ (see [Reference Barth, Hulek, Peters and Van de Ven2, 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 $F_i\preceq 3F_i^s+3\sum _{j=1}^n E_j\preceq 3K_X+3D$ , 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 4.3.
By Lemma 4.17, the general fiber F of f has genus $0$ , and in particular, we have $p_g(X)=0$ . Since $K_XF=2$ , 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
Since $D_0$ is reduced and connected, we have $p_a(D_0)=h^0(D_0,\omega _{D_0})\le h^0(D,\omega _D) = \overline P_1(V)=1$ (cf. Lemma 2.11 and 2.12) 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 4.14 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
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 $\rho (\Gamma )^2\le 0$ 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 4.17) $\rho ^*(\rho (F_2^s))\leq F_2^s+\sum _{j=1}^n E_j$ , 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 4.17, 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 4.18. The hypothesis $\overline P_3(V)=1$ in Proposition 4.3 is necessary. A K3 surface with an elliptic fibration with three singular fibers of type $IV^*$ (cf. the proof of Lemma 4.17) 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 $(x,y)\mapsto (hx,h^2y)$ , and denote by $X_0$ the quotient surface. Then $X_0$ has nine 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 nine 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 three fibers of type IV*. Then V has $\overline P_1(V)=\overline P_2(V)=1$ and $\overline q(V)=2$ (see Proposition 2.1), and its quasiAlbanese map is the restriction of the elliptic fibration $X\to \mathbb P^1$ , and so it is not dominant.
5 Proof of Theorem A
Thanks to what we have proven in the previous section, we know that in the assumptions of Theorem A the quasiAlbanese 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 2.3. Now, our proof boils down to the following two facts

1. the morphism g has degree 1;

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 3.2. The second statement can be proven by contradiction (see §5.2).
The situation is more involved when $q(V)<2$ . We start by proving a slightly weaker version of assertion 2 (Lemma 5.5), 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 quasiAlbanese map implies that g has indeed degree 1. Finally, we complete the proof of assertion 2 by means of a local computation.
5.1 Preliminary steps
Notation 5.1. 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 $q(V)\ge 1$ and $\overline P_1(V)=\overline P_3(V)=1$ if $q(V)=0$ .
If $q(V)\le 1$ fix a compactification Z with boundary $\Delta $ of the quasiAlbanese 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 2.4, 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 quasiAlbanese map $a_V\colon V\to A(V)$ extends to a morphism $g\colon X \to Z$ . In addition, we write $H:=g^{1} (\Delta )$ (settheoretic 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