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 [12].

We give a notion of ordinary Enriques surfaces and their canonical lifts in any positive characteristic, and we prove Torelli-type results for this class of Enriques surfaces.

We study products of irreducible theta divisors from two points of view. On the one hand, we characterize them as normal subvarieties of abelian varieties such that a desingularization has holomorphic Euler characteristic $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}1$. On the other hand, we identify them up to birational equivalence among all varieties of maximal Albanese dimension. We also describe the structure of varieties $X$ of maximal Albanese dimension, with holomorphic Euler characteristic $1$ and irregularity $2\dim X-1$.