Effective characterization of quasi-abelian surfaces

Abstract

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 [12].