On algebraic fiber spaces over varieties of maximal Albanese dimension

Abstract

We study algebraic fiber spaces $f : X \longrightarrow Y$ where $Y$ is of maximal Albanese dimension. In particular, we give an effective version of a theorem of Y. Kawamata: If $P_m(X)=1$ for some $m\geq 2$, then the Albanese map of $X$ is surjective. Combining this with [1], it follows that $X$ is birational to an abelian variety if and only if $P_2(X)=1$ and $q(X)=\dim(X)$.