On algebraic fiber spaces over varieties of maximal Albanese dimension

Abstract

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