On algebraic fiber spaces over varieties of maximal Albanese dimension

摘要

We study algebraic fiber spaces f : X --> 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 greater than or equal to 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). [References: 16]