Let $X$ be a smooth complex projective variety. From Kawamata's
classical work, we know that the properties of the Albanese morphism
of $X$ are closely related to the birational invariants of X, for instance,
if $k(X)=0$, the Albanese morphism is an algebraic fiber space. Hacon
and Pardini made part of Kawamata's result effective by proving that
if some plurigenus $P_m(X)$ is ”small” for some $m ≥ 3$, the Albanese
morphism of X should be surjective. I will first present better bounds
on $P_m(X)$ for the Albanese morphism to be surjective (resp. a fiber
space). Then I will use these bounds to classify (birationally) some
varieties with maximal Albanese dimension and small plurigenera. This
application is an extension of a work of Chen and Hacon.