  For an irregular variety $X$ of general type, we show that if a general fiber
$F$ of the Albanese morphism of $X$ satisfies certain Hodge theoretic
condition, the $0$-th cohomological support loci of $K_X$ generates the Picard
variety of $X$ . We then show that the condition that the $0$-th cohomological
support loci of $K_X$ generates the Picard variety of $X$ can often be applied
to prove the birationality of certain pluricanonical maps of $X$.
