The cohomological support locus of pluricaonical sheaves and the Iitaka fibration

Let $al{b}_X:X\to A$ be the Albanese map of a smooth projective variety and $f:X\to Y$ the fibration from the Stein factorization of $al{b}_X$. For a positive integer $m$, if $f$ and $m$ satisfy the assumptions AS(1,2), then the translates through the origin of all components of cohomological locus $V^0({\omega }_X^m,al{b}_X)$ generates $I^{\ast }Pi{c}^0(S)$ where $I: X \rightarrow S$ denotes the Iitaka fibration. This result applies to studying pluricanonical maps. We also considered the problem about whether a fibration is isotrivial and isogenous to a product.