The cohomological support locus of pluricaonical sheaves and the Iitaka fibration

TL;DR

This paper investigates the structure of cohomological support loci of pluricanonical sheaves on smooth projective varieties, revealing their relation to the Iitaka fibration and implications for pluricanonical maps and isotriviality.

Contribution

It establishes a connection between cohomological support loci components and the Iitaka fibration under certain conditions, advancing understanding of pluricanonical sheaves.

Findings

The translates of cohomological locus components generate the pullback of Pic^0(S).

Results apply to the study of pluricanonical maps.

Addresses the isotriviality and product structure of fibrations.

Abstract

Let $alb_X: X \rightarrow A$ be the Albanese map of a smooth projective variety and $f: X \rightarrow Y$ the fibration from the Stein factorization of $alb_X$. 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 $V^0(\omega_X^m, alb_X)$ generates $I^*Pic^0(S)$ 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.