Cohomological support loci of varieties of Albanese fiber dimension one

TL;DR

This paper investigates the structure of cohomological support loci for certain algebraic varieties and demonstrates that the fourth pluricanonical map induces a birational equivalence.

Contribution

It establishes that the components of the cohomological support loci generate the Picard variety and proves the birationality of the fourth pluricanonical map for these varieties.

Findings

Components of $V^0(\omega_X)$ generate $ ext{Pic}^0(X)$

The map $|4K_X|$ is birational

Provides new insights into the geometry of varieties with Albanese fiber dimension one

Abstract

Let $X$ be a smooth projective variety of Albanese fiber dimension 1 and of general type. We prove that the translates through 0 of all components of $V^0(\omega_X)$ generate $\Pic^0(X)$. We then study the pluricanonical maps of $X$. We show that $|4K_X|$ induces a birational map.