Generic Vanishing Index and the Birationality of the Bicanonical Map of Irregular Varieties

TL;DR

This paper establishes a link between the generic vanishing index of irregular varieties and the birationality of their bicanonical maps, providing new criteria for when these maps are birational.

Contribution

It proves that varieties with a generic vanishing index of at least 2 have birational bicanonical maps, advancing understanding of the birational geometry of irregular varieties.

Findings

Varieties with generic vanishing index ≥ 2 have birational bicanonical maps.

If the bicanonical map is not birational, the Albanese image is fibered by subvarieties of codimension ≤ 1.

The results connect the generic vanishing index to the birational properties of the bicanonical map.

Abstract

We prove that any smooth complex projective variety with generic vanishing index bigger or equal than 2 has birational bicanonical map. Therefore, if X is a smooth complex projective variety X with maximal Albanese dimension and non-birational bicanonical map, then the Albanese image of X is fibered by subvarieties of codimension at most 1 of an abelian subvariety of Alb X.