On positivity and base loci of vector bundles

TL;DR

This paper explores the relationships among various positivity notions for vector bundles using asymptotic base loci defined directly on the variety, helping to clarify and unify existing concepts in the field.

Contribution

It introduces new definitions of augmented and restricted base loci for vector bundles and demonstrates their usefulness in understanding positivity properties.

Findings

Base loci behave well under projectivization maps.

New base loci definitions generalize line bundle concepts.

Simplifies relationships among positivity notions.

Abstract

The aim of this note is to shed some light on the relationships among some notions of positivity for vector bundles that arose in recent decades. Our purpose is to study several of the positivity notions studied for vector bundles with some notions of asymptotic base loci that can be defined on the variety itself, rather than on the projectivization of the given vector bundle. We relate some of the different notions conjectured to be equivalent with the help of these base loci, and we show that these can help simplify the various relationships between the positivity properties present in the literature. In particular, we define augmented and restricted base loci $\mathbb{B}_+ (E)$ and $\mathbb{B}_- (E)$ of a vector bundle $E$ on the variety $X$, as generalizations of the corresponding notions studied extensively for line bundles. As it turns out, the asymptotic base loci defined here behave well with respect to the natural map induced by the projectivization of the vector bundle $E$.