Positivity of vector bundles and Dominance

TL;DR

This paper generalizes the relationship between the positivity properties of Schur functors applied to vector bundles, extending from ampleness to k-ampleness, semi-ampleness, and nefness, using algebraic and combinatorial methods.

Contribution

It extends previous results on ampleness dominance to broader positivity notions for vector bundles, employing Littlewood-Richardson rules and algebraic techniques.

Findings

Ampleness dominance extends to k-ample, semiample, and nef vector bundles.

The proof utilizes algebraic properties common to these positivity notions.

Investigation of Littlewood-Richardson rules underpins the generalization.

Abstract

Let $E$ be a vector bundle and $S_a$, $S_b$ the Schur functors associated to partitions $a$ and $b$. Previously we have shown that ampleness of $S_aE$ implies ampleness of $S_bE$ when $a$ is greater than $b$ in the dominance partial order. Here we prove that this result generalizes to $k$-ample, semiample and nef vector bundles. Our proof uses the common algebraic nature of these three properties and an investigation of the Littlewood-Richardson rules.