Positivity and vanishing theorems for ample vector bundles

TL;DR

This paper establishes new vanishing theorems for ample vector bundles by proving Nakano-positivity and dual-Nakano-positivity of certain associated bundles, leading to cohomology vanishing results on compact Kähler manifolds.

Contribution

It introduces novel positivity properties for adjoint bundles of ample vector bundles, resulting in new vanishing theorems and extending positivity results to Griffiths-positive bundles.

Findings

S^kE s ext{det} E is Nakano-positive and dual-Nakano-positive for ample E.

Vanishing of certain H^{p,q} cohomology groups for these bundles.

Positivity extends to Griffiths-positive bundles with induced Hermitian metrics.

Abstract

In this paper, we study the Nakano-positivity and dual-Nakano-positivity of certain adjoint vector bundles associated to ample vector bundles. As applications, we get new vanishing theorems about ample vector bundles. For example, we prove that if $E$ is an ample vector bundle over a compact K\"ahler manifold $X$, $S^kE\ts \det E$ is both Nakano-positive and dual-Nakano-positive for any $k\geq 0$. Moreover, $H^{n,q}(X,S^kE\ts \det E)=H^{q,n}(X,S^kE\ts \det E)=0$ for any $q\geq 1$. In particular, if $(E,h)$ is a Griffiths-positive vector bundle, the naturally induced Hermitian vector bundle $(S^kE\ts \det E, S^kh\ts \det h)$ is both Nakano-positive and dual-Nakano-positive for any $k\geq 0$.