$(k,s)$-positivity and vanishing theorems for compact Kahler manifolds

TL;DR

This paper introduces a unified framework for various positivity notions of holomorphic vector bundles on compact Kähler manifolds, establishing new vanishing theorems that generalize existing results for $k$-ample bundles.

Contribution

It develops a comprehensive theory of $(k,s)$-positivity, linking different positivity concepts and proving new vanishing theorems for these bundles on compact Kähler manifolds.

Findings

Unified theory for all positivity types of vector bundles.

New vanishing theorems for $(k,s)$-positive bundles.

Generalization of $k$-ample bundle results to Kähler manifolds.

Abstract

We study the $(k,s)$-positivity for holomorphic vector bundles on compact complex manifolds. $(0,s)$-positivity is exactly the Demailly $s$-positivity and a $(k,1)$-positive line bundle is just a $k$-positive line bundle in the sense of Sommese. In this way we get a unified theory for all kinds of positivities used for semipositive vector bundles. Several new vanishing theorems for $(k,s)$-positive vector bundles are proved and the vanishing theorems for $k$-ample vector bundles on projective algebraic manifolds are generalized to $k$-positive vector bundles on compact K\"ahler manifolds.