Effective vanishing theorems for ample and globally generated vector bundles

TL;DR

This paper develops new vanishing theorems for ample and globally generated vector bundles using an analytic approach based on curvature tensor formulas, extending classical results in complex geometry.

Contribution

It introduces a novel integral curvature formula for vector bundles and derives vanishing theorems without relying on spectral sequences.

Findings

Vanishing theorems for ample vector bundles

Vanishing theorems for nef vector bundles

Vanishing theorems for globally generated bundles

Abstract

By proving an integral formula of the curvature tensor of $E\ts \det E$, we observe that the curvature tensor of $E\ts \det E$ is very similar to that of a line bundle and obtain certain new Kodaira-Akizuki-Nakano type vanishing theorems for vector bundles. As special cases, we deduce vanishing theorems for ample, nef and globally generated vector bundles by analytic method instead of the Leray-Borel-Le Potier spectral sequence.