Splitting criteria for vector bundles on higher dimensional varieties

TL;DR

This paper extends Horrocks' splitting criterion for vector bundles from projective space to certain higher-dimensional smooth complex projective varieties where all line bundle extensions split.

Contribution

It introduces a new splitting criterion for vector bundles on higher-dimensional varieties, generalizing a classical result from projective space.

Findings

Established a splitting criterion for vector bundles on specific higher-dimensional varieties.

Proved that on these varieties, every extension of line bundles splits.

Generalized Horrocks' criterion beyond projective space.

Abstract

We generalize Horrocks' criterion for the splitting of vector bundles on projective space. We establish an analogous splitting criterion for vector bundles on a class of smooth complex projective varieties of dimension at least four, over which every extension of line bundles splits.