Positivity of extensions of vector bundles

TL;DR

This paper investigates how positivity properties of vector bundles are affected by extensions, establishing conditions under which positivity is preserved or not, with implications for the structure of vector bundles.

Contribution

It provides new criteria for the preservation of positivity in vector bundle extensions and clarifies limitations of positivity inheritance.

Findings

Extensions of big line bundles by ample vector bundles remain big.

Extensions of pseudo-effective line bundles by nef vector bundles remain pseudo-effective.

An almost nef vector bundle may not be pseudo-effective.

Abstract

In this paper, we study when positivity conditions of vector bundles are preserved by extension. We prove that an extension of a big (resp. pseudo-effective) line bundle by an ample (resp. a nef) vector bundle is big (resp. pseudo-effective). We also show that an extension of an ample line bundle by a big line bundle is not necessarily pseudo-effective. In particular, this implies that an almost nef vector bundle is not necessarily pseudo-effective.