Obstructions of extension of vector bundles

TL;DR

This paper investigates the cohomological obstructions that prevent extending a vector bundle from a subvariety to higher neighborhoods within a smooth ambient variety, focusing on the algebraic and holomorphic contexts.

Contribution

It provides a detailed analysis of the obstructions to extending vector bundles beyond a certain neighborhood, offering new insights into their cohomological properties.

Findings

Identifies specific cohomological obstructions to extension

Provides conditions under which extensions are possible

Enhances understanding of vector bundle extension theory

Abstract

In the holomorphic or algebraic setting we consider a vector bundle E on a smooth subvariety X in a smooth variety Y over a field of characteristic zero. Assuming E extends to the l-th neighborhood of X in Y, we study cohomological obstructions to extending it further to the k-th neighborhood, for k > l.