Extending Vector Bundles from Curves

TL;DR

This paper demonstrates conditions under which vector bundles on a curve can be extended to a smooth projective variety, focusing on stability and determinant extension.

Contribution

It establishes that vector bundles of rank at least the dimension of the variety can be extended from a curve to the entire variety if their determinants extend.

Findings

Extension is possible for vector bundles with rank ≥ dimension of X.

Stability (μ-stability) is preserved in the extension.

Determinant extension is a key condition for extension.

Abstract

Given a curve in a (smooth) projective variety $C \subset X$, we show that a vector bundle $V$ on C can be extended to a ($\mu$-stable) vector bundle on $X$ if $\text{rank}(V) \geq \text{dim}(X)$ and $\text{det}(V)$ extends to $X$.