An extension theorem of Ohsawa-Takegoshi type for sections of a vector bundle

TL;DR

This paper extends the Ohsawa-Takegoshi theorem to holomorphic sections of vector bundles on compact Kähler manifolds using $L^2$-methods, broadening the conditions under which such extensions are possible.

Contribution

It introduces an extension theorem for vector bundle sections, relaxing previous conditions and employing $L^2$-techniques for the $ar ext{d}$-equation on compact Kähler manifolds.

Findings

Extension theorem applies to vector bundles, not just line bundles.

Conditions for extension are more liberal than algebraic geometric reductions.

Uses $L^2$-methods to solve the $ar ext{d}$-equation in this context.

Abstract

Using $L^2$-methods for the $\bar\partial$-equation we prove that the Ohsawa-Takegoshi extension theorem also holds for holomorphic sections of a vector bundle, over compact K\"ahler manifolds. We then proceed to show that the conditions that are needed are more liberal than the ones one would need if one instead reduced the extension problem to line bundles through the usual algebraic geometric procedure of studying the projective bundle associated with the vector bundle.