$L^2$-extension theorems for jet sections of nef holomorphic vector bundles on compact K\"ahler manifolds and rational homogeneous manifolds, I

TL;DR

This paper develops $L^2$-extension theorems for jet sections of nef holomorphic vector bundles on compact Kähler manifolds and applies these results to characterize Fano manifolds with nef tangent bundles as rational homogeneous spaces.

Contribution

It introduces new $L^2$-extension theorems for jet sections of nef bundles and uses them to classify certain Fano manifolds as rational homogeneous spaces.

Findings

Extension theorem for holomorphic jet sections of nef bundles.

Fano manifolds with strongly Griffiths nef tangent bundles are rational homogeneous.

Advances understanding of the structure of nef vector bundles on Kähler manifolds.

Abstract

In this paper we study holomorphic vector bundles with singular Hermitian metrics whose curvature are Hermitian matrix currents. We obtain an extension theorem for holomorphic jet sections of nef holomorphic vector bundle on compact K\"ahler manifolds. Using it we prove that Fano manifolds with strong Griffiths nef tangent bundles are rational homogeneous spaces.