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In this paper we study holomorphic vector bundles with singular Hermitianmetrics whose curvature are Hermitian matrix currents. We obtain an extensiontheorem for holomorphic jet sections of nef holomorphic vector bundle oncompact K\"ahler manifolds. Using it we prove that Fano manifolds with strongGriffiths nef tangent bundles are rational homogeneous spaces.

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