Approximation and extension of Hermitian metrics on holomorphic vector bundles over Stein manifolds

TL;DR

This paper demonstrates that singular Hermitian metrics with Griffiths or Nakano negativity on holomorphic vector bundles over Stein manifolds can be approximated by smooth metrics with the same negativity, and such metrics can be extended from submanifolds.

Contribution

It introduces methods to approximate and extend Hermitian metrics with curvature negativity on vector bundles over Stein manifolds, preserving negativity properties.

Findings

Singular Hermitian metrics can be approximated by smooth metrics with the same negativity.

Negative metrics on submanifolds can be extended to the entire bundle maintaining negativity.

The results apply to both Griffiths and Nakano curvature notions.

Abstract

We show that a singular Hermitian metric on a holomorphic vector bundle over a Stein manifold which is negative in the sense of Griffiths (resp. Nakano) can be approximated by a sequence of smooth Hermitian metrics with the same curvature negativity. We also show that a smooth Hermitian metric on a holomorphic vector bundle over a Stein manifold restricted to a submanifold which is negative in the sense of Griffiths (resp. Nakano) can be extended to the whole bundle with the same curvature negativity.