Stability and Hermitian-Einstein metrics for vector bundles on framed manifolds

TL;DR

This paper extends the concepts of stability and Hermitian-Einstein metrics to vector bundles on framed manifolds, establishing existence and uniqueness results in this geometric setting.

Contribution

It adapts stability and Hermitian-Einstein metrics to canonically polarized framed manifolds, proving existence and uniqueness for stable bundles.

Findings

Degree with respect to polarization matches that with the Kähler-Einstein metric

Existence of Hermitian-Einstein metrics for stable bundles

Uniqueness of these metrics in an adapted sense

Abstract

We adapt the notions of stability of holomorphic vector bundles in the sense of Mumford-Takemoto and Hermitian-Einstein metrics in holomorphic vector bundles for canonically polarized framed manifolds, i.e. compact complex manifolds X together with a smooth divisor D such that K_X \otimes [D] is ample. It turns out that the degree of a torsion-free coherent sheaf on X with respect to the polarization K_X \otimes [D] coincides with the degree with respect to the complete K\"ahler-Einstein metric g_{X \setminus D} on X \setminus D. For stable holomorphic vector bundles, we prove the existence of a Hermitian-Einstein metric with respect to g_{X \setminus D} and also the uniqueness in an adapted sense.