On the Kobayashi-Hitchin correspondence for K\"{a}hler currents

TL;DR

This paper extends the Kobayashi-Hitchin correspondence to K"{a}hler currents with singularities, showing that slope polystable holomorphic bundles admit Hermitian-Yang-Mills metrics in this broader setting.

Contribution

It generalizes the correspondence to K"{a}hler currents with singularities, broadening the class of metrics and currents where the correspondence holds.

Findings

Holomorphic bundles with slope polystability admit Hermitian-Yang-Mills metrics with respect to K"{a}hler currents.

The proof extends to closed positive $(1,1)$-currents representing nef and big classes.

The results connect stability conditions with geometric structures in the presence of singularities.

Abstract

In this paper, we show that if a holomorphic vector bundle is slope polystable with respect to a K\"{a}hler class, then it admits a Hermitian-Yang-Mills metric with respect to a suitable K\"{a}hler current with singularities in higher codimension which represents the K\"{a}hler class. Most parts of the proof remains valid for closed positive $(1,1)$-currents representing a nef and big class.