A Kobayashi-Hitchin correspondence between Dirac-type singular mini-holomorphic bundles and HE-monopoles

TL;DR

This paper establishes a correspondence between certain singular mini-holomorphic bundles and HE-monopoles on fibered 3-folds, extending Kobayashi-Hitchin theory to new geometric contexts.

Contribution

It introduces a Kobayashi-Hitchin correspondence for Dirac-type singular mini-holomorphic bundles on fibered 3-folds, including Sasakian manifolds, with a new stability condition.

Findings

Existence of admissible BHE-metrics on stable mini-holomorphic bundles.

Extension of Kobayashi-Hitchin correspondence to orbifold fibered 3-folds.

Characterization of algebraic Dirac-type singularities.

Abstract

We prove an analogue of the Kobayashi-Hitchin correspondence oncompact connected 3-folds that is fibered on orbifold Riemann surfaces and satisfy an integrability condition, which contains compact connected Sasakian 3-folds. We define mini-holomorphic bundles on such 3-folds and the algebraic Dirac-type singularities on mini-holomorphic bundles, and prove that there exists a special Hermitian metric (admissible BHE-metric) on a Dirac-type singular mini-holomorphic bundle if the bundle satisfies a slope stability.