Kobayashi-Hitchin Correspondence for Saturated Reflexive Parabolic Sheaves on K\"ahler manifolds

TL;DR

This paper extends the Kobayashi-Hitchin correspondence to saturated reflexive parabolic sheaves on compact K"ahler manifolds, utilizing Hermitian-Yang-Mills flow to establish the relationship between stability and Hermitian-Einstein metrics.

Contribution

It introduces a new approach to the Kobayashi-Hitchin correspondence for parabolic sheaves with normal crossing divisors on K"ahler manifolds using Hermitian-Yang-Mills flow.

Findings

Established the correspondence for saturated reflexive parabolic sheaves.

Applied Hermitian-Yang-Mills flow to prove stability implies existence of Hermitian-Einstein metrics.

Extended classical results to more general sheaves on complex manifolds.

Abstract

In this paper, we study the Kobayashi-Hitchin correspondence in the setting of parabolic sheaves with a simple normal crossing divisor over a compact K\"ahler manifold using the method of Hermitian-Yang-Mills flow.