Hermitian Einstein Structure Research Articles

Semi-stable twisted holomorphic vector bundles over Gauduchon manifolds

In this paper, we study the semi-stable twisted holomorphic vector bundles over compact Gauduchon manifolds. By using Uhlenbeck–Yau's continuity method, we show that the existence of approximate Hermitian–Einstein structure and the semi-stability of twisted holomorphic vector bundles are equivalent over compact Gauduchon manifolds. As its application, we show that the Bogomolov type inequality is also valid for a semi-stable twisted holomorphic vector bundle.

Canonical Metrics on Holomorphic Filtrations over Compact Hermitian Manifolds

The purpose of this paper is twofold. We first solve the Dirichlet problem for $$\tau $$-Hermitian–Einstein equations on holomorphic filtrations over compact Hermitian manifolds. Secondly, by using Uhlenbeck–Yau’s continuity method, we prove the existence of approximate $$\tau $$-Hermitian–Einstein structure on holomorphic filtrations over closed Gauduchon manifolds.

Semi-stability for Holomorphic Pairs on Compact Gauduchon Manifolds

In this paper, we establish a generalized Hitchin–Kobayashi correspondence between the $$\tau $$ -semi-stability and the existence of approximate $$\tau $$ -Hermitian–Einstein structure on holomorphic pair $$(E,\phi )$$ over the compact Gauduchon manifold.

Semi-stable Higgs sheaves and Bogomolov type inequality

In this paper, we study semi-stable Higgs sheaves over compact Kahler manifolds. We prove that there is an admissible approximate Hermitian-Einstein structure on a semi-stable reflexive Higgs sheaf and consequently, the Bogomolov type inequality holds on a semi-stable reflexive Higgs sheaf.

Semistable Higgs Bundles Over Compact Gauduchon Manifolds

In this paper, we consider the existence of approximate Hermitian–Einstein structure and the semi-stability on Higgs bundles over compact Gauduchon manifolds. Using the continuity method, we show that they are equivalent.

Existence of approximate Hermitian–Einstein structures on semi-stable Higgs bundles

In this paper, using Donaldson’s heat flow, we show that the semi-stability of a Higgs bundle over a compact Kähler manifold implies the existence of approximate Hermitian–Einstein structure.

Existence of approximate Hermitian-Einstein structures on semi-stable bundles

The purpose of this paper is to investigate canonical metrics on a semi-stable vector bundle $E$ over a compact Kahler manifold $X$. It is shown that if $E$ is semi-stable, then Donaldson’s functional is bounded from below. This implies that $E$ admits an approximate Hermitian-Einstein structure, generalizing a classic result of Kobayashi for projective manifolds to the Kahler case. As an application some basic properties of semi-stable vector bundles over compact Kahler manifolds are established, such as the fact that semi-stability is preserved under certain exterior and symmetric products.

Progress on asymptotic behavior of the Yang-Mills-Higgs flow

In this note we will introduce our recent work on the existence of approximate Hermitian-Einstein structures on semi-stable Higgs bundles, and the asymptotic behavior of the Yang-Mills-Higgs flow for Higgs pairs at infinity.

Existence of approximate Hermitian–Einstein structures on semistable principal bundles

Let E_G be a principal G-bundle over a compact connected K\"ahler manifold, where G is a connected reductive complex linear algebraic group. We show that E_G is semistable if and only if it admits approximate Hermitian-Einstein structures.

HERMITIAN–EINSTEIN CONNECTIONS ON PRINCIPAL BUNDLES OVER FLAT AFFINE MANIFOLDS

Let M be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric g and a covariant constant volume form. Let G be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat principal G-bundle EG over M admits a Hermitian–Einstein structure if and only if EG is polystable. A polystable flat principal G-bundle over M admits a unique Hermitian–Einstein connection. We also prove the existence and uniqueness of a Harder–Narasimhan filtration for flat vector bundles over M. We prove a Bogomolov type inequality for semistable vector bundles under the assumption that the Gauduchon metric g is astheno-Kähler.