Abstract
Abstract We prove the Kobayashi—Hitchin correspondence and the approximate Kobayashi—Hitchin correspondence for twisted holomorphic vector bundles on compact Kähler manifolds. More precisely, if X is a compact manifold and g is a Gauduchon metric on X, a twisted holomorphic vector bundle on X is g−polystable if and only if it is g−Hermite-Einstein, and if X is a compact Kähler manifold and g is a Kähler metric on X, then a twisted holomorphic vector bundle on X is g−semistable if and only if it is approximate g−Hermite-Einstein.
Highlights
The Kobayashi—Hitchin correspondence for vector bundles is a nowadays well-established result in complex geometry, saying that a holomorphic vector bundle on a compact complex manifold X is polystable if and only if it admits a Hermite-Einstein metric
We prove the Kobayashi—Hitchin correspondence and the approximate Kobayashi—Hitchin correspondence for twisted holomorphic vector bundles on compact Kähler manifolds
We prove in particular that every twisted holomorphic vector bundle E over which we x a Hermitian metric h, carries a unique connection which is compatible with the holomorphic structure of E and the metric h
Summary
The Kobayashi—Hitchin correspondence for vector bundles is a nowadays well-established result in complex geometry, saying that a holomorphic vector bundle on a compact complex manifold X is polystable if and only if it admits a Hermite-Einstein metric. A holomorphic vector bundle is polystable if it is the direct sum of stable holomorphic vector bundles (where stability is the slope-stability, or Mumford-Takemoto stability) with the same slope, and a Hermite-Einstein metric is a Hermitian metric whose mean curvature is a constant multiple of the identity. This result was proved in an increasing order of generalization by several authors. A few years after that, Buchdahl proved in [9] that the Kobayashi—Hitchin correspondence holds on any com-
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