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

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Summary

Introduction

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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Connections and metrics
We consider
But then
Composing Di with the two projections we get
As in the proof of
Hence we have
We notice that as Hi
As fi is injective and hi is a Hermitian metric on
It is known that
Ri be the curvature
Now all the forms in the formula glue together to give
Ui and every a
We let
We then have b
By the very de nition of Λg we then see that
Let h
Written in another way we then get
Hj it follows that
Taking the trace we then get
We now de ne
Semistability for twisted vector bundles
To r
We recall that
The proof
But then we get
In a similar way one de nes
By de nition we have
It then follows that
Remark that the map
Kähler metric
We now let
But this implies that
We now consider the evolution equation
Since we get
Similar calculations give that
If we let
Kt of
Hence we get
Then we get
The matrix Fi representing fi is
HiS matrix representing hSi and
We used the notation g instead
Ch π
As a consequence we get
But now notice that
Rt be
Consider now the exact sequence
Conclusion of the proof
We know
Wk such
Full Text
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