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Hermitian-Einstein connections and stable vector bundles over compact complex surfaces. (English) Zbl 0617.32044
A theorem of P. Gauduchon states that an arbitrary hermitian metric on a compact complex surface has a conformal rescaling such that the associated Kähler form is then \({\bar \partial}\partial\)-closed. Given such a form, the degree of a holomorphic line bundle can be defined in the usual way and with that, the notion of stability in the sense of Mumford and Takemoto for torsion-free sheaves. It is proved here that an indecomposable holomorphic vector bundle on the surface is stable iff it admits an irreducible Hermitian-Einstein connection, where ”stable” and ”Hermitian-Einstein” are both with respect to a given positive \({\bar \partial}\partial\)-closed (1,1)-form. This generalizes a result of Donaldson, who proved this theorem in the case of algebraic surfaces in \({\mathbb{P}}_ N\) equipped with a Kähler metric whose Kähler form is cohomologous to that of the Fubini-Study metric.

MSC:
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
53C05 Connections, general theory
32J15 Compact complex surfaces
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