Extremal metrics and stabilities on polarized manifolds

Details Extremal metrics and stabilities on polarized manifolds Extremal metrics and stabilities on polarized manifolds

2006-03-21

arxiv.org/abs/math/0603493v3

Mathematics

Differential Geometry

to appear in Proc. ICM2006, Madrid; added references for Section 1; corrected careless mistakes

Scientific paper

The Hitchin-Kobayashi correspondence for vector bundles, established by Donaldson, Kobayashi, Luebke, Uhlenbeck and Yau, states that an indecomposable holomorphic vector bundle over a compact Kaehler manifold is stable in the sense of Takemoto-Mumford if and only if the vector bundle admits a Hermitian-Einstein metric. Its manifold analogue known as Yau's conjecture, which originated from Calabi's conjecture, asks whether ``stability'' and ``existence of extremal metrics'' for polarized manifolds are equivalent. In this note, the recent progress of this subject, by Donaldson, Tian and our group, together with its relationship to algebraic geometry will be discussed.

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