Extremal metrics and stabilities on polarized manifolds

Abstract

The Hitchin–Kobayashi correspondence for vector bundles, established by Donaldson, Kobayashi, Lübke, Uhlenbeck andYau, states that an indecomposable holomorphic vector bundle over a compact Kähler 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.