K-stability

This article covers the general theory of K-stability of projective varieties. The specific case of Fano varieties is covered in K-stability of Fano varieties.

In mathematics, and especially differential and algebraic geometry, K-stability is an algebro-geometric stability condition, for complex manifolds and complex algebraic varieties. The notion of K-stability was first introduced by Gang Tian[1] and reformulated more algebraically later by Simon Donaldson.[2] The definition was inspired by a comparison to geometric invariant theory (GIT) stability. In the special case of Fano varieties, K-stability precisely characterises the existence of Kähler–Einstein metrics. More generally, on any compact complex manifold, K-stability is conjectured to be equivalent to the existence of constant scalar curvature Kähler metrics (cscK metrics).

  1. ^ Tian, Gang (1997). "Kähler–Einstein metrics with positive scalar curvature". Inventiones Mathematicae. 130 (1): 1–37. Bibcode:1997InMat.130....1T. doi:10.1007/s002220050176. MR 1471884. S2CID 122529381.
  2. ^ Donaldson, Simon K. (2002). "Scalar curvature and stability of toric varieties". Journal of Differential Geometry. 62 (2): 289–349. doi:10.4310/jdg/1090950195.