Constant scalar curvature Kähler metric

In differential geometry, a constant scalar curvature Kähler metric (cscK metric), is (as the name suggests) a Kähler metric on a complex manifold whose scalar curvature is constant. A special case is Kähler–Einstein metric, and a more general case is extremal Kähler metric.

Donaldson (2002), Tian and Yau conjectured that the existence of a cscK metric on a polarised projective manifold is equivalent to the polarised manifold being K-polystable. Recent developments in the field suggest that the correct equivalence may be to the polarised manifold being uniformly K-polystable . When the polarisation is given by the (anti)-canonical line bundle (i.e. in the case of Fano or Calabi–Yau manifolds) the notions of K-stability and K-polystability coincide, cscK metrics are precisely Kähler-Einstein metrics and the Yau-Tian-Donaldson conjecture is known to hold .

References

• Biquard, Olivier (2006), "Métriques kählériennes à courbure scalaire constante: unicité, stabilité", Astérisque, Séminaire Bourbaki. Vol. 2004/2005 Exp. No. 938 (307): 1–31, ISSN 0303-1179, MR 2296414
• Donaldson, S. K. (2001), "Scalar curvature and projective embeddings. I", Journal of Differential Geometry, 59 (3): 479–522, ISSN 0022-040X, MR 1916953
• Donaldson, S. K. (2002), "Scalar curvature and stability of toric varieties", Journal of Differential Geometry, 62 (2): 289–349, ISSN 0022-040X, MR 1988506