Auteurs : Thibaut Delcroix

We express notions of K-stability of polarized spherical varieties in terms of combinatorial data, vastly generalizing the case of toric varieties. We then provide a combinatorial sufficient condition of G-uniform K-stability by studying the corresponding convex geometric problem. Thanks to recent work of Chi Li and a remark by Yuji Odaka, this provides an explicitly checkable sufficient condition of existence of constant scalar curvature Kahler metrics.
As a side effect, we show that, on several families of spherical varieties, G-uniform K-stability is equivalent to K-polystability with respect to G-equivariant test configurations for polarizations close to the anticanonical bundle.

Comment: with an appendix by Yuji Odaka, v2 taking into account referee's comments and correcting minor mistakes, v3 EPIGA journal version

• 14M27 - Compactifications; symmetric and spherical varieties
• 32Q20 - Kähler-Einstein manifolds
• 32Q26 - Notions of stability for complex manifolds
• 53C25 - Special Riemannian manifolds (Einstein, Sasakian, etc.)

• 1 zbMATH Open