Kähler–Einstein metrics on group compactifications

Abstract

We obtain a necessary and sufficient condition of existence of a Kähler-Einstein metric on a G×G-equivariant Fano compactification of a complex connected reductive group G in terms of the associated polytope. This condition is not equivalent to the vanishing of the Futaki invariant. The proof relies on the continuity method and its translation into a real MongeAmpère equation, using the invariance under the action of a maximal compact subgroup K × K.

Ricci flat Kähler metrics on rank two complex symmetric spaces

“…Section 4 is devoted to the proof of this criterion by a continuity method following the usual steps for complex Monge-Ampère equations. The 0 estimates are obtained using essentially Wang and Zhu's method, slightly modified as in [Del17a]. In Section 5, we build an asymptotic solution to the Ricci flat equation on a rank two symmetric space, using as essential ingredients Stenzel's metrics and the positive Kähler-Einstein metrics obtained in Section 3.…”

Volume minimization and obstructions to solving some problems in Kähler geometry

“…It is usually difficult to check whether a manifold is K-stable since there are infinitely many test configurations. However, in the cases with large symmetry groups checking K-stability can be easier, see [31], [17], [18]. For alternate proofs for the Yau-Tian-Donaldson conjecture for the Fano case, other important contributions, recent further developments and applications, the reader is referred to the two survey papers of Donaldson [21], [22].…”

Greatest lower bounds on Ricci curvature for Fano T-manifolds of complexity one

“…Using this same method we obtain an effective formula for manifolds with a torus action of complexity one, in terms of the combinatorial data of its divisorial polytope. Previously has been calculated for group compactifications by Delcroix and for homogeneous toric bundles by Yao .…”