Symmetries, quotients and Kähler-Einstein metrics

Abstract

We consider Fano manifolds M that admit a collection of finite automorphism groups G 1 , ..., G k , such that the quotients M/G i are smooth Fano manifolds possessing a Kähler-Einstein metric. Under some numerical and smoothness assumptions on the ramification divisors, we prove that M admits a Kähler-Einstein metric too.

Blowing up and desingularizing constant scalar curvature Kähler manifolds

“…The maximum principle applied in {z∈C Γ :|z|∈ [1,2]} then allows us to fill in the gap in the estimate, and we conclude that…”

“…This means any manifold with negative first Chern class and many families of examples of positive first Chern class [2], [45], [55]. We should note that there are no Futaki-nondegenerate Kähler-Einstein manifolds except the ones with discrete automorphisms, as observed by LeBrun-Simanca [36].…”

“…In given examples, we will see in §7 that these potentials can arise with various orders and decays, and we will show (Lemma 7.2) that, for zero scalar curvature metrics and under reasonable growth assumptions on the potential, one can change suitably the potential in order to get a potential for which (2) holds.…”

On K-stability of finite covers

“…We give some rather elementary applications of Theorem 1.1, mostly by comparison with the work of Arezzo-Ghigi-Pirola [2]. In most examples, we take the base X to either be projective space or a product of projective spaces.…”

Kähler–Einstein metrics on orbifolds and Einstein metrics on spheres

“…By the Galois condition on X 0 , there are g n ∈ Gal(π) such that z n = g n .z n . As Gal(π) is finite, we can extract a subsequence with g n ≡ g. Since lim z n = x 2 as π −1 (y) ∩ U 2 = {x 2 }, we get x 2 = g.x 1 .…”