Exposé Bourbaki 1112 : Approche variationnelle pour les équations de Monge-Ampère complexes et applications géométriques
Exposé Bourbaki 1112 : Variational approach for complex Monge-Ampère equations and geometric applications
Astérisque | Exposés Bourbaki | 2017
Anglais
Solutions of Monge-Ampère equations on compact Kähler manifolds can be obtained by a variational method independent of Yau's theorem. The technique relies on the study of certain functionals (Ding-Tian, Mabuchi) on the space of Kähler metrics, and on their geodesic convexity, due to Berndtsson-Berman in its general form. Applications include the existence and uniqueness of Kähler-Einstein metrics on $\mathbb Q$-Fano varieties with log terminal singularities and a new proof of a uniform version of the Yau-Tian-Donaldson conjecture.
Variational method, Monge-Ampère equation, Kähler-Einstein metric, plurisubharmonic potential, Fano variety, Tian-Yau-Donaldson conjecture, Donaldson-Futaki invariant, $K$-stability, space of Kähler potentials, Aubin functional, Mabuchi functional, Ding functional.
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