The Einstein-Hilbert functional in Kähler and Sasaki geometry

Abstract: Given a polarised K\"ahler manifold $(M,L)$, we consider the circle bundle associated to the polarization with the induced transversal holomorphic structure. The space of contact structures compatible with this transversal structure is naturally identified with a bundle, of infinite rank, over the space of K\"ahler metrics in the first Chern class of $L$. We show that the Einstein--Hilbert functional of the associated Tanaka--Webster connections is a functional on this bundle, whose critical points are constant scalar curvature Sasaki structures. In particular, when the group of automorphisms of $(M,L)$ is discrete, these critical points correspond to constant scalar curvature K\"ahler metrics in the first Chern class of $L$. We show that the Einstein--Hilbert functional satisfies some monotonicity properties along some one-parameter families of CR-contact structures that are naturally associated to test configurations, and that its limit on the central fiber of a test configuration is related to the Donaldson--Futaki invariant. As a by-product, we show that the existence of cscK metrics on a polarized manifold implies K-semistability. This is a joint work with Eveline Legendre and Carlo Scarpa.

algebraic geometryanalysis of PDEsalgebraic topologycomplex variablesdifferential geometrygeneral topologygeometric topologyK-theory and homologymetric geometrysymplectic geometry

Audience: researchers in the topic

CRM - Séminaire du CIRGET / Géométrie et Topologie

Series comments: Hybrid seminar of geometry and topology. Laboratory : CIRGET - www.cirget.uqam.ca The homepage of the seminar is www.cirget.uqam.ca/fr/seminaires.html

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