Conférencier: Abdellah Lahdili, UQAM

Résumé: Given a polarised Kähler 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 Kähler 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 Kähler 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.