The Einstein-Hilbert functional in Kähler and Sasaki geometry

Início: 01/12/2023 17:00

Término: 01/12/2023 18:00

Palestrante: Eveline Legendre (U. Lyon)

E-mail do Palestrante: eveline.legendre@univ-lyon1.fr

Resumo: In this talk I will present a recent joint work with Abdellah Lahdilli and Carlo Scarpa where, 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 $L$. If time permits, I will explain how we associate a two real parameters family of these contact structures to any ample test configuration and relate the limit, on the central fibre, to a primitive of the Donaldson-Futaki invariant. As a by-product, we show that the existence of cscK metrics on a polarized manifold implies K-semistability