Éveline Legendre (Lyon) will speak in the geometry seminar on 11th March at 2pm in the Salle des Profs (9th floor building NO). Éveilne’s title is “The Einstein-Hilbert functional in Kähler geometry” and her abstract is below.
In this talk I will present a joint work with Abdellah Lahdilli (UQAM, Canada) and Carlo Scarpa (UQAM, Canada) where we study the existence problem of constant scalar Kähler metrics on polarised Kähler manifolds. Given such a polarised Kähler manifold (M,L), we translate the problem on the natural circle bundle associated to the polarization L, which comes equipped with a transversal holomorphic structure and an infinite family of CR structures each corresponding to a single Kähler metric of (M,L). Applying the theory developed around the CR-Yamabe problem in the eighties, we show that the Einstein-Hilbert functional, defined on this bundle of CR-contact structures, detects the constant scalar curvature Kähler metrics in the first Chern class of L. If time permits, I will also explain how we associate a two real parameters family of these contact structures to any ample test configuration and relate the limit, to the central fibre, of the Einstein-Hilbert functional 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.