5 dec 2019 -- 14:30
Aula 1, DiMaI, Firenze
Existence of constant scalar curvature Kähler (cscK) metrics on compact Kähler manifolds is a central question in complex geometry. Following the variational approach pioneered by Mabuchi in the 1980's it was recently proven (by X.X. Chen and J. Cheng) that existence of cscK metrics is equivalent to coercivity of a certain energy functional on the space of Kähler metrics. In this talk I will present new coercivity estimates directly related to this problem. In particular, I will present an explicit formula for the optimal coercivity constants appearing in the variational picture mentioned above, thus characterizing explicitly solvability of Donaldson's J-equation (an equation related both to the cscK equation and to the Hermitian Yang Mills equation in geometry and physics). We will focus in this talk on the applications of our estimates to the cscK problem; in particular we obtain new and improved existence and non-existence criteria this way.