Canonical Kähler metrics with cone singularities

Abstract

This dissertation consists of some results on the existence and regularity of canonical Kähler metrics with cone singularities. First, a much shorter proof is provided for a result of H. Guenancia and M. Paun, that solutions to some complex Monge-Ampère equations with conical singularities along effective simple normal crossing divisors are uniformly equivalent to a conical metric along that divisor. It is also shown that such metrics can always be approximated, in the Gromov-Hausdorff topology, by smooth metrics with a uniform Ricci lower bound and uniform diameter bound. As an application, it is proved that the regular set of these metrics is convex. Next, the existence of conical Kähler-Einstein metrics and conical Kähler-Ricci solitons on toric manifolds is studied in relation to the greatest lower bounds for the Ricci and the Bakry-Emery Ricci curvatures. It is also shown that any two toric manifolds of the same dimension can be connected by a continuous path of toric manifolds with conical Kähler-Einstein metrics in the Gromov-Hausdorff topology. In the final chapter, the greatest lower bound for the Bakry-Emery Ricci curvature is studied on Fano manifolds. In particular, it is related to the solvability of some soliton-type complex Monge-Ampère equations and the properness of a twisted Mabuchi energy, extending previous work of Székelyhidi on the greatest lower bound for Ricci curvature.