OF THE DISSERTATION Canonical Kähler metrics with cone singularities by Ved V. Datar Dissertation Director: Jian Song 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. P˘ aun, that solutions to some complex Monge-Ampère equations with conical singularities along effective simple normal crossing divisors are uniformly equivalent to a conical
more » ... ric 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 soli-tons 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. ii Acknowledgements First and foremost, I would like thank my advisor Jian Song, for his guidance and support, without which this dissertation would not have been possible. I am deeply indebted to him for all that he has taught me, for supplying a continuous stream of new problems and directions, and above all, for his generosity in sharing his ideas and vision. Most of the complex geometry I know was learnt through the informal seminars organized by him, with carefully chosen topics and an emphasis on detailed presentations. But more importantly, his energy and enthusiasm for the subject were very infectious, always driving me to work harder. In addition to all this, I will also remember his occasional ramblings on mathematics, life, and other random topics, with great fondness.