On the moduli space of constant scalar curvature Kähler metrics on complex surfaces

Abstract

This thesis focuses on the existence of constant scalar curvature Kahler metrics on complex surfaces and does so in two main ways. The first uses the approach of Tonnesen-Friedman and the momentum construction of Hwang-Singer to construct scalar-flat, asymptotically locally Euclidean Kahler metrics on O(-k) by taking limits of momentum profiles of extremal metrics on the k-th Hirzebruch surfaces. What's more, the limit metrics are in fact the metrics LeBrun constructed in his seminal paper. The second focus of this thesis is trying to better understand which regions of the Kahler cone admit constant scalar curvature metrics and which do not. We employ parabolic stability introduced by Rollin-Singer in order to obtain regions of existence and slope stability of Ross-Thomas to obtain regions of non-existence.