Kahler Geometry with a view towards the Calabi Conjecture

Abstract

Complex manifolds provide a fertile ground for studying Riemannian geometry as well as algebraic geometry. Many complex manifolds admit K¨ahler metrics. K¨ahler metrics are Riemannian metrics which tie in well with the complex structure and have a compatible symplectic structure. In the 1930s, E. Calabi conjectured the existence of K¨ahler metrics with good curvature properties on some compact complex manifolds. This conjecture was resolved by Aubin and Yau in the 70s. In parallel, Yau also proved the existence of K¨ahler metrics that are Einstein (Ric(!) = "!) in many cases (c1(M) > 0 and c1(M) = 0). In the case of Fano manifolds (c1(M) > 0), the existence of K¨ahler-Einstein metrics is not always true and is a much harder question. It was only recently completed thanks to the works of Chen, Donaldson, Sun, and Tian (among others). The primary aim of the present thesis is to study Yau’s proof of the Calabi conjecture (Chapter 4), as a part of which we study the basics of complex and K¨ahler geometry (Chapter 2) and the theory of the Monge-Amp`ere equation (Chapter 3). We will also look into a couple of applications of the Calabi conjecture, and discuss about K¨ahler-Einstein metrics (Chapter 5). The necessary preliminaries are presented in Chapter 1.