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.
