Kähler-Einstein metrics on algebraic manifolds

Abstract

It is one of fundamental problems in differential geometry to find a distinguished metric on a smooth manifold. H. Poincare's Uniformization theorem settles this problem for Riemann surfaces. That is, there is a unique metric with constant curvature in each Kahler class on a Riemann surface. Trying to generalize it to higher dimensions, E. Calabi conjectured in the 50s the existence of KahlerEinstein metrics on a compact Kahler manifold with its first Chern class definite. A Kahler-Einstein metric is a Kahler metric with constant Ricci curvature. In the middle of the 70s, this conjecture was solved by S. T. Yau in case the first Chern class is vanishing and Aubin and Yau, independently, in case the Chern class is negative (cf. [Yl]). The uniqueness in these two cases was done by E. Calabi himself in the 50s. Such Kahler-Einstein metrics were then applied to studying projective manifolds. For instance, Yau used these metrics to show the Miyaoka-Yau inequality on surface of general type, its generalized version in higher dimensions and the characterization of the quotients of the complex hyperbolic spaces (cf. [Y2]). We also refer readers to [CY, Ko, Ts, TY1] for the generalizations of these to quasi-projective manifolds. However, this conjecture of Calabi still remains open in general in case the first Chern class is positive. In this paper, we will survey the recent progress on this part of Calabi's conjecture, including the uniqueness and the existence of Kahler-Einstein metrics with positive scalar curvature, the outline of the complete solution for Calabi's conjecture in case of complex dimension two, etc. Some related problems will also be discussed. From now on, we always denote by M a compact Kahler manifold with positive first Chern class C\(M), that is, M is a smooth Fano variety. Then we can choose a Kahler metric g with its Kahler class cog representing C\(M). In local coordinates (z\,'--,zn) of M with dime M = n, if g is represented by positive hermitian metrices {g (z)}i /,y5<H,