International Congress of Mathematicians

274 Xiuxiong ChenFor simplicity, in the following, we will often denote by OJ the corresponding Kahlermetric. The Kahler class of w is its cohomology class [OJ] in H 2 (M, R). It followsfrom the Hodge-Dolbeault theorem that any other Kahler metric in the same Kahlerclass is of the formï£>0. . -, dz i dzi*J=Ifor some real valued function tp on M.Given a Kahler metric OJ, its volume form is—^ = (>/=!)" det (W) dz 1 A dz T A • • • A dz n Adz".Its Ricci (curvature) form is:Ric(oj) = y/^ÏRfj dwi dwj = —y^ïdd log det oj n .Note also that R(OJ) = g^Rfj corresponds to one half times the scalar curvatureas it is usually defines in Riemannian geometry. We say that the first Chern classof M is positive of negative definite, if there exists a real valued function ip onfi' 2 M such that Rß + dw,ß W . is, respectively, positive of negative definite. A Kahlermetric is Kähler-Einstein, if the Ricci form is proportional to the Kahler form bya constant factor. A Kahler metric is called extremal in the sense of E. Calabi [3],if it is a critical point of the functional / \Ric(oj)\ 2 oj n , or, equivalently, if theJ Mcomplex gradient vector field of the scalar curvature function g a/3 (OJ) d^' ^f- is aholomorphic vector field.0.2. Existence of extremal Kahler metricsIt is well known that a Kähler-Einstein metric satisfies a Monge-Ampere equationwhere [Ric(ojj] = X [OJ] andOJ^log det —- = —A tp+ h u0J nRic(oj) — À OJ = idd h w .In Calabi's work in the 1950s, he made conjectures about the existence of Kähler-Einstein metrics on compact Kahler manifolds with definite first Chern class. In1976, Aubin and Yau independently obtained existence when the first Chern classis negative. Around the same time, Yau proved also the existence of a Kähler-Einstein metric when the first Chern class vanishes. This is a celebrated work; andany Kahler manifold admit such a metric is called "Calabi-Yau" manifold. Thepositive case remains open, but significant progress has been made in the last twodecades. G. Tian proved in [29] the existence of Kähler-Einstein metrics on anycomplexsurface with positive first Chern class and reductive automorphism group.