International Congress of Mathematicians

ICM 2002 • Vol. II • 273-282Recent Progress in Kahler Geometry*Xiuxiong Chen^AbstractIn recent years, there are many progress made in Kahler geometry. Inparticular, the topics related to the problems of the existence and uniquenessof extremal Kahler metrics, as well as obstructions to the existence of suchmetrics in general Kahler manifold. In this talk, we will report some recentdevelopments in this direction. In particular, we will discuss the progressrecently obtained in understanding the metric structure of the infinite dimensionalspace of Kaehler potentials, and their applications to the problemsmentioned above. We also will discuss some recent on Kaehler Ricci flow.2000 Mathematics Subject Classification: 53, 35.Keywords and Phrases: Extremal Kahler metrics, Kähler-Einstein metrics,Holomorphic vector field, Holomorphic invariant, Kahler Ricci flow.In the last few years, we have witnessed a rapid progress in Kahler geometry.In particular, the topic related to the existence, to the uniqueness of extremal Kahlermetrics, and to obstructions to the existence of such metrics. In this talk, we willgive a brief survey of these exciting progress made in this direction.0.1. Some backgroundLet (M,OJ) be a polarized n-dimensional compact Kahler manifold, where OJ isa Kahler form on M. In local coordinates z\, • • • ,z n , we havenOJ = sf^i ^2 9fjdz % Adz j > 0,where {gn} is a positive definite Hermitian matrix function. The Kahler conditionrequires that a; is a closed positive (l,l)-form. The Kahler metric corresponding toOJ is given byn9u = Yl 9aß dza ®dz$.a,/3=l*Paritally supported by NSF research grant DMS-0110321 (2001-2004).'Department of Mathematics, Princeton University. Department of Mathematics, Universityof Wisconsin at Madison, USA. E-mail: xiu@math.princeton.edu