Hong Kong Geometry Colloquium

March 31, 2007 (Saturday)

Room 517, Meng Wah Complex, HKU

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Professor Alexander Isaev

Australian National University, Canberra

Proper group actions in complex geometry

Abstract

In their celebrated paper of 1939 Myers and Steenrod showed that the group of isometries of a Riemannian manifold acts properly on the manifold. This fact has many important consequences. In particular, it implies that the group of isometries is a Lie group in the compact-open topology. This result triggered extensive studies of closed subgroups of the isometry groups of Riemannian manifolds. The peak of activities in this area occurred in the 1950's-70's, with many outstanding mathematicians involved: Kobayashi, Nagano, Yano, H.-C. Wang, Egorov, to name a few. In particular, Riemannian manifolds whose isometry groups possess subgroups of sufficiently high dimensions were explicitly determined.

I will speak about proper actions in the complex-geometric setting. In this setting (real) Lie groups act properly by holomorphic transformations on complex manifolds. My general aim is to build a theory parallel to the theory that exists in the Riemannian case. In my lecture I will survey recent classification results for complex manifolds that admit proper actions of high-dimensional groups.