Alexander Isaev, Canberra
Title: 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 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, Egorov, to name a few. In particular, Riemannian manifolds with isometry groups of sufficiently high dimensions were explicitly determined.
I will speak about proper actions in the complex-geometric setting. In this setting groups act properly by holomorphic transformations on complex manifolds and are no longer assumed to be the full isometry groups with respect to some Riemannian metric. My general aim is to build a theory parallel to the theory that exists in the Riemannian case. In my lecture I will survey my recent classification results for complex manifolds that admit proper actions of high-dimensional groups.