Seminar in Real & Complex Geometry

Wednesday, 17.09.2014, 11:00-13:00, Schreiber building, room 209

Alex Isaev, Australian National University

1st lecture: On the CR-embeddability of a family of real hypersurfaces
in an n-dimensional complex affine quadric into complex Euclidean n-space


We discuss a family of real hypersurfaces M_t^n, t>1, in a complex affine n-dimensional quadric arising in connection with the classification of homogeneous compact simply-connected real-analytic hypersurfaces in C^n due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the embeddability of M_t^n in C^n for n=3,7. We show that M_t^7 is not embeddable in C^7 for every t and that M_t^3 is embeddable in C^3 for all t\in(1,1+10^{-6}). For all other values of t the embeddability question remains open. This question is closely related to the well-known problem of the existence of a totally real embedding of the sphere S^n to C^n.

2nd lecture: Proper group actions in complex geometry


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 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 1950s-70s, with many outstanding mathematicians involved: Kobayashi, Nagano, Yano, H.-C. Wang, I. P. 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.