DIFFERENTIAL GEOMETRY/PDE SEMINAR

Wednesday, March 14 Padelford C-401 (Note: Special room) 3:45 PM

CR construction of the versal deformation of normal isolated singularties

Kimio Miyajima (Kagoshima University, Japan)

As a Hartogs-Bochner phenomenon in Several Complex Variables, the germ of a normal isolated singularity of an analytic variety is uniquely determined by the CR structure on its boundary. In particular, a compact strongly pseudo-convex CR manifold of dim_{R} ≥ 5 is realized as a boundary of a Stein analytic variety. Based on this correspondence, M. Kuranishi proposed a program to construct the versal deformations of normal isolated singularities (of dim_{C} ≥ 3) in terms of deformations of CR structures on its boundary. The main problem in this program is to clear the following difficulties;

(1) there exist differences between the deformation theory of CR structures and the one of normal isolated singularities,

(2) the integrability condition of CR structure is a non-elliptic PDE.

In this talk, I will give a survey on the solution of the Kuranishi’s program, which clears these two difficulties in all dimensions of dim_{C} ≥ 2. Detailed argument appeared in J. Alg. Geom. 8 (1999), 403–470. In addition to the general theory, application to some typical examples might be mentioned.