Colloquium: "On several rigidity problems in several complex variables and complex geometry"
Abstract: In 1907, Poincare proved that any non-constant holomorphic map sending a piece of the boundary of the unit ball in a complex space of dimension two into the boundary of the unit ball extends to an automorphism of the ball. This result fails for holomorphic functions of one variable and reveals a strong rigidity property for holomorphic functions of several variables. The program of Poincare that started in 1907 was later pursued further by Segre (1930's), Cartan (1940's) and Chern-Moser (1974). The theory of Segre-Cartan-Chern-Moser shows that real hypersyrfaces in a complex space of higher dimension possese many holomorphic invariants and holomorphic maps between hypersurfaces need to preserve those invariants and thus must be very rigid. Based on these invariants, Webster and the speaker established a several complex analysis version of the W. L. Chow theorem in 1977 (equi-dimensional case) and in 1994 (any codimensional case), respectively. These invariant and rigidity results have found many applications in the study of other rigidity problems in several complex variables, CR geometry, isolated complex singularity theory and complex geometry. The main goal of this talk is to give a survey on these lines of investigation in which the author has been involved in the past 20 years. The topics include: Gap rigidity for proper holomorphic maps between balls; rigidity problems for Milnor links and isolated normal complex singularities; rigidity of local holomorphic volume preserving maps between Hermitian symmetric spaces.
Host: Quo-Shin Chi