Geometry and Topology Seminar

In complex geometry, the Bergman metric plays a very important role as a canonical metric as a pullback metric of the Fubini-Study metric of complex projective ambient space. This work is trying to do something really new to find a whole new approach of studying hyperbolic complex geometry, especially for a bounded domain in C^n, we replace the infinite dimensional complex projective ambient space to the collection of probability distributions defined on a bounded domain. We prove that in this new framework, the Bergman metric is given as a pullback metric of the Fisher-Information metric considered in information geometry, and from this, a new perspective on the contraction property and biholomorphic invariance of the Bergman metric will be discussed. As an application of this framework, in the case of bounded hermitian symmetric domains, we will discuss about the existence of a sequence of i.i.d random variables in which the covariance matrix converges to a distribution sense with a normal distribution given by the Bergman metric, and if more time is left, we will talk about recent progresses on stochastic differential geometry.