Tag: Jihun Yum

For a bounded domain Ω in Cⁿ, let P(Ω) be the set of all (real) probability distributions on Ω. Then, in general, P(Ω) becomes an infinite-dimensional smooth manifold and it always admit a natural Riemannian pseudo-metric, called the Fisher information metric, on P(Ω). Information geometry studies a finite-dimensional submanifold M, which is called a statistical model, in P(Ω) using geometric concepts such as Riemannian metric, distance, connection, and curvature, to better understand the properties of statistical models M and provide insights into the behavior of learning algorithms and optimization methods.

In this talk, we first introduce a map Φ : Ω → P(Ω) and prove that the pull-back of the Fisher information metric on P(Ω) is exactly same as the Bergman metric of Ω. This map provides a completely new perspective that allows us to view Bergman geometry from a stochastical viewpoint. We will discuss the following 4 things.

1. The relation between Φ and the Kobayashi map ι : Ω → CP^∞.

2. A Stochastic formula for the holomorphic sectional curvature of the Bergman metric.

3. A Stochastic condition for injectivity of a proper holomorphic surjective map between two bounded domains.

4. The central limit theorem on Ω.

This is a joint work with Gunhee Cho at UC Santa Barbara University.

The Bergman kernel B_X, which is by the definition the reproducing kernel of the space of L² holomorphic n-forms on a n-dimensional complex manifold X, is one of the important objects in complex geometry. In this talk, we observe the asymptotics of the Bergman kernels, as well as the Bergman metric, on a tower of coverings. More precisely, we show that, for a tower of finite Galois coverings {ϕ_j : X_j → X} of compact Kähler manifold X converging to an infinite Galois covering ϕ : X^~ → X, the sequence of push-forward Bergman kernels ϕ_j*B_Xj locally uniformly converges to ϕ_*B_X^~. Also, we show that if the canonical line bundle K_X^~ of X^~ is very ample, then the canonical line bundle K_Xj of X_j is also very ample for sufficiently large j. This is a joint work with S. Yoo in IBS-CCG.

Let Ω be a relatively compact pseudoconvex domain in a complex manifold X with smooth boundary ∂Ω. The Diederich-Fornaess index and the Steinness index of Ω are defined by

DF(Ω) := sup_ρ { 0 < η < 1 : -(-ρ)^η is strictly plurisubharmonic on Ω ∩ U for some neighborhood U of ∂Ω },

S(Ω) := inf_ρ { η > 1 : ρ^η is strictly plurisubharmonic on Ω^c ∩ U for some neighborhood U of ∂Ω },

where ρ is a defining function for Ω.

In the previous talk, we have seen that two indices are completely characterized by D’Angelo 1-form when the ambient space is X = Cⁿ. In this talk, we generalize the formulas for a relatively compact pseudoconvex domains in a (general) complex manifold X. Since the formulas do not hold anymore in general, unfortunately, we introduce 4 kinds of each of the Diederich-Fornaess and Steinness indices. Then we give some non-degeneracy conditions for these indices agree. Also, we exam the geometric meaning of the D’Angelo 1-form when the boundary ∂Ω is Levi-flat.

Let Ω be a bounded pseudoconvex domain in Cⁿ with smooth boundary ∂Ω. The Diederich-Fornaess index and the Steinness index of Ω are defined by

DF(Ω) := sup_ρ { 0 < η < 1 : -(-ρ)^η is strictly plurisubharmonic on Ω ∩ U for some neighborhood U of ∂Ω },

S(Ω) := inf_ρ { η > 1 : ρ^η is strictly plurisubharmonic on Ω^c ∩ U for some neighborhood U of ∂Ω },

where ρ is a defining function for Ω.

First, we see basic properties and known results about the Diederich-Fornaess and Steinness indices. Also, we see the relation between two indices on a 1-parameter family of domains in C², called worm domains, constructed by Diederich and Fornaess.