Algebraic Bergman kernels and finite type domains in $\mathbb{C}^2$

TL;DR

This paper investigates the relationship between algebraic Bergman kernels and the finite type of boundary points in smoothly bounded pseudoconvex domains in a72, establishing bounds on the type and exploring conditions for rational kernels.

Contribution

It proves that algebraic Bergman kernels imply finite type boundaries with bounds, and characterizes domains with rational kernels, showing they are mostly the unit ball or of specific types.

Findings

Boundary b4G is of finite type with type 60a; r d   2d.

A rational Bergman kernel of degree    6 implies the domain is biholomorphic to the unit ball.

Domains with algebraic kernels cannot have rational kernels unless they are the ball.

Abstract

Let $G \subset \mathbb{C}^2$ be a smoothly bounded pseudoconvex domain and assume that the Bergman kernel of $G$ is algebraic of degree $d$. We show that the boundary $\partial G $ is of finite type and the type $r$ satisfies $r\leq 2d$. The inequality is optimal as equality holds for the egg domains $\{|z|^2+|w|^{2s}<1\},$ $s \in \mathbb{Z}_+$, by D'Angelo's explicit formula for their Bergman kernels. Our results imply, in particular, that a smoothly bounded pseudoconvex domain $G \subset \mathbb{C}^2$ cannot have rational Bergman kernel unless it is strongly pseudoconvex and biholomorphic to the unit ball by a rational map. Furthermore, we show that if the Bergman kernel of $G$ is rational of the form $\frac{p}{q}$, reduced to lowest degrees, then its rational degree $\max\{\text{deg } p, \text{deg } q \}\geq 6$. Equality is achieved if and only if $G$ is biholomorphic to the unit ball by a complex affine transformation of $\mathbb{C}^2$.