Local algebraicity and localization of the Bergman kernel on Stein spaces with finite type boundaries

TL;DR

This work develops a localization theory for Bergman kernels on 2D Stein spaces with isolated singularities and smooth finite-type boundaries, introducing a new negative Sobolev-type norm to extend Fefferman’s asymptotics to complex manifolds. It proves a sharp boundary-type bound $r(\xi) \le 2d$ when the Bergman kernel form $K_{\Omega}$ is algebraic of degree $d$, and shows that locally rational kernels (degree $d=1$) force $\Omega$ to be biholomorphic to a ball quotient $\mathbb{B}^2/\Gamma$. A central technical advance is a localization theorem for Bergman kernels: near a boundary point, $K_M(z,\bar z)$ decomposes as $K_D(z,\bar z)+\phi(z,\bar z)$ with a model domain $D$, enabling Fefferman-type expansions and subelliptic estimates to be carried over to the manifold setting. The results connect algebraicity properties of Bergman kernels with precise geometric boundary invariants and yield a geometric classification in dimension two via ball quotients.

Abstract

On a two dimensional Stein space with isolated, normal singularities, smooth finite type boundary, and locally algebraic Bergman kernel, we establish an estimate on the type of the boundary in terms of the local algebraic degree of the Bergman kernel. As an application, we characterize two dimensional ball quotients as the only Stein spaces with smooth finite type boundary and locally rational Bergman kernel. A key ingredient in the proof of the degree estimate is a new localization result for the Bergman kernel of a pseudoconvex, finite type domain in a complex manifold.

Theorems & Definitions (35)

• Theorem 1
• Theorem 2
• Lemma 2.1
• Theorem 3
• proof
• Claim 2.1
• Remark 2.1
• proof : Proof of Claim \ref{['claim21']}:
• Claim 2.2
• proof