On the singularities of the Bergman projections for lower energy forms on complex manifolds with boundary

TL;DR

This paper investigates the singularities of Bergman projections on complex manifolds with boundary, providing asymptotic expansions of the spectral kernel and applications to weakly pseudoconvex domains and symmetric embeddings.

Contribution

It establishes full asymptotic expansions of the spectral kernel and Bergman projection near the boundary, extending understanding to weakly pseudoconvex and symmetric cases.

Findings

Spectral kernel admits full asymptotic expansion near boundary

Bergman projection has asymptotic expansion under local closed range condition

Derived embedding theorems for domains with holomorphic $S^1$-action

Abstract

Let $M$ be a complex manifold of dimension $n$ with smooth boundary $X$. Given $q\in\{0,1,\ldots,n-1\}$, let $\Box^{(q)}$ be the $\ddbar$-Neumann Laplacian for $(0,q)$ forms. We show that the spectral kernel of $\Box^{(q)}$ admits a full asymptotic expansion near the non-degenerate part of the boundary $X$ and the Bergman projection admits an asymptotic expansion under some local closed range condition. As applications, we establish Bergman kernel asymptotic expansions for some domains with weakly pseudoconvex boundary and $S^1$-equivariant Bergman kernel asymptotic expansions and embedding theorems for domains with holomorphic $S^1$-action.