Semi-classical asymptotics of partial Bergman kernels on $\mathbb{R}$-symmetric complex manifolds with boundary

TL;DR

This paper derives the asymptotic behavior of partial Bergman kernels on complex manifolds with boundary under an $R$-action, and applies results to boundary extension of biholomorphic maps and embedding of pseudoconcave manifolds.

Contribution

It establishes the asymptotic expansion of partial Bergman kernels for Fourier modes on $R$-symmetric complex manifolds with boundary, extending classical boundary extension results.

Findings

Asymptotic expansion of partial Bergman kernels for high-frequency Fourier modes.

An $R$-equivariant boundary extension theorem for biholomorphic maps.

Results on embedding pseudoconcave manifolds.

Abstract

Let $M$ be a relatively compact connected open subset with smooth connected boundary of a complex manifold $M'$. Let $(L,h^L)\rightarrow M'$ be a positive line bundle over $M'$. Suppose that $M'$ admits a holomorphic $\mathbb{R}$-action which preserves the boundary of $M$ and lifts to $L$. We establish the asymptotic expansion of a partial Bergman kernel associated to a package of Fourier modes of high frequency with respect to the $\mathbb{R}$-action in the high powers of $L$. As an application, we establish an $\mathbb{R}$-equivariant analogue of Fefferman's and Bell-Ligocka's result about smooth extension up to the boundary of biholomorphic maps between weakly pseudoconvex domains in $\mathbb{C}^n$. Another application concerns the embedding of pseudoconcave manifolds.