Embedding theorems for quantizable pseudo-K\"ahler manifolds

TL;DR

This paper explores embedding theorems for quantizable pseudo-Kähler manifolds, demonstrating how Bergman kernel asymptotics lead to classical results like Kodaira embedding and Tian's almost-isometry theorem in this setting.

Contribution

It establishes new embedding theorems for pseudo-Kähler manifolds using Bergman kernel asymptotics, extending classical results to the pseudo-Kähler context.

Findings

Asymptotic expansion of Bergman kernels for quantizable pseudo-Kähler manifolds.

Derivation of Kodaira embedding theorem analogues.

Derivation of Tian's almost-isometry theorem analogues.

Abstract

Given a compact quantizable pseudo-K\"ahler manifold $(M,\omega)$ of constant signature, there exists a Hermitian line bundle $(L,h)$ over $M$ with curvature $-2\pi i\,\omega$. We shall show that the asymptotic expansion of the Bergman kernels for $L^{\otimes k}$-valued $(0,q)$-forms implies more or less immediately a number of analogues of well-known results, such as Kodaira embedding theorem and Tian's almost-isometry theorem.