Berezin-Toeplitz quantization for lower energy forms

TL;DR

This paper develops a Berezin-Toeplitz quantization framework for complex manifolds using spectral spaces of the Kodaira Laplace operator, establishing asymptotic expansions and star-products, with applications to semi-positive and big line bundles.

Contribution

It introduces a new quantization method based on spectral spaces of the Kodaira Laplace operator, extending Berezin-Toeplitz quantization to lower energy forms and semi-positive line bundles.

Findings

Established asymptotic expansion of Toeplitz operators as k→∞

Defined a star-product corresponding to the quantization

Applied the method to semi-positive and big line bundles

Abstract

Let $M$ be an arbitrary complex manifold and let $L$ be a Hermitian holomorphic line bundle over $M$. We introduce the Berezin-Toeplitz quantization of the open set of $M$ where the curvature on $L$ is non-degenerate. The quantum spaces are the spectral spaces corresponding to $[0,k^{-N}]$ ($N>1$ fixed), of the Kodaira Laplace operator acting on forms with values in tensor powers $L^k$. We establish the asymptotic expansion of associated Toeplitz operators and their composition as $k\to\infty$ and we define the corresponding star-product. If the Kodaira Laplace operator has a certain spectral gap this method yields quantization by means of harmonic forms. As applications, we obtain the Berezin-Toeplitz quantization for semi-positive and big line bundles.