Berezin-Toeplitz quantization for lower energy forms

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.

44 pages; v.2 is a final update to agree with the published paper

References in corpus (6)

• Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles
• Toeplitz operators on symplectic manifolds
• Berezin-Toeplitz quantization on Kaehler manifolds
• Berezin-Toeplitz quantization and its kernel expansion
• Quantisation and the Hessian of Mabuchi energy
• An explicit formula for the Berezin star product

Cited by in corpus (5)

• Generalized Bergman kernels on symplectic manifolds of bounded geometry
• Semiclassical spectral analysis of Toeplitz operators on symplectic manifolds: the case of discrete wells
• Berezin-Toeplitz quantization for eigenstates of the Bochner-Laplacian on symplectic manifolds
• Geometric quantization results for semi-positive line bundles on a Riemann surface
• Hankel and Berezin type operators on weighted Besov spaces of holomorphic functions on polydiscs