Generalized Bergman kernels on symplectic manifolds of bounded geometry

TL;DR

This paper investigates the asymptotic properties of generalized Bergman kernels on symplectic manifolds with bounded geometry, providing new estimates, expansions, and applications to quantization on orbifolds.

Contribution

It establishes off-diagonal exponential estimates, full asymptotic expansions, and extends Berezin-Toeplitz quantization theory to symplectic orbifolds.

Findings

Off-diagonal exponential decay of the Bergman kernel

Full off-diagonal asymptotic expansion with improved remainder estimates

Development of Berezin-Toeplitz quantization on symplectic orbifolds

Abstract

We study the asymptotic behavior of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle on a symplectic manifold of bounded geometry. First, we establish the off-diagonal exponential estimate for the generalized Bergman kernel. As an application, we obtain the relation between the generalized Bergman kernel on a Galois covering of a compact symplectic manifold and the generalized Bergman kernel on the base. Then we state the full off-diagonal asymptotic expansion of the generalized Bergman kernel, improving the remainder estimate known in the compact case to an exponential decay. Finally, we establish the theory of Berezin-Toeplitz quantization on symplectic orbifolds associated with the renormalized Bochner-Laplacian.