# On asymptotic expansions of generalized Bergman kernels on symplectic manifolds

## TL;DR

This paper derives detailed asymptotic expansions for generalized Bergman kernels on symplectic manifolds and constructs the associated Toeplitz operator algebra, advancing understanding in geometric analysis.

## Contribution

It provides the first full off-diagonal asymptotic expansion for these kernels and constructs the Toeplitz algebra related to the renormalized Bochner Laplacian.

## Key findings

- Established full off-diagonal asymptotic expansion for generalized Bergman kernels.
- Constructed the algebra of Toeplitz operators on symplectic manifolds.
- Enhanced tools for geometric quantization and analysis on symplectic manifolds.

## Abstract

A full off-diagonal asymptotic expansion is established for the generalized Bergman kernels of the renormalized Bochner Laplacians associated with high tensor powers of a positive line bundle over a compact symplectic manifold. As an application, the algebra of Toeplitz operators on the symplectic manifold associated with the renormalized Bochner Laplacian is constructed.