# Optimal convergence speed of Bergman metrics on symplectic manifolds

## TL;DR

This paper proves that embeddings of symplectic manifolds via eigensections of the Bochner Laplacian converge rapidly to the symplectic form, generalizing Donaldson's bounds to almost-Kähler manifolds.

## Contribution

It establishes a $1/p^2$ convergence rate of Fubini-Study forms to the symplectic form, extending Donaldson's bounds to the almost-Kähler setting.

## Key findings

- Fubini-Study forms converge at rate $1/p^{2}$ to the symplectic form.
- Generalization of Donaldson's bounds on the Calabi functional to almost-Kähler manifolds.
- Provides a quantitative convergence result for symplectic embeddings.

## Abstract

It is known that a compact symplectic manifold endowed with a prequantum line bundle can be embedded in the projective space generated by the eigensections of low energy of the Bochner Laplacian acting on high $p$-tensor powers of the prequantum line bundle. We show that the Fubini-Study forms induced by these embeddings converge at speed rate $1/p^{2}$ to the symplectic form. This result implies the generalization to the almost-K\"ahler case of the lower bounds on the Calabi functional given by Donaldson for K\"ahler manifolds, as shown by Lejmi and Keller.