The Simanca metric admits a regular quantization

TL;DR

This paper demonstrates that the Simanca metric on the blow-up of complex two-space admits a regular quantization, leading to vanishing coefficients in its Tian-Yau-Zelditch expansion and the existence of a Berezin quantization on a dense subset.

Contribution

It proves the regular quantization of the Simanca metric and shows the vanishing of all coefficients in its Tian-Yau-Zelditch expansion, establishing Berezin quantization for a dense subset.

Findings

The Simanca metric admits a regular quantization.

All coefficients in the Tian-Yau-Zelditch expansion vanish.

A dense subset admits a Berezin quantization.

Abstract

Let $g_S$ be the Simanca metric on the blow-up $\tilde{\mathbb{C}}^2$ of $\mathbb{C}^2$ at the origin. We show that $(\tilde{\mathbb{C}}^2,g_S)$ admits a regular quantization. We use this fact to prove that all coefficients in the Tian-Yau-Zelditch expansion for the Simanca metric vanish and that a dense subset of $(\tilde{\mathbb{C}}^2, g_S)$ admits a Berezin quantization