The Simanca metric admits a regular 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