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

arXiv comment17 pages. A new theorem and a new remark have been added

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• On holomorphic isometries into blow-ups of ${\mathbb{C}}^n$
• On weak and strict relatives Kähler manifolds