Balanced metrics on \(\mathbb C^n\)
type: Item
instance of: scholarly article
title: Balanced metrics on \(\mathbb C^n\)
zbMATH Open document ID: 1115.53051
author: Fabrizio Cuccu
author: Andrea Loi
published in: Journal of Geometry and Physics
publication date: 2007-03-15T00:00:00Z
review text: Let \(g\) be a Kähler metric on \({\mathbb C}^n\) with a Kähler potential \(\Phi\colon{\mathbb C}^n\to{\mathbb R}\) of the form \(\Phi(z_1,\dots,z_n)=g_1(| z_1|^2)+\cdots+g_n(| z_n|^2)\) and assume that all the holomorphic polynomial functions on \({\mathbb C}^n\) belong to the corresponding Hilbert space \({\mathcal H}_\Phi=\{f\in{\mathcal O}({\mathbb C}^n)\mid \int_{{\mathbb C}^n}| f|^2\text{e}^{-\Phi}\text{d}\mu(z)<\infty\}\), where \(\text{d}\mu(z)\) stands for the Lebesgue measure on \({\mathbb C}^n\). In this case any orthonormal basis \(\{f_j\}_{0\leq j<N}\) in \({\mathcal H}_\Phi\) gives rise to a holomorphic map \(z\mapsto[f_0(z),f_1(z),\dots]\) from \({\mathbb C}^n\) into the complex projective space \({\mathbb C}P^N\) (where \(N=\dim{\mathcal H}_\Phi\in\{1,2,\dots,\aleph_0\}\)). The Kähler metric \(g\) is said to be balanced if it is the pull-back of the corresponding Fubini-Study metric by the above map. If this is the case, then the main theorem of the paper under review says that the metric \(g\) is equal to the Euclidean metric on \({\mathbb C}^n\) up to a biholomorphic isometry. The proof of this fact relies on works by \textit{E.~Calabi} [Ann. Math. (2) 58, 1--23 (1953; Zbl 0051.13103)] and \textit{J.~Miles} and \textit{F.~Wiliamson} [J. Lond. Math. Soc., II. Ser. 33, 110--116 (1986; Zbl 0545.30017)].
reviewed by: Daniel Beltiţă
Mathematics Subject Classification ID: 53D05
Mathematics Subject Classification ID: 53D50
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 58C25
zbMATH DE Number: 5134191
zbMATH Keywords: Kähler metric
zbMATH Keywords: diastasis function
zbMATH Keywords: geometric quantization
MaRDI profile type: Publication
full work available at URL: j.geomphys.2006.09.006
OpenAlex ID: W2028871622
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Wikidata QID: Q125932488
DOI: 10.1016/J.GEOMPHYS.2006.09.006
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description: scientific article; zbMATH DE number 5134191