On homothetic balanced metrics
type: Item
instance of: scholarly article
title: On homothetic balanced metrics
zbMATH Open document ID: 1241.53061
author: Claudio Arezzo
author: Andrea Loi
author: Fabio Zuddas
published in: Annals of Global Analysis and Geometry
publication date: 2012-03-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1105.5315
review text: Let \(g\) be a Kähler metric on a compact complex manifold \(M\). In the terminology of quantum mechanics a compact Kähler manifold \((M, g)\) is said to be quantizable if the Kähler form \(\omega\) associated to g is integral, i.e., there exists a holomorphic line bundle \(L\) over \(M\), called the \textit{quantum line bundle}, whose first Chern class equals the second De-Rham cohomology class of \(\omega\), that is \(c_1(L)=[\omega]_{dR}\). By Kodaira's theory this is equivalent to saying that \(M\) is a projective algebraic manifold and \(L\) is a positive (or ample) line bundle over \(M\). In terms of algebraic-geometric \(L\) is called a \textit{polarization} of \(M\), \(g\) a \textit{polarized} metric and the pair \((M, L)\) a \textit{polarized manifold}. Fix a polarization \(L\) over \(M\). Then there exists a Hermitian metric \(h\) on \(L\), uniquely defined up to the multiplication with a positive constant, such that its Ricci curvature equals \(\omega\). The pair \((L, h)\) is called a \textit{geometric quantization} of the Kähler manifold \((M, \omega)\). Let \(s_0, \dots , s_{N}\) be an orthonormal basis of \(H^0(L)\) (the space of global holomorphic sections of \(L\)) with respect to the \(L^2\) scalar product induced by \(\omega\). Consider the non-negative smooth function \(T_{g}\) on \(M\) given by \(T_{g} (x) =\sum_{j=0}^{N}h(s_j(x), s_j(x)).\) The metrics for which \(T_{g}\) is constant are called balanced in the terminology introduced by \textit{S. K. Donaldson} in [J. Differ. Geom. 59, No. 3, 479--522 (2001; Zbl 1052.32017)]. In the paper under consideration the authors study the set \({B}(L)\) consisting of all balanced metrics \(g_B\) and its subset \({B}_{g_B} = \{mg_{B}\text{ is balanced}| m \in \mathbb N^+\} \), for any \(g_B \in {B}(L)\). The quotient of \({B}(L)\) by the equivalence relation (which identifies two balanced metrics if they belong to the same cohomology class) is denoted by \({B}_c(L)\). The main results are: 1) If L is a po…
reviewed by: Cornelia-Livia Bejan
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 58E11
zbMATH DE Number: 6018063
zbMATH Keywords: Kähler manifolds
zbMATH Keywords: balanced metrics
zbMATH Keywords: regular quantization
zbMATH Keywords: Tian-Yau-Zelditch asymptotic expansion
zbMATH Keywords: constant scalar curvature metrics
MaRDI profile type: Publication
OpenAlex ID: W2064637868
arXiv ID: 1105.5315
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Wikidata QID: Q126123936
DOI: 10.1007/S10455-011-9295-8
description: scientific article