Regularized determinants of Laplacians for Hermitian line bundles over projective spaces
type: Item
instance of: scholarly article
title: Regularized determinants of Laplacians for Hermitian line bundles over projective spaces
zbMATH Open document ID: 0847.14002
author: Lin Weng
published in: Journal of Mathematics of Kyoto University
publication date: 1996-10-15T00:00:00Z
review text: Regularized determinants of Laplacians for Hermitian vector bundles play a very important role in modern mathematics. While a lot of progress has been made in the qualitative study of such a concept, comparably few results in the quantitative study are known. In this paper, we give the precise values for the regularized determinant for all line bundles \({\mathcal O}(m)\) over projective spaces \(\mathbb{P}^n\) with respect to the Fubini-Study metric, in terms of \(m \) and \(n\). The value of the regularized determinant, by definition, depends on the eigenvalues and their multiplicities of the associated Laplacians. In practice, there are two ways to compute it. One is done by a certain general formalism. The other is done by using the precise eigenvalues and their multiplicities, which are, in general, hard to find. In this paper, we adopt the first method. More precisely, we will consider the projective space as an arithmetic variety. Then we use the arithmetic Riemann-Roch theorem to evaluate the regularized determinant in question. For doing so, we need to know a certain arithmetic intersection and the \(L^2\) contribution to the Quillen metric on the associated determinant line bundle. During this process, one then needs to use the classical Bott-Chern secondary characteristic class. When \(m=0\), the result was first obtained by Gillet, Soulé and Zagier, by using the second method. For the corresponding result on the sphere in Riemannian geometry, we refer to the paper: ``Analytic torsion of spheres'', by the author and \textit{Y. You} in Int. J. Math. 7, No. 1, 109-125 (1996; Zbl 0854.58043).
reviewed by: Lin Weng
Mathematics Subject Classification ID: 14C40
Mathematics Subject Classification ID: 14M12
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 32L10
Mathematics Subject Classification ID: 58J52
zbMATH DE Number: 880133
zbMATH Keywords: regularized determinants
zbMATH Keywords: arithmetic Riemann-Roch theorem
zbMATH Keywords: Quillen metric
zbMATH Keywords: Bott-Chern secondary characteristic class
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518700
OpenAlex ID: W1503638726
DOI: 10.1215/KJM/1250518700
description: scientific article; zbMATH DE number 880133