Determinants of the Laplacians on complex projective spaces \(\mathbf{P}^n(\mathbb{C})(n\geq 1)\)
type: Item
instance of: scholarly article
title: Determinants of the Laplacians on complex projective spaces \(\mathbf{P}^n(\mathbb{C})(n\geq 1)\)
zbMATH Open document ID: 1404.33005
author: Richard Olu Awonusika
published in: Journal of Number Theory
publication date: 2018-05-29T00:00:00Z
review text: The purpose of this well-written article is to determine explicitly the determinants of the Laplacians on the complex projective space by studying the asymptotic expansion of their heat coefficients. These asymptotic expansions for the corresponding Dirichlet series are given by the quite general so-called Minakshisundaram-Pleijel formula. All important formulas distinguish between even and odd values of \(n\), for the heat coefficients \(a_k^n\). First, the spectral zeta function of \(M\), i.e., the Mellin transform of the heat trace, is introduced. The heat kernel on the complex projective \(n\)-space admits a spectral representation in terms of the spherical function, which can be expressed in terms of the normalised Jacobi polynomials. The first main theorem expresses the heat coefficients \(a_k^n\) in terms of Bernoulli numbers. The proof uses the fact that the asymptotic expansion the first theta function can be expressed as a sum with coefficients Bernoulli numbers. As examples, the heat coefficients for the special cases \(n = 1,2, 3, 4, 5\) are explicitly computed. Then, by using the Hurwitz zeta function, the heat coefficients are described by the residues of the spectral zeta functions. Finally, the determinant of the Laplacian is expressed via the spectral zeta function. These two formulas are quite complicated powers of products of exponentials etc. The last proof seems quite simple and uses the digamma function and Bernoulli polynomials. As before, various special cases of \(n\) are considered.
reviewed by: Thomas Ernst
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 35A08
Mathematics Subject Classification ID: 35C05
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 6876041
zbMATH Keywords: complex projective space
zbMATH Keywords: determinant of Laplacian
zbMATH Keywords: Minakshisundaram-Pleijel coefficients
zbMATH Keywords: spectral zeta function
MaRDI profile type: Publication
full work available at URL: j.jnt.2018.02.007
OpenAlex ID: W2792425398
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DOI: 10.1016/J.JNT.2018.02.007
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description: scientific article; zbMATH DE number 6876041