On the residue of the spectral zeta functions of Kähler manifolds with conical singularities
type: Item
instance of: scholarly article
title: On the residue of the spectral zeta functions of Kähler manifolds with conical singularities
zbMATH Open document ID: 0849.58072
author: Ken-Ichi Yoshikawa
published in: Journal of Differential Geometry
publication date: 1996-11-10T00:00:00Z
review text: We show that there is a nontrivial relation among the special values of the spectral zeta functions for Kähler manifolds with conical singularities. To be precise, let \((M^n,g)\) be a Kähler manifold with conical singularities, and \(\zeta_q (s)\) the spectral zeta function of the Laplacian acting on \((0,q)\)-forms. We consider its Friedrichs extension. Then, under some assumptions on curvature, we prove the following formula: \[ \sum^n_{q = 0} (- 1)^q q \cdot \text{Res}_{s = 0} \zeta_q (s) = 0. \] As a corollary, the Ray-Singer analytic torsion is defined by the usual manner for Kähler manifolds with conical singularities. To prove the formula, we use the theory of the Quillen metrics developped by Bismut-Gillet-Soulé, and the theory of manifolds with conical singularities developed by Cheeger.
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 32S65
zbMATH DE Number: 895030
zbMATH Keywords: spectral zeta functions
zbMATH Keywords: Kähler manifold with conical singularities
zbMATH Keywords: analytic torsion
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214458112
OpenAlex ID: W1576482330
Wikidata QID: Q115177614
DOI: 10.4310/JDG/1214458112
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description: scientific article; zbMATH DE number 895030