On the residue of the spectral zeta functions of Kähler manifolds with conical singularities
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.
Kähler manifold with conical singularities
scientific article; zbMATH DE number 895030