The Riemann-Roch theorem for manifolds with conical singularities
type: Item
instance of: scholarly article
title: The Riemann-Roch theorem for manifolds with conical singularities
zbMATH Open document ID: 0964.58021
author: B.-W. Schulze
author: N. N. Tarkhanov
published in: Osaka Journal of Mathematics
publication date: 2001-07-08T00:00:00Z
review text: In [Adv. Sov. Math. 16, 211-241 (1993; Zbl 0802.58051)], \textit{M. Gromov} and \textit{M. A. Shubin} generalized the Riemann-Roch theorem to solutions with zeros and poles of prescribed orders of arbitrary elliptic operators on compact manifolds. In the present paper, the result of Gromov and Shubin is generalized to manifolds with isolated conical singularities. Let \(M\) be a manifold with a finite set of conical point singularities, let \(V\), \(V'\to M\) be vector bundles, let \(A\) be an elliptic operator of order \(a\) between weighted Sobolev spaces \(H^{\infty,\gamma}(M,V)\) and \(H^{\infty,a-\gamma}(M,V')\), and let \(A'\) be the transpose of \(A\). If \(\delta=p_1^{m_1}\cdots p_N^{m_N}\) is a point divisor on the regular part of \(M\), consider the spaces \(L(\delta,A)\subset H^{\infty,\gamma}_{loc} (M\setminus\{p_1,\dots,p_N\},V)\) and \(L(\delta^{-1},A')\subset H^{\infty,a-\gamma}_{loc} (M\setminus\{p_1,\dots,p_N\},V)\) of \(A\)-harmonic (\(A'\)-harmonic) generalized sections with singularities of prescribed order at \(p_1\), \dots, \(p_N\). Then the authors show that \[ \dim L(\delta,A) =\text{ind}A+\deg\delta+\dim L(\delta^{-1},A') , \] where \(\deg\delta\) is defined as in the paper of Gromov and Shubin. A formula for \(\text{ind}(A)\) for certain operators is given by \textit{B. Fedosov} and the authors in [Sel. Math., New Ser. 5, No. 4, 467-506 (1999; Zbl 0951.58026)].
reviewed by: Sebastian Goette
Mathematics Subject Classification ID: 58J20
Mathematics Subject Classification ID: 58J40
zbMATH DE Number: 1455448
zbMATH Keywords: Riemann-Roch theorem
zbMATH Keywords: manifolds with isolated conical singularities
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1455448