The Riemann-Roch theorem for manifolds with conical singularities
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)].
manifolds with isolated conical singularities
scientific article; zbMATH DE number 1455448