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\(L^2\)-properties of the \(\overline{\partial }\) and the \(\overline{\partial }\)-Neumann operator on spaces with isolated singularities

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Abstract

Let \(X\) be a Hermitian complex space of pure dimension with only isolated singularities and \(\pi : M\rightarrow X\) a resolution of singularities. Let \(\Omega \subset \subset X\) be a domain with no singularities in the boundary, \(\Omega ^*=\Omega {\setminus }\!{{\mathrm{Sing}}}X\) and \(\Omega '=\pi ^{-1}(\Omega )\). We relate \(L^2\)-properties of the \(\overline{\partial }\) and the \(\overline{\partial }\)-Neumann operator on \(\Omega ^*\) to properties of the corresponding operators on \(\Omega '\) (where the situation is classically well understood). Outside some middle degrees, there are compact solution operators for the \(\overline{\partial }\)-equation on \(\Omega ^*\) exactly if there are such operators on the resolution \(\Omega '\), and the \(\overline{\partial }\)-Neumann operator is compact on \(\Omega ^*\) exactly if it is compact on \(\Omega '\).

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Notes

  1. A Hermitian complex space \((X,g)\) is a reduced complex space \(X\) with a metric \(g\) on the regular part such that the following holds: if \(x\in X\) is an arbitrary point there exists a neighborhood \(U=U(x)\) and a biholomorphic embedding of \(U\) into a domain \(G\) in \(\mathbb {C}^N\) and an ordinary smooth Hermitian metric in \(G\) whose restriction to \(U\) is \(g|_U\).

  2. When we talk about a resolution of singularities \(\pi : M\rightarrow X\), we require that the exceptional set \(E=\pi ^{-1}({{\mathrm{Sing}}}X)\) is a (reduced) divisor in \(M\).

  3. By a little abuse of notation, we write \(N=\Box ^{-1}\) for \(\overline{\partial }\)-Neumann operators though \(N\) is an inverse to the \(\overline{\partial }\)-Laplacian \(\Box =\overline{\partial }\overline{\partial }^*+\overline{\partial }^*\overline{\partial }\) only on the range \(\text{ Im } \Box \).

  4. As \(U'\) is str. pseudoconvex, we can also use Serre duality: \(H^n(U',\mathcal {O})\cong H^0_{cpt}(U',\Omega ^n)=0\).

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Acknowledgments

J. Ruppenthal was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), grant RU 1474/2 within DFG’s Emmy Noether Programme. The authors thank the unknown referee for several suggestions which helped to improve the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Oslo, P.B 1053, Blindern, 0316 , Oslo, Norway

    N. Øvrelid

  2. Department of Mathematics, University of Wuppertal, Gaußstr. 20, 42119 , Wuppertal, Germany

    J. Ruppenthal

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  1. N. Øvrelid
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Øvrelid, N., Ruppenthal, J. \(L^2\)-properties of the \(\overline{\partial }\) and the \(\overline{\partial }\)-Neumann operator on spaces with isolated singularities. Math. Ann. 359, 803–838 (2014). https://doi.org/10.1007/s00208-014-1016-8

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  • DOI: https://doi.org/10.1007/s00208-014-1016-8

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