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\(L^2\)-Serre Duality on Singular Complex Spaces and Applications

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Complex Analysis and Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 144))

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Abstract

In this survey, we explain a version of topological \(L^2\)-Serre duality for singular complex spaces with arbitrary singularities. This duality can be used to deduce various \(L^2\)-vanishing theorems for the \(\overline{\partial }\)-equation on singular spaces. As one application, we prove Hartogs’ extension theorem for \((n-1)\)-complete spaces. Another application is the characterization of rational singularities. It is shown that complex spaces with rational singularities behave quite tame with respect to some \(\overline{\partial }\)-equation in the \(L^2\)-sense. More precisely: a singular point is rational if and only if the appropriate \(L^2\)-\(\overline{\partial }\)-complex is exact in this point. So, we obtain an \(L^2\)-\(\overline{\partial }\)-resolution of the structure sheaf in rational singular points.

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Notes

  1. 1.

    The interest in this setting goes back to the invention of intersection (co-)homology by Goresky and MacPherson which has very tight connections to the \(L^2\)-deRham cohomology of the regular part of a singular variety. We refer here to the solution of the Cheeger-Goresky-MacPherson conjecture [CGM] for varieties with isolated singularities by Ohsawa [O] (see [PS] for more details).

  2. 2.

    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 \(g|_U\).

  3. 3.

    This is what we mean by square-integrable up to the singular set.

  4. 4.

    Note that different Hermitian metrics lead to \(\overline{\partial }\)-complexes which are equivalent on relatively compact subsets. So, one can put any Hermitian metric on X in many of the results below.

  5. 5.

    The notation w / s refers either to the index w or the index s in the whole statement.

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Acknowledgments

The author was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), grant RU 1474/2 within DFG’s Emmy Noether Programme.

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Authors and Affiliations

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

    J. Ruppenthal

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  1. J. Ruppenthal
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Correspondence to J. Ruppenthal .

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Editors and Affiliations

  1. Department of Mathematics, Università di Roma Tor Vergata, Rome, Italy

    Filippo Bracci

  2. Dept. of Mathematics Education, Kyungnam University, Changwon, Korea (Republic of)

    Jisoo Byun

  3. Université Joseph Fourier Grenoble, Institut Fourier, Grenoble, France

    Hervé Gaussier

  4. Graduate School of Mathematical Sciences, The University of Tokyo, Meguro-ku, Tokyo, Japan

    Kengo Hirachi

  5. Department of Mathematics, POSTECH, Pohang, Korea (Republic of)

    Kang-Tae Kim

  6. Department of Mathematics, Bergische Universität Wuppertal, Wuppertal, Nordrhein-Westfalen, Germany

    Nikolay Shcherbina

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Ruppenthal, J. (2015). \(L^2\)-Serre Duality on Singular Complex Spaces and Applications. In: Bracci, F., Byun, J., Gaussier, H., Hirachi, K., Kim, KT., Shcherbina, N. (eds) Complex Analysis and Geometry. Springer Proceedings in Mathematics & Statistics, vol 144. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55744-9_23

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