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On nearly smooth complex spaces

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Abstract

We introduce a class of normal complex spaces having only mild singularities (close to quotient singularities) for which we generalize the notion of a (analytic) fundamental class for an analytic cycle and also the notion of a relative fundamental class for an analytic family of cycles. We also generalize to these spaces the geometric intersection theory for analytic cycles with rational positive coefficients and show that it behaves well with respect to analytic families of cycles. We prove that this intersection theory has most of the usual properties of the standard geometric intersection theory on complex manifolds, but with the exception that the intersection cycle of two cycles with positive integral coefficients that intersect properly may have rational coefficients.

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Notes

  1. This means that the space is locally isomorphic to a quotient of a complex manifold by a finite group of automorphisms.

  2. For the notion of a current on a reduced complex space see for example [4].

  3. See theorem IV.3.3.1 in [4]; “very general” means outside a countable union of locally closed analytic subsets with no interior point.

  4. Holomorphic simply means that the family of cycles \((\varphi (y))_{y \in N}\) is analytic; see [4] ch.IV for a definition.

  5. See [4] chIV §2.1 or [6] §2.

  6. The space \({\tilde{Z}}\) is pure dimensional, and the map \(\pi \) is proper, finite and surjective but not geometrically flat in general.

  7. This open set is constructed by removing finitely many times a nowhere dense analytic subset in the previous open dense subset. A section of the sheaf \(\underline{H}_{Y}^{p}(\omega _{M}^{p})\) is determined by its restriction to an open set of this kind.

  8. This reduces to show that the natural map \((\mathcal {O}_{S,s}\hat{\otimes }_{{{\mathrm{\mathbb {C}}}}}\mathcal {O}_{M,q(x)})\otimes _{\mathcal {O}_{M,q(x)}}\mathcal {O}_{{\tilde{M}},x} \longrightarrow \mathcal {O}_{S,s}\hat{\otimes }_{{{\mathrm{\mathbb {C}}}}}\mathcal {O}_{{\tilde{M}},x}\) is an isomorphism for every (s, x) in \(S\times {\tilde{M}}\), which is straightforward. Here \(\hat{\otimes }_{{{\mathrm{\mathbb {C}}}}}\) denotes the completed tensor product.

  9. They will be found in the book [5].

  10. Note that, as N is normal, this just means that f is equidimensional.

References

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Acknowledgements

The authors would like to thank the referee for comments that lead to improvements of the paper.

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Authors and Affiliations

  1. Institut Elie Cartan, Géomètrie, Université de Lorraine, CNRS UMR 7502 and Institut Universitaire de France, Nancy, France

    Daniel Barlet

  2. Department of Mathematics, School of Engineering and Physical Sciences, University of Iceland, Reykjavík, Iceland

    Jón Magnússon

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  1. Daniel Barlet
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  2. Jón Magnússon
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Correspondence to Jón Magnússon.

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Barlet, D., Magnússon, J. On nearly smooth complex spaces. Math. Z. 292, 317–341 (2019). https://doi.org/10.1007/s00209-018-2202-2

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  • DOI: https://doi.org/10.1007/s00209-018-2202-2

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