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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Isosingular loci and the Cartesian product structure of complex analytic singularities

Author: Robert Ephraim
Journal: Trans. Amer. Math. Soc. 241 (1978), 357-371
MSC: Primary 32B10; Secondary 32C40
MathSciNet review: 492307
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Abstract: Let X be a (not necessarily reduced) complex analytic space, and let V be a germ of an analytic space. The locus of points q in X at which the germ $ {X_q}$ is complex analytically isomorphic to V is studied. If it is nonempty it is shown to be a locally closed submanifold of X, and X is locally a Cartesian product along this submanifold. This is used to define what amounts to a coarse partial ordering of singularities. This partial ordering is used to show that there is an essentially unique way to completely decompose an arbitrary reduced singularity as a cartesian product of lower dimensional singularities. This generalizes a result previously known only for irreducible singularities.

References [Enhancements On Off] (What's this?)

  • [1] J. Becker and R. Ephraim, Real and complex products of singularities (in preparation).
  • [2] R. Ephraim, The Cartesian product structure and $ {C^\infty }$ equivalences of singularities, Trans. Amer. Math. Soc. 224 (1976), 299-311. MR 0422676 (54:10662)
  • [3] G. Fischer, Complex analytic geometry, Lecture Notes in Math., vol. 538, Springer-Verlag, Berlin and New York, 1976. MR 0430286 (55:3291)
  • [4] D. Mumford, Introduction to algebraic geometry (preliminary version of the first three chapters), Lecture notes, Harvard Univ., Cambridge, Mass.
  • [5] A. Seidenberg, Analytic products, Amer. J. Math. 91 (1969), 577-590. MR 0254048 (40:7261)
  • [6] -, On analytically equivalent ideals, Inst. Hautes Etude Sci. Publ. Math. No. 36 (1969), 69-74. MR 0260726 (41:5350)
  • [7] J. Wavrik, A theorem on solutions of analytic equations with applications to deformations of complex structures, Math. Ann. 216 (1975), 127-142. MR 0387649 (52:8488)
  • [8] H. Whitney, Local properties of analytic varieties, Differential and Combinatorial Topology, Princeton Univ. Press, Princeton, N. J., 1965. MR 0188486 (32:5924)
  • [9] -, Complex analytic varieties, Addison-Wesley, Reading, Mass., 1972.

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Keywords: Nonreduced complex space, cartesian product, derivations, complex analytic isomorphism, reduced singularity
Article copyright: © Copyright 1978 American Mathematical Society

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