Isosingular loci and the Cartesian product structure of complex analytic singularities
Author:
Robert Ephraim
Journal:
Trans. Amer. Math. Soc. 241 (1978), 357371
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 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.
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 [1]
 J. Becker and R. Ephraim, Real and complex products of singularities (in preparation).
 [2]
 R. Ephraim, The Cartesian product structure and equivalences of singularities, Trans. Amer. Math. Soc. 224 (1976), 299311. MR 0422676 (54:10662)
 [3]
 G. Fischer, Complex analytic geometry, Lecture Notes in Math., vol. 538, SpringerVerlag, 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), 577590. MR 0254048 (40:7261)
 [6]
 , On analytically equivalent ideals, Inst. Hautes Etude Sci. Publ. Math. No. 36 (1969), 6974. 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), 127142. 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, AddisonWesley, Reading, Mass., 1972.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719780492307X
PII:
S 00029947(1978)0492307X
Keywords:
Nonreduced complex space,
cartesian product,
derivations,
complex analytic isomorphism,
reduced singularity
Article copyright:
© Copyright 1978
American Mathematical Society
