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Intersection cohomology monodromy and the Milnor fiber. (English) Zbl 1183.32009

A complex analytic space \( X\) is said to be an intersection cohomology manifold if the shifted constant sheaf on \( X\) is isomorphic to the intersection cohomology. Given an analytic function \( f\) on an intersection cohomology manifold, the author describes a simple relation between \( V(f)\) being an intersection cohomology manifold and the vanishing cycle Milnor monodromy of \( f\). He then describes how the Sebastiani-Thom isomorphism allows one to produce intersection cohomology manifolds with arbitrary singular sets. Finally, he obtains restrictions on the cohomology of the Milnor fiber of a hypersurface with a special type of one-dimensional critical locus.

MSC:

32C35 Analytic sheaves and cohomology groups
32C18 Topology of analytic spaces
32B15 Analytic subsets of affine space
32B10 Germs of analytic sets, local parametrization
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[1] Barlet D., Ann. Sci. École Norm. Sup. (4) 24 pp 401–
[2] Beilinson A. A., Astérisque 100 pp 5–
[3] Borho W., Astérisque 101 pp 23–
[4] DOI: 10.1090/S0002-9904-1978-14527-3 · Zbl 0418.57005
[5] DOI: 10.1090/S0002-9947-1969-0233814-9
[6] DOI: 10.1016/0040-9383(80)90003-8 · Zbl 0448.55004
[7] Goresky M., Invent. Math. 71 pp 77–
[8] Goresky M., Astérisque 101 pp 135–
[9] Grothendieck A., Lecture Notes in Mathematics 288, in: Séminaire de Géométrie Algébrique (SGA VII-1) (1972) · Zbl 0234.00007
[10] Hamm H., Progress in Mathematics 87 (1990)
[11] Kashiwara M., Astérisque 128 pp 235–
[12] DOI: 10.1007/978-3-662-02661-8
[13] DOI: 10.1090/S0002-9947-1973-0344248-1
[14] Lê D. T., Ann. Inst. Fourier (Grenoble) 23 pp 261–
[15] Lê D. T., Indag. Math. 35 pp 403–
[16] D. T. Lê, C. P. Ramanujam – A Tribute, Tata Institute of Fundametal Research Studies in Mathematics 8 (Springer-Verlag, 1978) pp. 157–173.
[17] Lê D. T., Pure Appl. Math. Quart. 2 pp 893–
[18] DOI: 10.1023/A:1002608716514 · Zbl 0986.32004
[19] Massey D., Mem. Amer. Math. Soc. 778
[20] Milnor J., Annals of Mathematics Studies, in: Singular Points of Complex Hypersurfaces (1968) · Zbl 0184.48405
[21] Némethi A., Compositio Math. 80 pp 1–
[22] DOI: 10.2969/jmsj/04320213 · Zbl 0739.32034
[23] DOI: 10.1016/0040-9383(73)90019-0 · Zbl 0263.32005
[24] DOI: 10.2969/jmsj/02640714 · Zbl 0286.32010
[25] Schürmann J., Monografie Matematyczne 63, in: Topology of Singular Spaces and Constructible Sheaves (2004)
[26] DOI: 10.1007/BF01390095 · Zbl 0233.32025
[27] DOI: 10.1016/0040-9383(91)90025-Y · Zbl 0746.32014
[28] DOI: 10.2307/1970239 · Zbl 0099.39202
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