[Une remarque sur les cycles évanescents à deux strates]
Supposons que le lieu critique Σ dʼune fonction analytique complexe f sur un espace affine soit un espace avec un point singulier isolé à lʼorigine 0, et que le nombre de Milnor de la fonction f restreinte à des sections transverses à soit constant. Alors, la théorie générale des faisceaux pervers impose des conditions strictes sur la cohomologie de la fibre de Milnor de f en 0 et, de façon encore plus surprenante, des restrictions sur la cohomologie de la fibre de Milnor dʼune section hyperplane générique.
Suppose that the critical locus Σ of a complex analytic function f on affine space is, itself, a space with an isolated singular point at the origin 0, and that the Milnor number of f restricted to normal slices of is constant. Then, the general theory of perverse sheaves puts severe restrictions on the cohomology of the Milnor fiber of f at 0, and even more surprising restrictions on the cohomology of the Milnor fiber of generic hyperplane slices.
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@article{CRMATH_2012__350_3-4_217_0,
author = {L\^e, D\~{u}ng Tr\'ang and Massey, David B.},
title = {A remark on vanishing cycles with two strata},
journal = {Comptes Rendus. Math\'ematique},
pages = {217--220},
publisher = {Elsevier},
volume = {350},
number = {3-4},
year = {2012},
doi = {10.1016/j.crma.2012.01.008},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2012.01.008/}
}
TY - JOUR AU - Lê, Dũng Tráng AU - Massey, David B. TI - A remark on vanishing cycles with two strata JO - Comptes Rendus. Mathématique PY - 2012 SP - 217 EP - 220 VL - 350 IS - 3-4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2012.01.008/ DO - 10.1016/j.crma.2012.01.008 LA - en ID - CRMATH_2012__350_3-4_217_0 ER -
%0 Journal Article %A Lê, Dũng Tráng %A Massey, David B. %T A remark on vanishing cycles with two strata %J Comptes Rendus. Mathématique %D 2012 %P 217-220 %V 350 %N 3-4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2012.01.008/ %R 10.1016/j.crma.2012.01.008 %G en %F CRMATH_2012__350_3-4_217_0
Lê, Dũng Tráng; Massey, David B. A remark on vanishing cycles with two strata. Comptes Rendus. Mathématique, Tome 350 (2012) no. 3-4, pp. 217-220. doi : 10.1016/j.crma.2012.01.008. http://www.numdam.org/articles/10.1016/j.crma.2012.01.008/
[1] Faisceaux pervers, Astérisque, Volume 100 (1982) (Société Mathématique de France, Paris)
[2] Remarque sur les cycles évanouissants dʼune hypersurface analytique à singularité non isolée, C. R. Acad. Sci. Paris, Ser. A–B, Volume 277 (1973), p. A599-A600
[3] Sheaves in Topology, Springer-Verlag, 2004
[4] On the connectivity of the Milnor fiber of a holomorphic function at a critical point, Tokyo, 1973, Univ. Tokyo Press, Tokyo (1975), pp. 131-136
[5] Calcul du nombre de cycles évanouissants dʼune hypersurface complexe, Ann. Inst. Fourier (Grenoble), Volume 23 (1973), pp. 261-270
[6] Une application dʼun théorème dʼAʼCampo à lʼéquisingularité, Nederl. Akad. Wetensch. Proc. Ser. A 76, Indag. Math., Volume 35 (1973), pp. 403-409
[7] Hypersurface singularities and Milnor equisingularity, Pure Appl. Math. Q., Volume 2 (2006) no. 3, pp. 893-914
[8] Isolated Singular Points on Complete Intersection, Cambridge Univ. Press, Cambridge, 1984
[9] Lê Cycles Hypersurface Singularities, Lecture Notes in Math., vol. 1615, Springer-Verlag, 1995
[10] Vanishing cycles and Thomʼs condition, Bull. Lond. Math. Soc., Volume 39 (2007) no. 4, pp. 591-602
[11] Singular Points of Complex Hypersurfaces, Ann. of Math. Stud., vol. 77, Princeton Univ. Press, 1968
[12] Singularities with critical locus an complete intersection and transversal type http://www.mpim-bonn.mpg.de/preblob/4141 (Preprint 2010-16, Max Planck Institute for Mathematics preprints)
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