[Quelques conséquences de la perversité des cycles évanescents]
Pour une fonction holomorphe sur une variété lisse complexe, nous montrons que l'annulation de la cohomologie en bas degré en un point est déterminée par celle aux points voisins, via la perversité du complexe des cycles évanescents. Nous calculons explicitement cet ordre d'annulation lorsque les singularités voisines sont à croisements normaux. Nous en déduisons des résultats sur la taille des blocs de Jordan de la monodromie.
For a holomorphic function on a complex manifold, we show that the vanishing cohomology of lower degree at a point is determined by that for the points near it, using the perversity of the vanishing cycle complex. We calculate this order of vanishing explicitly in the case the hypersurface has simple normal crossings outside the point. We also give some applications to the size of Jordan blocks for monodromy.
Keywords: Milnor fibration, perverse sheaf, vanishing cycles
Mot clés : fibration de Milnor, faisceau pervers, cycles évanescents
@article{AIF_2004__54_6_1769_0,
author = {Dimca, Alexandru and Saito, Morihiko},
title = {Some consequences of perversity of vanishing cycles},
journal = {Annales de l'Institut Fourier},
pages = {1769--1792},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {54},
number = {6},
year = {2004},
doi = {10.5802/aif.2065},
mrnumber = {2134223},
zbl = {1070.14011},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/aif.2065/}
}
TY - JOUR AU - Dimca, Alexandru AU - Saito, Morihiko TI - Some consequences of perversity of vanishing cycles JO - Annales de l'Institut Fourier PY - 2004 SP - 1769 EP - 1792 VL - 54 IS - 6 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2065/ DO - 10.5802/aif.2065 LA - en ID - AIF_2004__54_6_1769_0 ER -
%0 Journal Article %A Dimca, Alexandru %A Saito, Morihiko %T Some consequences of perversity of vanishing cycles %J Annales de l'Institut Fourier %D 2004 %P 1769-1792 %V 54 %N 6 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2065/ %R 10.5802/aif.2065 %G en %F AIF_2004__54_6_1769_0
Dimca, Alexandru; Saito, Morihiko. Some consequences of perversity of vanishing cycles. Annales de l'Institut Fourier, Tome 54 (2004) no. 6, pp. 1769-1792. doi : 10.5802/aif.2065. http://archive.numdam.org/articles/10.5802/aif.2065/
[1] Interaction de strates consécutives pour les cycles évanescents, Ann. Sci. École Norm. Sup. (4), Volume 24 (1991), pp. 401-505 | Numdam | MR | Zbl
[2] Faisceaux pervers, Astérisque, 100, Soc. Math. France, Paris, 1982 | MR | Zbl
[3] Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques, Géométrie et analyse microlocales (Astérisque), Volume 140-141 (1986), pp. 3-134 | Numdam | Zbl
[4] Multiplier ideals, -filtration, and spectrum (e-print, math.AG/0305118) | Zbl
[5] On Milnor fibrations of arrangements, J. London Math. Soc., Volume 51 (1995), pp. 105-119 | MR | Zbl
[6] Théorie de Hodge I, Actes Congrès Intern. Math., Volume 1 (1970), pp. 425-430 | MR | Zbl
[6] Théorie de Hodge II, Publ. Math. IHES, Volume 40 (1971), pp. 5-57 | Numdam | MR | Zbl
[6] Théorie de Hodge III (ibid), Publ. Math. IHES, Volume 44 (1974), pp. 5-77 | Numdam | MR | Zbl
[7] Le formalisme des cycles évanescents, SGA7 (Lect. Notes in Math. 340), Volume Exposé XIII (1973), pp. 82-115 | Zbl
[7] Le formalisme des cycles évanescents, SGA7 (Lect. Notes in Math. 340), Volume Exposé XIV (1973), pp. 116-164 | Zbl
[8] Sheaves in Topology, Universitext, Springer, Berlin, 2004 | MR | Zbl
[9] Local topology of reducible divisors (e-print, math.AG/0303215)
[10] Monodromy at infinity and the weights of cohomology, Compos. Math., Volume 138 (2003), pp. 55-71 | MR | Zbl
[11] Intersection theory, Springer, Berlin, 1984 | MR | Zbl
[12] Vanishing cycle sheaves and holonomic systems of differential equations, Algebraic geometry (Tokyo/Kyoto, 1982) (Lect. Notes in Math.), Volume 1016 (1983), pp. 134-142 | Zbl
[13] Polynôme de Bernstein-Sato et cohomologie évanescente, Analysis and topology on singular spaces, II-III (Luminy, 1981) (Astérisque), Volume 101-102 (1983), pp. 243-267 | Numdam | Zbl
[14] Sur la théorie de Hodge-Deligne, Inv. Math., Volume 90 (1987), pp. 11-76 | MR | Zbl
[15] The Milnor fiber of a generic arrangement, Ark. Mat., Volume 31 (1993), pp. 71-81 | MR | Zbl
[16] Modules de Hodge polarisables, Publ. RIMS, Kyoto Univ., Volume 24 (1988), pp. 849-995 | MR | Zbl
[17] Mixed Hodge Modules, Publ. RIMS, Kyoto Univ., Volume 26 (1990), pp. 221-333 | MR | Zbl
[18] Variation mappings on singularities with a 1-dimensional critical locus, Topology, Volume 30 (1991), pp. 445-469 | MR | Zbl
[19] The vanishing topology of non isolated singularities, New Developments in Singularity Theory (2001), pp. 447-472 | Zbl
[20] Mixed Hodge structure on the vanishing cohomology, Real and Complex Singularities (Proc. Nordic Summer School, Oslo, 1976) (1977), pp. 525-563 | Zbl
Cité par Sources :
