[Cohomologie évanescente et nombres de Betti pour les hypersurfaces projectives]
Nous utilisons le formalisme des cycles évanescents et des faisceaux pervers pour introduire et étudier la cohomologie évanescente des hypersurfaces projectives. Nous déduisons des majorants pour les nombres de Betti des hypersurfaces projectives, en généralisant ceux obtenus avec des méthodes différentes par Dimca dans le cas des singularités isolées, et par Siersma–Tibăr dans le cas des hypersurfaces avec lieu singulier de dimension 1. Nous prouvons aussi un complément au théorème de la section hyperplane de Lefschetz pour les hypersurfaces qui tient compte de la dimension du lieu singulier, et nous l’utilisons pour donner une nouvelle preuve du résultat de Kato.
We employ the formalism of vanishing cycles and perverse sheaves to introduce and study the vanishing cohomology of complex projective hypersurfaces. As a consequence, we give upper bounds for the Betti numbers of projective hypersurfaces, generalizing those obtained by different methods by Dimca in the isolated singularities case, and by Siersma–Tibăr in the case of hypersurfaces with a 1-dimensional singular locus. We also prove a supplement to the Lefschetz hyperplane theorem for hypersurfaces, which takes the dimension of the singular locus into account, and we use it to give a new proof of a result of Kato.
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Keywords: singular projective hypersurface, vanishing cycles, vanishing cohomology, Betti numbers, Milnor fiber, Lefschetz hyperplane theorem
Mot clés : hypersurfaces projectives singulières, cycles évanescents, cohomologie évanescente, nombre de Betti, fibre de Milnor, théorème de Lefschetz
CC-BY-ND 4.0
@article{AIF_2022__72_4_1705_0,
author = {Maxim, Lauren\c{t}iu G. and P\u{a}unescu, Lauren\c{t}iu and Tib\u{a}r, Mihai},
title = {Vanishing cohomology and {Betti} bounds for complex projective hypersurfaces},
journal = {Annales de l'Institut Fourier},
pages = {1705--1731},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {72},
number = {4},
year = {2022},
doi = {10.5802/aif.3486},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3486/}
}
TY - JOUR AU - Maxim, Laurenţiu G. AU - Păunescu, Laurenţiu AU - Tibăr, Mihai TI - Vanishing cohomology and Betti bounds for complex projective hypersurfaces JO - Annales de l'Institut Fourier PY - 2022 SP - 1705 EP - 1731 VL - 72 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3486/ DO - 10.5802/aif.3486 LA - en ID - AIF_2022__72_4_1705_0 ER -
%0 Journal Article %A Maxim, Laurenţiu G. %A Păunescu, Laurenţiu %A Tibăr, Mihai %T Vanishing cohomology and Betti bounds for complex projective hypersurfaces %J Annales de l'Institut Fourier %D 2022 %P 1705-1731 %V 72 %N 4 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3486/ %R 10.5802/aif.3486 %G en %F AIF_2022__72_4_1705_0
Maxim, Laurenţiu G.; Păunescu, Laurenţiu; Tibăr, Mihai. Vanishing cohomology and Betti bounds for complex projective hypersurfaces. Annales de l'Institut Fourier, Tome 72 (2022) no. 4, pp. 1705-1731. doi : 10.5802/aif.3486. https://aif.centre-mersenne.org/articles/10.5802/aif.3486/
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