no. 4
Vanishing cohomology and Betti bounds for complex projective hypersurfaces
[Cohomologie évanescente et nombres de Betti pour les hypersurfaces projectives]
Maxim, Laurenţiu G.1 ; Păunescu, Laurenţiu2 ; Tibăr, Mihai3
1 Department of Mathematics, University of Wisconsin-Madison 480 Lincoln Drive, Madison WI 53706-1388 (USA)
2 Department of Mathematics, University of Sydney, Sydney, NSW, 2006, (Australia)
3 Université de Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé, F-59000 Lille (France)
Annales de l'Institut Fourier, Tome 72 (2022) no. 4, pp. 1705-1731.

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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DOI : 10.5802/aif.3486
Classification : 32S30, 32S50, 55R55, 58K60
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

Maxim, Laurenţiu G. 1 ; Păunescu, Laurenţiu 2 ; Tibăr, Mihai 3

1 Department of Mathematics, University of Wisconsin-Madison 480 Lincoln Drive, Madison WI 53706-1388 (USA)
2 Department of Mathematics, University of Sydney, Sydney, NSW, 2006, (Australia)
3 Université de Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé, F-59000 Lille (France)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Vanishing cohomology and {Betti} bounds for complex projective hypersurfaces},
     journal = {Annales de l'Institut Fourier},
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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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