Vanishing cohomology and Betti bounds for complex projective hypersurfaces

Abstract

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.

arXiv commentfinal version, to be published in Annales de l'Institut Fourier (Grenoble)

References in corpus (5)

• Characteristic Classes of Hypersurfaces and Characteristic Cycles
• Natural Commuting of Vanishing Cycles and the Verdier Dual
• The vanishing cohomology of non-isolated hypersurface singularities
• Vanishing homology of projective hypersurfaces with 1-dimensional singularities
• Milnor Fibre Homology via Deformation

Cited by in corpus (2)

• The vanishing cohomology of non-isolated hypersurface singularities
• Polar degree of hypersurfaces with 1-dimensional singularities