Jean-Paul Brasselet

Vector fields on singular varieties

The Case of Manifolds.- The Schwartz Index.- The GSV Index.- Indices of Vector Fields on Real Analytic Varieties.- The Virtual Index.- The Case of Holomorphic Vector Fields.- The Homological Index and Algebraic Formulas.- The Local Euler Obstruction.- Indices for 1-Forms.- The Schwartz Classes.- The Virtual Classes.- Milnor Number and Milnor Classes.- Characteristic Classes of Coherent Sheaves on Singular Varieties.

Hodge-Riemann Relations for Polytopes: A Geometric Approach

The key to the Hard Lefschetz Theorem for combinatorial intersection cohomology of polytopes is to prove the Hodge-Riemann bilinear relations. In these notes, we strive to present an easily accessible proof. The strategy essentially follows the original approach of [Ka], applying induction a la [BreLu2], but our guiding principle here is to emphasize the geometry behind the algebraic arguments by consequently stressing polytopes rather than fans endowed with a strictly convex conewise linear function. It is our belief that this approach makes the exposition more transparent since polytopes are more appealing to our geometric intuition than convex functions on a fan.

Interpolation of characteristic classes of singular hypersurfaces

Abstract We show that the Chern–Schwartz–MacPherson class of a hypersurface X in a nonsingular variety M ‘interpolates’ between two other notions of characteristic classes for singular varieties, provided that the singular locus of X is smooth and that certain numerical invariants of X are constant along this locus. This allows us to define a lift of the Chern–Schwartz–MacPherson class of such ‘nice’ hypersurfaces to intersection homology. As another application, the interpolation result leads to an explicit formula for the Chern–Schwartz–MacPherson class of X in terms of its polar classes.

Singularity Theory: Differential Forms on Singular Varieties and Cyclic Homology

A classical result of A. Connes asserts that the Frechet algebra of smooth functions on a smooth compact manifold X provides, by a purely algebraic procedure, the de Rham cohomology of X. Namely the procedure uses Hochschild and cyclic homology of this algebra. In the situation of a Thom-Mather stratified variety, we construct a Frechet algebra of functions on the regular part and a module of poles along the singular part. We associate to these objects a complex of differential forms and an Hochschild complex, on the regular part, both with poles along the singular part. The de Rham cohomology of the first complex and the cylic homology of the second one are related to the intersection homology of the variety, the corresponding perversity is determined by the orders of poles.

Invariante Divisoren und Schnitthomologie von torischen Varietäten

In this article, we complete the interpretation of groups of classes of invariant divisors on a complex toric variety X of dimension n in terms of suitable (co-) homology groups. In [BBFK], we proved the following result (see Satz 1 below): Let Cl DivC(X) and Cl Div T W(X) denote the groups of classes of invariant Cartier resp. Weil divisors on X. If X is non degenerate (i.e., not equivariantly isomorphic to the product of a toric variety and a torus of positive dimension), then the natural homomorphisms Cl DivC(X) → H 2(X) and Cl DivW(X) → H cld 2n−2(X) are isomorphisms, the inclusion Cl DivC(X) ↪→ Cl Div T W(X) corresponds to the Poincare duality 1991 Mathematics Subject Classification: Primary 14M25, 14C20, 55N33; Secondary 52B20, 32M12. Unterstutzt durch das Programm ”PROCOPE“ zur Forderung der deutsch-franzosischen Kooperation in der Forschung. Avec le concours du programme «PROCOPE» pour l’avancement de la cooperation francoallemande dans la recherche scientifique. The paper is in final form and no version of it will be published elsewhere.

Residues for flags of holomorphic foliations

Abstract In this work we prove a Baum–Bott type residue theorem for flags of holomorphic foliations. We prove some relations between the residues of the flag and the residues of their correspondent foliations. We define the Nash residue for flags and we give a partial answer to the Baum–Bott type rationality conjecture in this context.

Motivic and derived motivic Hirzebruch classes

In this paper we give a formula for the Hirzebruch

Regular differential forms on stratified spaces

\chi_y

Teleman localization of Hochschild homology in a singular setting

-genus

Characteristic Classes of Coherent Sheaves on Singular Varieties

\chi_y(X)