arXiv subjects

Jean-Paul Brasselet

Publications and source records attributed to Jean-Paul Brasselet.

22 records · Page 2Linked to original sources

arXiv2000

Combinatorial Intersection Cohomology for Fans

We continue the approach toward a purely combinatorial "virtual" intersection cohomology for possibly non-rational fans, based on our investigation of equivariant intersection cohomology for toric varieties (see math.AG/9904159). Fundamental objects of study are "minimal extension sheaves" on "fan spaces". These are flabby sheaves of graded modules over a sheaf of polynomial rings, satisfying three relatively simple axioms that characterize the properties of the equivariant intersection cohomology sheaf on a toric variety, endowed with the finite topology given by open invariant subsets. These sheaves are models for the "pure" objects of a "perverse category"; a "Decomposition Theorem" is shown to hold. -- Formalizing those fans that define "equivariantly formal" toric varieties (where equivariant and non-equivariant intersection cohomology determine each other by Kunneth type formulae), we study "quasi-convex" fans (including fans with convex or with "co-convex" support). For these, there is a meaningful "virtual intersection cohomology". We characterize quasi-convex fans by a topological condition on the support of their boundary fan and prove a generalization of Stanley's "Local-Global" formula realizing the intersection Poincare polynomial of a complete toric variety in terms of local data. Virtual intersection cohomology of quasi-convex fans is shown to satify Poincare duality. To describe the local data in terms of virtual intersection cohomology of lower-dimensional complete polytopal fans, one needs a "Hard Lefschetz" type theorem. It requires a vanishing condition that is known to hold for rational cones, but yet remains to be proven in the general case.

math.AG↗

arXiv1999

Equivariant Intersection Cohomology of Toric Varieties

We investigate the equivariant intersection cohomology of a toric variety. Considering the defining fan of the variety as a finite topological space with the subfans being the open sets (that corresponds to the "toric" topology given by the invariant open subsets), equivariant intersection cohomology provides a sheaf (of graded modules over a sheaf of graded rings) on that "fan space". We prove that this sheaf is a "minimal extension sheaf", i.e., that it satisfies three relatively simple axioms which are known to characterize such a sheaf up to isomorphism. In the verification of the second of these axioms, a key role is played by "equivariantly formal" toric varieties, where equivariant and "usual" (non-equivariant) intersection cohomology determine each other by Kunneth type formulae. Minimal extension sheaves can be constructed in a purely formal way and thus also exist for non-rational fans. As a consequence, we can extend the notion of an equivariantly formal fan even to this general setup. In this way, it will be possible to introduce "virtual" intersection cohomology for equivariantly formal non-rational fans.

math.AG↗

arXiv1997

A canonical lift of Chern-Mather classes

There are several ways to generalize characteristic classes for singular algebraic varieties. The simplest ones to describe are Chern-Mather classes obtained by Nash blow up. They serve as an ingredient to construct Chern-MacPherson-Schwartz classes. Unfortunately, they all are defined in homology. There are examples showing, that they do not lie in the image of Poincaré morphism. On the other hand they are represented by an algebraic cycles. Barthel, Brasselet, Fiesler, Kaup and Gabber have shown that, any algebraic cycle can be lifted to intersection homology. Nevertheless, a lift is not unique. The Chern-Mather classes are represented by polar varieties. We show how to define a canonical lift of Chern-Mather classes to intersection homology. Instead of the polar variety alone, we consider it as a term in the whole sequence of inclusions of polar varieties. The inclusions are of codimension one. In this case the lifts are unique.

alg-geom↗

arXiv1996

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.

alg-geom↗