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Caractérisation des variétés homologiques à l'aide des invariants d'homologie d'intersection

Jean-Paul Brasselet, Karl-Heinz Fieseler, Ludger Kaup (1996)

Banach Center Publications

La conférence de J. P. Brasselet au Symposium de Varsovie a eu pour thème les problèmes actuels de l’homologie d’intersection. Nous en présentons ici l’un des aspects, résultat d’un travail commun réalisé dans le cadre du programme Procope et pendant lequel le second auteur a été chercheur associé au CNRS.

Decomposition numbers for perverse sheaves

Daniel Juteau (2009)

Annales de l’institut Fourier

The purpose of this article is to set foundations for decomposition numbers of perverse sheaves, to give some methods to calculate them in simple cases, and to compute them concretely in two situations: for a simple (Kleinian) surface singularity, and for the closure of the minimal non-trivial nilpotent orbit in a simple Lie algebra.This work has applications to modular representation theory, for Weyl groups using the nilpotent cone of the corresponding semisimple Lie algebra, and for reductive...

Intersection cohomology of reductive varieties

Roy Joshua, Michel Brion (2004)

Journal of the European Mathematical Society

We extend the methods developed in our earlier work to algorithmically compute the intersection cohomology Betti numbers of reductive varieties. These form a class of highly symmetric varieties that includes equivariant compactifications of reductive groups. Thereby, we extend a well-known algorithm for toric varieties.

On the geometry of polynomial mappings at infinity

Anna Valette, Guillaume Valette (2014)

Annales de l’institut Fourier

We associate to a given polynomial map from ℂ 2 to itself with nonvanishing Jacobian a variety whose homology or intersection homology describes the geometry of singularities at infinity of this map.

Parity sheaves, moment graphs and the p -smooth locus of Schubert varieties

Peter Fiebig, Geordie Williamson (2014)

Annales de l’institut Fourier

We show that the Braden-MacPherson algorithm computes the stalks of parity sheaves. As a consequence we deduce that the Braden-MacPherson algorithm may be used to calculate the characters of tilting modules for algebraic groups and show that the p -smooth locus of a (Kac-Moody) Schubert variety coincides with the rationally smooth locus, if the underlying Bruhat graph satisfies a GKM-condition.

Rational BV-algebra in string topology

Yves Félix, Jean-Claude Thomas (2008)

Bulletin de la Société Mathématique de France

Let M be a 1-connected closed manifold of dimension m and L M be the space of free loops on M . M.Chas and D.Sullivan defined a structure of BV-algebra on the singular homology of L M , H * ( L M ; k ) . When the ring of coefficients is a field of characteristic zero, we prove that there exists a BV-algebra structure on the Hochschild cohomology H H * ( C * ( M ) ; C * ( M ) ) which extends the canonical structure of Gerstenhaber algebra. We construct then an isomorphism of BV-algebras between H H * ( C * ( M ) ; C * ( M ) ) and the shifted homology H * + m ( L M ; k ) . We also prove that the...

Rational string topology

Yves Félix, Jean-Claude Thomas, Micheline Vigué-Poirrier (2007)

Journal of the European Mathematical Society

We use the computational power of rational homotopy theory to provide an explicit cochain model for the loop product and the string bracket of a simply connected closed manifold M . We prove that the loop homology of M is isomorphic to the Hochschild cohomology of the cochain algebra C * ( M ) with coefficients in C * ( M ) . Some explicit computations of the loop product and the string bracket are given.

Weights in cohomology and the Eilenberg-Moore spectral sequence

Matthias Franz, Andrzej Weber (2005)

Annales de l’institut Fourier

We show that in the category of complex algebraic varieties, the Eilenberg–Moore spectral sequence can be endowed with a weight filtration. This implies that it degenerates if all spaces involved have pure cohomology. As application, we compute the rational cohomology of an algebraic G -variety X ( G being a connected algebraic group) in terms of its equivariant cohomology provided that H G * ( X ) is pure. This is the case, for example, if X is smooth and has only finitely many orbits. We work in the category...

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