Michel Brion

Residue formulae, vector partition functions and lattice points in rational polytopes

We obtain residue formulae for certain functions of several vari- ables. As an application, we obtain closed formulae for vector partition func- tions and for their continuous analogs. They imply an Euler-MacLaurin sum- mation formula for vector partition functions, and for rational convex poly- topes as well: we express the sum of values of a polynomial function at all lattice points of a rational convex polytope in terms of the variation of the integral of the function over the deformed polytope. Institut Fourier, B.P. 74, 38402 Saint-Martin d’Heres Cedex, France E-mail address: mbrion@fourier.ujf-grenoble.fr Ecole Normale Superieure, 45 rue d’Ulm, 75005 Paris Cedex 05, France E-mail address: vergne@dmi.ens.fr

Equivariant Chow groups for torus actions

We study Edidin and Grahams equivariant Chow groups in the case of torus actions. Our main results are: (i) a presentation of equivariant Chow groups in terms of invariant cycles, which shows how to recover usual Chow groups from equivariant ones; (ii) a precise form of the localization theorem for torus actions on projective, nonsingular varieties; (iii) a construction of equivariant multiplicities, as functionals on equivariant Chow groups; (iv) a construction of the action of operators of divided differences on theT-equivariant Chow group of any scheme with an action of a reductive group with maximal torusT. We apply these results to intersection theory on varieties with group actions, especially to Schubert calculus and its generalizations. In particular, we obtain a presentation of the Chow ring of any smooth, projective spherical variety.

Equivariant cohomology and equivariant intersection theory

This text is an introduction to equivariant cohomology, a classical tool for topological transformation groups, and to equivariant intersection theory, a much more recent topic initiated by D. Edidin and W. Graham.

Toric degenerations of spherical varieties

Abstract.We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by mirror symmetry, we give conditions for the limit toric variety to be a Gorenstein Fano, and provide many examples. We also provide an explanation for the limits as boundary points of the moduli space of stable pairs whose existence is predicted by the Minimal Model Program.

Arrangement of hyperplanes. I : Rational functions and Jeffrey-Kirwan residue

Consider the space RΔ of rational functions of several variables with poles on a fixed arrangement Δ of hyperplanes. We obtain a decomposition of RΔ as a module over the ring of differential operators with constant coefficients. We generalize the notions of principal part and of residue to the space RΔ, and we describe their relations to Laplace transforms of locally polynomial functions. This explains algebraic aspects of the work by L. Jeffrey and F. Kirwan about integrals of equivariant cohomology classes on Hamiltonian manifolds. As another application, we will construct multidimensional versions of Eisenstein series in a subsequent article, and we will obtain another proof of a residue formula of A. Szenes for Witten zeta functions.

Positivity in the Grothendieck group of complex flag varieties

Abstract We prove a conjecture of A.S. Buch concerning the structure constants of the Grothendieck ring of a flag variety with respect to its basis of Schubert structure sheaves. For this, we show that the coefficients in this basis of the structure sheaf of any subvariety with rational singularities have alternating signs. Equivalently, the class of the dualizing sheaf of such a subvariety is a nonnegative combination of classes of dualizing sheaves of Schubert varieties.

Moduli of affine schemes with reductive group action

For a connected reductive group G and a finite-dimensional G-module V, we study the invariant Hilbert scheme that parameterizes closed G-stable subschemes of V affording a fixed, multiplicity-finite representation of G in their coordinate ring. We construct an action on this invariant Hilbert scheme of a maximal torus T of G, together with an open T-stable subscheme admitting a good quotient. The fibers of the quotient map classify affine G-schemes having a prescribed categorical quotient by a maximal unipotent subgroup of G. We show that V contains only finitely many multiplicity-free G-subvarieties, up to the action of the centralizer of G in GL(V). As a consequence, there are only finitely many isomorphism classes of affine G-varieties affording a prescribed multiplicity-free representation in their coordinate ring. Final version, to appear in Journal of Algebraic Geometry

On orbit closures of spherical subgroups in flag varieties

Abstract. Let

Generic singularities of certain Schubert varieties

\cal F

Large Schubert varieties

be the flag variety of a complex semi-simple group G, let H be an algebraic subgroup of G acting on