Tom Braden

Perverse Sheaves on Grassmannians

We compute the category of perversesheaves on Hermitian symmetric spaces in types A and D, constructible with respect to the Schubert stratification. The calculation is microlocal, and uses the action of the Borel group to study the geometry of the conormal variety�.

The hypertoric intersection cohomology ring

We present a functorial computation of the equivariant intersection cohomology of a hypertoric variety, and endow it with a natural ring structure. When the hyperplane arrangement associated with the hypertoric variety is unimodular, we show that this ring structure is induced by a ring structure on the equivariant intersection cohomology sheaf in the equivariant derived category. The computation is given in terms of a localization functor which takes equivariant sheaves on a sufficiently nice stratified space to sheaves on a poset.

On the reducibility of characteristic varieties

We show that some monodromies in the Morse local systems of a conically stratified perverse sheaf imply that other Morse local systems for smaller strata do not vanish. This result is then used to explain the examples of reducible characteristic varieties of Schubert varieties given by Kashiwara and Saito in type A and by Boe and Fu for the Lagrangian Grassmannian.

Koszul duality for toric varieties

We show that certain categories of perverse sheaves on affine toric varieties X σ and X σ v defined by dual cones are Koszul dual in the sense of Beilinson, Ginzburg and Soergel (1996). The functor expressing this duality is constructed explicitly by using a combinatorial model for mixed sheaves on toric varieties.

Matroidal Schur algebras

Motivated by a geometric description of the Schur algebra due to the second author, we define for any matroid M and principal ideal domain k, a quasi-hereditary algebra R(M) defined over k which we call a matroidal Schur algebra. We show that the Ringel dual of R(M) is the matroidal Schur algebra

From moment graphs to intersection cohomology

R(M^*)

Quantizations of conical symplectic resolutions II: category

R(M∗) associated with the dual matroid