Publications

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.

We give a presentation for the (integral) torus-equivariant Chow ring of the quot scheme, a smooth compactification of the space of rational curves of degree d in the Grassmannian. For this presentation, we refine Evain's extension of the method of Goresky, Kottwitz, and MacPherson to express the torus-equivariant Chow ring in terms of the torus-fixed points and explicit relations coming from the geometry of families of torus-invariant curves. As part of this calculation, we give a complete description of the torus-invariant curves on the quot scheme and show that each family is a product of projective spaces.

Examples of torsion of intersection cohomology of Schubert varieties, in preparation.

Soergel has asked for which N does the decomposition theorem hold with coefficients in Z[1/N] for a Bott-Samelson resolution of a Schubert variety X_w in a flag variety G/B. Failure of the decomposition theorem implies the existence of torsion in intersection cohomology of X_w with coefficients in Z or Z/nZ for certain n. Examples where this happens have been known for some time in the case where G is not simply laced. We give examples with G of type A₇ and D₄ where the decomposition theorem fails with coefficients in any ring in which 2 is not a unit.

This partly expository paper reviews the theory of combinatorial intersection cohomology of fans developed by Barthel-Brasselet-Fieseler-Kaup, Bressler-Lunts, and Karu. This theory gives a substitute for the intersection cohomology of toric varieties which has all the expected formal properties but makes sense even for non-rational fans, which do not define a toric variety. As a result, a number of interesting results on the toric g and h polynomials have been extended from rational polytopes to general polytopes. We present explicit complexes computing the combinatorial IH in degrees one and two; the degree two complex gives the rigidity complex previously used by Kalai to study g₂. We present several new results which follow from these methods, as well as previously unpublished proofs of Kalai that g_k(P) = 0 implies g_k(P*) = 0 and g_k+1(P) = 0.

For a pair of affine toric varieties X and Y defined by dual cones, we define an equivalence between two triangulated categories. The first is a mixed version of the equivariant derived category of X and the second is a mixed version of the derived category of sheaves on Y which are locally constant with unipotent monodromy on each orbit. This equivalence satisfies the Koszul duality formalism of Beilinson, Ginzburg, and Soergel. A similar duality was constructed in the following paper; this new approach is more canonical.

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

We give a lower bound for the value at q=1 of a Kazhdan-Lustig polynomial in a Weyl group in terms of ``patterns''. These patterns correspond with special subgroups of the Weyl group generated by reflections. This notion generalizes the concept of patterns and pattern avoidance for permutations to all Weyl groups. The main tool of the proof is a "hyperbolic localization" on intersection cohomology, proved in the next paper.

For a complex variety X with an action of a one-dimensional algebraic torus T, we consider the ``hyperbolic localization'' functor to the set of T-fixed points of X. It takes an object of the derived category of Q-sheaves on X and localizes using closed supports in the directions flowing into the fixed points, and compact supports in the directions flowing out. We show that if X embeds into a smooth T-variety, then the hyperbolic localization of the intersection cohomology sheaf of X is a direct sum of intersection cohomology sheaves.

We compute the category of perverse sheaves 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.

We show that certain 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 the full flag variety for type A and by Boe and Fu for the Lagrangian Grassmannian.

We describe a method of computing equivariant and ordinary intersection cohomology of certain varieties with actions of algebraic tori, in terms of structure of the zero- and one-dimensional orbits. The class of varieties to which our formula applies includes Schubert varieties in flag varieties and affine flag varieties. We also prove a monotonicity result on local intersection cohomology stalks.

We prove an inequality, conjectured by Kalai, relating the g-polynomials of a polytope P, a face F, and the quotient polytope P/F, in the case where P is rational. We introduce a new family of polynomials g(P,F), which measures the complexity of the part of P ``far away" from the face F; Kalai's conjecture follows from the nonnegativity of these polynomials. This nonnegativity comes from showing that the restriction of the intersection cohomology sheaf on a toric variety to the closure of an orbit is a direct sum of intersection homology sheaves.

For three families of stratified spaces: n x n complex matrices, n x n complex symmetric matrics, and 2n x 2n complex antisymmetric matrices, we give a quiver description of the category of peverse sheaves constructible with respect to the stratification, compute the structure of the nearby cycles of the determinant (Pfaffian in the antisymmetric case), and calculate the functor of global hypercohomology.

Tom Braden: publications