From Moment Graphs to Intersection Cohomology

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 zeroand 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. 1. Statement of the Main Results To a variety X with an appropriate torus action (§1.1), we will associate a moment graph (§1.2), a combinatorial object which reflects the structure of the 0 and 1-dimensional orbits. There is a canonical sheaf (§1.3) on the moment graph, combinatorially constructed from it (§1.4), which we denote byM. The main result (§1.5) uses the sheafM to compute the local and global equivariant and ordinary intersection cohomology of X functorially. 1.1. Assumptions on the Variety X. We assume that X is an irreducible complex algebraic variety endowed with two structures: 1. An action of an algebraic torus T ∼= (C). We assume that (a) for every fixed point x ∈ X there is a one-dimensional subtorus which is contracting near x, i.e. there is a homomorphism i : C → T and a Zariski open neighborhood U of x so that limα→0 i(α)y = x for all y ∈ U (this implies X is finite), and (b) X has finitely many one-dimensional orbits 2. A T -invariant Whitney stratification by affine spaces. It follows that each stratum contains exactly one fixed point, since a contracting C action on an affine space must act linearly with respect to some coordinate system (see [2], Theorem 2.5). Let Cx denote the stratum containing the fixed point x, so X = ⋃ x∈XT Cx. Every one dimensional orbit L has exactly two distinct limit points: the T fixed 1991 Mathematics Subject Classification. Primary 32S60, secondary 14M15, 58K70.