Combinatorial intersection cohomology for fans

Abstract

Intersection cohomology IH • (X ∆ ; R) of a complete toric variety X ∆ , associated to a fan ∆ in R n and with the action of an algebraic torus T ∼ = (C * ) n , is best computed using equivariant intersection cohomology IH • T (X ∆ ). The reason is that X ∆ is IH-"equivariantly formal" and equivariant intersection cohomology provides a sheaf on X ∆ , equipped with its T-invariant topology. An axiomatic description of that sheaf leads to the notion of a "minimal extension sheaf" E • on the fan ∆ and a surprisingly simple, completely combinatorial approach which immediately applies to non-rational fans ∆. These sheaves are the model for a larger class of "pure" sheaves, for which we prove a "Decomposition Theorem". For a certain class of fans (including fans with convex or co-convex support), called "quasi-convex", one can define a meaningful "virtual" intersection cohomology IH • (∆). We characterize quasi-convex fans by a purely topological condition on the support of their boundary fan ∂∆, and then deal with the question whether virtual intersection Betti numbers agree with the components of Stanley's generalized h-vector even for non-rational fans ∆, i.e. we try to prove that they satisfy the same computation algorithm. For quasi-convex fans, we prove a generalization of Stanley's formula realizing the intersection Poincaré polynomial of a complete toric variety in terms of local data. In order to show that the local data may be obtained from the virtual intersection cohomology of complete fans in lower dimensions, we have to assume that the virtual intersection cohomology of a cone σ satisfies a certain vanishing condition, analoguous to the vanishing axiom for local intersection cohomology on the closed orbit of the affine toric variety X σ for a rational cone σ. That assumption applied to cones in dimension n + 1 together with Poincaré duality which we show to hold for virtual intersection cohomology leads to a Hard Lefschetz theorem for polytopal fans and to the desired second step in the computation algorithm for virtual intersection Poincaré polynomials.

Hard Lefschetz theorem for nonrational polytopes

“…It follows from the decomposition theorem [1,2] that if s is a subdivision then L ∆ is a direct summand of s * L Σ as sheaves of A ∆ -modules; in particular,…”

“…As in [1] we consider a (possibly noncomplete) fan Σ such that all maximal cones of Σ have dimension n. Then the boundary ∂Σ of Σ consists of all faces of those n − 1 dimensional cones in Σ that are contained in only one n dimensional cone. We further restrict ourselves to the case of quasi-convex fans [1] where the boundary ∂Σ is a real homology manifold. The examples we wish to consider are complete fans (with ∂Σ = ∅), and fans of the type Σ = Star ∆ (δ) for some cone δ in a complete fan ∆.…”

“…The goal of this paper is to prove the Hard Lefschetz theorem for a general polytope, generalizing McMullen's purely combinatorial proof of the Hard Lefschetz theorem for a simple polytope [7]. Our proof is based on the description of the intersection cohomology of a fan (e.g., the normal fan of a polytope) given in [1,2] . When the fan is simplicial, the torus-equivariant cohomology of the corresponding toric variety can be identified with the space of conewise polynomial functions on the fan.…”

“…1 gives us the non-equivariant intersection cohomology. In order to generalize this, one views the conewise polynomial functions as global sections of a sheaf on the fan with respect to a certain topology.…”

Recent developments in algebraic combinatorics

“…The first step in the above program was the definition of IH(P) due to Barthel, Brasslet, Fieseler, and Kaup [1] and to Bressler and Lunts [6]. The precise definition is rather technical; we include it here (following [19]) so that even readers without the necessary background will have some idea of its flavor.…”