Towards a combinatorial Intersection Cohomology for Fans

TL;DR

This paper introduces a purely combinatorial and algebraic approach to intersection cohomology of fans, generalizing Stanley's f- and g-vectors for polytopes and fans, with implications for toric varieties.

Contribution

It defines a minimal extension sheaf on fans as an axiomatic algebraic model for equivariant intersection cohomology, extending previous geometric methods.

Findings

Provides a combinatorial description of intersection cohomology for fans.

Offers an algebraic interpretation of Stanley's generalized f- and g-vectors.

Establishes a new framework for studying toric varieties and polytopes.

Abstract

The real intersection cohomology of a toric variety is described in a purely combinatorial way using methods of elementary commutative algebra only. We define, for arbitrary fans, the notion of a ``minimal extension sheaf'' on the fan as an axiomatic characterization of the equivariant intersection cohomology sheaf. This provides a purely algebraic interpretation of Stanley's generalized f- and g-vector of an arbitrary polytope or complete fan under a natural vanishing condition. -- The results presented in this note originate from joint work with G.Barthel, J.-P.Brasselet and L.Kaup, continuing earlier research (see math.AG/9904159). A detailed exposition will appear elsewhere (see math.AG/0002181).