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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.

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We investigate the equivariant intersection cohomology of a toric variety. Considering the defining fan of the variety as a finite topological space with the subfans being the open sets (that corresponds to the "toric" topology given by the invariant open subsets), equivariant intersection cohomology provides a sheaf (of graded modules over a sheaf of graded rings) on that "fan space". We prove that this sheaf is a "minimal extension sheaf ", i.e., that it satisfies three relatively simple axioms which are known to characterize such a sheaf up to isomorphism. In the verification of the second of these axioms, a key role is played by "equivariantly formal" toric varieties, where equivariant and "usual" (non-equivariant) intersection cohomology determine each other by Künneth type formulae. Minimal extension sheaves can be constructed in a purely formal way and thus also exist for nonrational fans. As a consequence, we can extend the notion of an equivariantly formal fan even to this general setup. In this way, it will be possible to introduce "virtual" intersection cohomology for equivariantly formal non-rational fans.

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