Combinatorial intersection cohomology for fans

By G. Barthel, Jean-Paul Brasselet, K. Fieseler, Ludger Kaup Konstanz, Iml Marseille-Luminy, Uppsala. 2000

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Intersection cohomology IH•(X∆;R) of a complete toric variety X∆, associated to a fan ∆ in R and with the action of an algebraic torus T ∼= (C), 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 Poincare 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 Xσ for a rational cone σ. That assumption applied to cones in dimension n+ 1 together with Poincare 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 Poincare polynomials.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/885972cf0f46f6c22854f682e6cfe66541f51317

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Citations in the corpus

Higher algebraic K-theory for actions of diagonalizable groups (2001, background): “For example, consider the following theorem of Brion, inspired in turn by results in equivariant cohomology due to Atiyah ([ Ati74 ]), Bredon ([Bre74]), Hsiang ([Hsi75]), Chang and Skjelbred ([Ch-Sk74]), Kirwan ([Kir84]), Goresky, Kottwitz and MacPherson ([G-K-MP98]); see also the very useful discus”
From moment graphs to intersection cohomology (2000, background): “The second follows by an easy induction from the following statement: if S is a pure A-module on Γ , and S|Γ>x splits as a direct sum of two pure sheaves, then this splitting can be extended to Γ≥x (see [ 7 ], Theorem 2.3 for the analogous result for toric varieties).”
Hard Lefschetz theorem for nonrational polytopes (2001, methodology): “It follows from the decomposition theorem [1,3] that if s”
Exact cohomology sequences with integral coefficients for torus actions (2005, result): “This approach was inspired by Barthel–Brasselet–Fieseler–Kaup, who prove a result analogous to the rational version of Theorem 1.1 in the context of (intersection) cohomology for fans [ BBFK , Theorem 4.3].”