Combinatorial Intersection Cohomology for Fans
| dc.creator | Barthel, Gottfried | |
| dc.creator | Brasselet, Jean-Paul | |
| dc.creator | Fieseler, Karl-Heinz | |
| dc.creator | Kaup, Ludger | |
| dc.date | 2000-02-22 | |
| dc.date.accessioned | 2026-07-07T04:34:00Z | |
| dc.date.available | 2026-07-07T04:34:00Z | |
| dc.description | We continue the approach toward a purely combinatorial "virtual" intersection cohomology for possibly non-rational fans, based on our investigation of equivariant intersection cohomology for toric varieties (see math.AG/9904159). Fundamental objects of study are "minimal extension sheaves" on "fan spaces". These are flabby sheaves of graded modules over a sheaf of polynomial rings, satisfying three relatively simple axioms that characterize the properties of the equivariant intersection cohomology sheaf on a toric variety, endowed with the finite topology given by open invariant subsets. These sheaves are models for the "pure" objects of a "perverse category"; a "Decomposition Theorem" is shown to hold. -- Formalizing those fans that define "equivariantly formal" toric varieties (where equivariant and non-equivariant intersection cohomology determine each other by Kunneth type formulae), we study "quasi-convex" fans (including fans with convex or with "co-convex" support). For these, there is a meaningful "virtual intersection cohomology". We characterize quasi-convex fans by a topological condition on the support of their boundary fan and prove a generalization of Stanley's "Local-Global" formula realizing the intersection Poincare polynomial of a complete toric variety in terms of local data. Virtual intersection cohomology of quasi-convex fans is shown to satify Poincare duality. To describe the local data in terms of virtual intersection cohomology of lower-dimensional complete polytopal fans, one needs a "Hard Lefschetz" type theorem. It requires a vanishing condition that is known to hold for rational cones, but yet remains to be proven in the general case. | |
| dc.description | 41 pages, Plain TeX ("private" macros included), submitted for publication | |
| dc.identifier | https://arxiv.org/abs/math/0002181 | |
| dc.identifier | http://arxiv.org/abs/math/0002181 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/58739 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Algebraic Topology | |
| dc.subject | Combinatorics | |
| dc.subject | 06Axx, 14M25, 14F32, 52Bxx, 55N25, 55N33 (1991) | |
| dc.title | Combinatorial Intersection Cohomology for Fans | |
| dc.type | text |