1999/04/28 by Gottfried Barthel, Barthel, Gottfried, Jean-Paul Brasselet +5 · 1 citation
Mathematics · #14M25 #32S60 #52B20 #55N25 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #math.AG #math.AT #math.CO #msc:14M25 #msc:32S60 #msc:52B20 #msc:55N25
paper · pdf · doi:10.48550/arxiv.math/9904159
31 pages, AMS-Latex (all "private" macros included), to be published in "Algebraic Geometry - Hirzebruch 70" (Proceedings of the conference at the Banach Centre, Warszawa, May 1998), Contemporary Mathematics, AMS
arxiv created 1999/04/28 · arxiv updated 2009/11/30
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 Kunneth type formulae. Minimal extension sheaves can be constructed in a purely formal way and thus also exist for non-rational 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.