Equivariant Intersection Cohomology of Toric Varieties

Gottfried Barthel; Jean-Paul Brasselet; Karl-Heinz Fieseler; Ludger Kaup

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Equivariant Intersection Cohomology of Toric Varieties

Gottfried Barthel·Jean-Paul Brasselet·Karl-Heinz Fieseler·Ludger Kaup

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Abstract

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.

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BibTeX RIS

Gottfried Barthel, Jean-Paul Brasselet, Karl-Heinz Fieseler, Ludger Kaup. 1999-04-28. Equivariant Intersection Cohomology of Toric Varieties. https://arxiv.org/abs/math/9904159

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