Describing toric varieties and their equivariant cohomology

TL;DR

This paper provides a detailed topological description of both compact and non-compact toric varieties, extending known cohomological results and exploring torsion phenomena in their integral cohomology.

Contribution

It extends the topological and cohomological understanding of toric varieties to non-compact cases and describes their equivariant cohomology via piecewise polynomials.

Findings

Equivariant integral cohomology can be described by piecewise polynomials when cohomology is even-degree.

The topological construction of toric varieties as quotients is detailed and extended.

Torsion phenomena in integral cohomology are investigated.

Abstract

Topologically, compact toric varieties can be constructed as identification spaces: they are quotients of the product of a compact torus and the order complex of the fan. We give a detailed proof of this fact, extend it to the non-compact case and draw several, mostly cohomological conclusions. In particular, we show that the equivariant integral cohomology of a toric variety can be described in terms of piecewise polynomials on the fan if the ordinary integral cohomology is concentrated in even degrees. This generalizes a result of Bahri-Franz-Ray to the non-compact case. We also investigate torsion phenomena in integral cohomology.