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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm121-1-1,
author = {Matthias Franz},
title = {Describing toric varieties and their equivariant cohomology},
journal = {Colloquium Mathematicae},
volume = {120},
year = {2010},
pages = {1-16},
zbl = {1258.14058},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm121-1-1}
}
Matthias Franz. Describing toric varieties and their equivariant cohomology. Colloquium Mathematicae, Tome 120 (2010) pp. 1-16. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm121-1-1/