Describing toric varieties and their equivariant cohomology

Volume 121 / 2010

Matthias Franz Colloquium Mathematicum 121 (2010), 1-16 MSC: Primary 14M25; Secondary 55N91, 57Q15. DOI: 10.4064/cm121-1-1


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.


  • Matthias FranzDepartment of Mathematics
    University of Western Ontario
    London, Ontario N6A 5B7, Canada

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