Describing toric varieties and their equivariant cohomology

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.

13 pages; minor changes

References in corpus (2)

• Some examples in toric geometry
• Topology of Kempf-Ness sets for algebraic torus actions

Cited by in corpus (12)

• The cohomology rings of smooth toric varieties and quotients of moment-angle complexes
• Moment Angle Complexes and Big Cohen-Macaulayness
• Lickorish type construction of manifolds over simple polytopes
• Integral cohomology of quotients via toric geometry
• Torus orbifolds with two fixed points
• Equivariant cohomology of torus orbifolds
• GKM theory for orbifold stratified spaces and application to singular toric varieties
• Torsion in the cohomology of torus orbifolds
• The cohomology rings of real toric spaces and smooth real toric varieties
• The equivariant $K$-theory and cobordism rings of divisive weighted projective spaces
• Shift operators and connections on equivariant symplectic cohomology
• Cohomology of smooth toric varieties: naturality