On the integral cohomology of smooth toric varieties

Details On the integral cohomology of smooth toric varieties On the integral cohomology of smooth toric varieties

2003-08-27

arxiv.org/abs/math/0308253v1

Proc. Steklov Inst. Math. 252 (2006), 53-62 [substantially revised version]

Mathematics

Algebraic Topology

10 pages

Scientific paper

10.1134/S008154380601007X

Let $X_\Sigma$ be a smooth, not necessarily compact toric variety. We show that a certain complex, defined in terms of the fan $\Sigma$, computes the integral cohomology of $X_\Sigma$, including the module structure over the homology of the torus. In some cases we can also give the product. As a corollary we obtain that the cycle map from Chow groups to integral Borel-Moore homology is split injective for smooth toric varieties. Another result is that the differential algebra of singular cochains on the Borel construction of $X_\Sigma$ is formal.

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