On the integral cohomology of smooth toric varieties

Abstract

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.

10 pages

References in corpus (2)

• On Davis-Januszkiewicz homotopy types I; formality and rationalisation
• Koszul duality and equivariant cohomology for tori

Cited by in corpus (8)

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• Cohomology of smooth toric varieties: naturality
• Construction of logarithmic cohomology theories II: On Chow groups
• A unifying view toward polyhedral products through panel structures
• Equivariant cohomology of moment-angle complexes with respect to coordinate subtori