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The title pretty much says it all. I am looking for references (lecture notes, books, readable articles, suggestions), preferably example laden, that explain how to compute the rational cohomology of a smooth toric variety. I am particularly interested in methods for doing this when the variety is not compact. Implicit here is the assumption that this can be done in a practical way (please correct me if I am wrong, I do not know much about the subject).

Some background: I have a bunch of varieties whose cohomology I would really like to compute. I have reasonably explicit descriptions of these varieties as subsets of $\mathbb{C}^n$. These descriptions are along the lines of (but more involved) the variety in this question:

A cohomology computation request.

I realized earlier today that my varieties are toric, and I am hoping this observation will be the answer to my prayers.

If it helps, I can compute the equivariant cohomology of of my varieties (mainly because the Hodge structure on it is pure), but I am presuming this doesn't really completely determine the ordinary cohomology (apart from Hodge-Euler polynomials etc.).

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2 Answers 2

I have the following reference in mind:

Franz, M. "The integral cohomology of toric manifolds." Tr. Mat. Inst. Steklova 252 (2006), Geom. Topol., Diskret. Geom. i Teor. Mnozh., 61--70.

the paper is available on arxiv. In this paper the author explains how to compute the cohomology thanks to the Stanley-Reisner ring (theorem 1.2). This is nice but for the cup product structure theorem 1.2 gives the right algebra structure only in some special cases, the general case is a conjecture.

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Here is a (co-)homology computation for singular and noncompact toric varieties. It gives in particular a generalization of a Jurkiewicz-Danilov theorem (smooth compact case): Jordan,Arno : Homology and cohomology of toric varieties. (Konstanz:Hartong-Gorre (1998),108 p. ; see Zentralblatt 0919.14033).

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