Computing rational cohomology of smooth (not necessarily compact) toric varieties - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T12:00:29Z http://mathoverflow.net/feeds/question/115359 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/115359/computing-rational-cohomology-of-smooth-not-necessarily-compact-toric-varieties Computing rational cohomology of smooth (not necessarily compact) toric varieties Reladenine Vakalwe 2012-12-04T05:47:53Z 2013-01-15T08:22:00Z <p>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).</p> <p><b>Some background:</b> 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 <code>\$\mathbb{C}^n\$</code>. These descriptions are along the lines of (but more involved) the variety in this question:</p> <p><a href="http://mathoverflow.net/questions/115141/a-cohomology-computation-request" rel="nofollow">http://mathoverflow.net/questions/115141/a-cohomology-computation-request</a></p> <p>I realized earlier today that my varieties are toric, and I am hoping this observation will be the answer to my prayers.</p> <p>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.).</p> http://mathoverflow.net/questions/115359/computing-rational-cohomology-of-smooth-not-necessarily-compact-toric-varieties/115362#115362 Answer by David C for Computing rational cohomology of smooth (not necessarily compact) toric varieties David C 2012-12-04T06:37:13Z 2012-12-04T06:37:13Z <p>I have the following reference in mind:</p> <p>Franz, M. "The integral cohomology of toric manifolds." Tr. Mat. Inst. Steklova 252 (2006), Geom. Topol., Diskret. Geom. i Teor. Mnozh., 61--70.</p> <p>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.</p>