Talk:Chow group of a stack

There needs to be explicit examples of virtual fundamental classes with intersection theory computations giving the correct intersection numbers.

Absolutely. -- Taku (talk) 23:15, 9 May 2017 (UTC)[reply]

A good reference to look at is the first chapter of "Quantum Cohomology" by Behrend, Gomez, Tarasov, Tian. They compute the chow ring for $B\mathbb {G} _{m}$ . — Preceding unsigned comment added by Algebraic guidance (talk • contribs) 17:42, 7 June 2017 (UTC)[reply]

I believe you're referring to 3 lectures by Behrend; that would be https://www.math.ubc.ca/~behrend/cet.pdf Lecture 2 computes the Chow ring of the classifying stack. Stuff like this should definitely apppear in the article. -- Taku (talk) 04:10, 8 June 2017 (UTC)[reply]

Another good reference for this stuff is http://www1.mat.uniroma1.it/people/manetti/DT2011/marco2.pdf — Preceding unsigned comment added by 97.122.75.155 (talk) 04:45, 18 July 2017 (UTC)[reply]

It looks nice (very similar to Behrend's two papers on differential graded schemes, but how is it relevant for this article? -- Taku (talk) 22:53, 18 July 2017 (UTC)[reply]

On the nLab page for the virtual fundamental class you can define it as follows: given a derived scheme $(X,{\mathcal {O}}_{X}^{\bullet })$ if you take the K-theory class $[X]\sum (-1)^{k}\pi _{k}({\mathcal {O}}_{X})\in K((X,t_{0}({\mathcal {O}}_{X}^{\bullet })))$ and evaluate the todd class of the tangent complex pulled back to the truncated space you get the virtual fundamental class. In short

[X]^{vir}=\int _{[X]}{\text{td}}(\mathbf {T} ^{\bullet })

I referenced Manetti's article because it shows you how to compute the cotangent complex for a derived scheme constructed from a hypersurface. — Preceding unsigned comment added by 97.122.75.155 (talk) 05:17, 20 July 2017 (UTC)[reply]

Unfortunately, nLab cannot be used as a source (since it's not reliable). I think the calculation like this works because of Riemann-Roch, which allows one to equivalently see a virtual fundamental class either as a class in a Chow group or in thr K-theory. -- Taku (talk) 12:10, 20 July 2017 (UTC)[reply]

They reference a paper by Toen: check out page 23 of https://arxiv.org/pdf/1401.1044v1.pdf — Preceding unsigned comment added by 67.161.205.32 (talk) 19:26, 20 July 2017 (UTC)[reply]

Except Toen writes: "The point of view taken is to present as much as possible mathematical facts, without insisting too much nor on formal or definitional aspects neither on technical aspects (e.g. no proofs will be given or even sketched)." (italics in the original text). It's important to distinguish between what has been established and what is expected to be established. Wikipedia cannot be presenting fake math :) That's why it's very important to have reliable sources. Having said this, it is perfectly fine and should be encouraged to present some speculations as such. -- Taku (talk) 21:11, 20 July 2017 (UTC)[reply]

Another good reference for chow rings of stacky points is https://web.stanford.edu/~tonyfeng/top_AG.pdf — Preceding unsigned comment added by 71.212.185.82 (talk) 05:06, 24 August 2017 (UTC)[reply]

Computing Virtual Fundamental Classes[edit]

Check out theorem 26.1.1 of http://www.claymath.org/library/monographs/cmim01c.pdf for computing the pushforward of the virtual funamental class. In addition theorem 26.1.2 gives more motivation for a perfect obstruction theory.

Explanation of VFC[edit]

Check out the appendix of https://people.math.ethz.ch/~rahul/13curves.pdf to get an intuitive idea about the VFC package. In addition, https://www.maths.gla.ac.uk/~nnabijou/VFCProject.pdf has some useful information — Preceding unsigned comment added by Wundzer (talk • contribs) 20:03, 12 February 2020 (UTC)[reply]