I want to know where I can find the proof of the following theorem : if $X$ is a smooth toric variety, then $H^{\bullet}(X) \cong CH^{\bullet}(X)$.

Thanks in advance !

  • $\begingroup$ Could you clarify what is $CH^*$? Also, what exactly cohomology groups are you considering? For projective varieties there are many choices. $\endgroup$ – Moishe Kohan Jan 31 '17 at 1:44
  • $\begingroup$ @MoisheCohen : thanks for your comment ! $CH^*$ is the Chow ring and $H^*$ is the cohomology ring (de Rham, singular) which should all coincide in this nice case. I think I probably forgot to put projective in the hypothesis as well. $\endgroup$ – user378546 Jan 31 '17 at 8:20

In Section 5.2 of Fulton there is a theorem that says that for $X$ a smooth complete toric variety, $CH_*X \cong H_*X$, freely generated by the orbit closures.

Your statement follows by Poincare duality and universal coefficient theorem.

  • $\begingroup$ Ok, this is great. Thanks a lot !!! $\endgroup$ – user378546 Jan 31 '17 at 11:39
  • $\begingroup$ Sorry for the stupid question : are Grassmanians toric varieties ? Because their cohomology rings seems similar. $\endgroup$ – user378546 Jan 31 '17 at 11:52
  • $\begingroup$ Not stupid at all. No, Grassmannians are usually not toric. (For example, one can prove that the only smooth toric varieties of Picard number 1 are the projective spaces.) However, Grassmannians do belong to a wider class called cellular varieites, and these also have the property you asked about. Feel free to ask more questions if you want details. $\endgroup$ – Nefertiti Jan 31 '17 at 12:19
  • $\begingroup$ I will surely have more questions but probably later since I have soon an exam about toric varieties and this was by curiosity. Thanks again for the explanations ! $\endgroup$ – user378546 Jan 31 '17 at 12:22
  • $\begingroup$ Dear @Nefertiti, can I ask you about more informations about cellular varieties ? More precisely, I know that with Grothendieck ring allows us to compute the cohomology of smooth projective varieties which are polynomials in $L := [\mathbb A^1]$. Is this related to the definition of cellular varieties ? Where can I find information about it ? Thanks in advance ! $\endgroup$ – user378546 Mar 13 '17 at 17:34

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