# Reference for a result in toric geometry

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

• Could you clarify what is $CH^*$? Also, what exactly cohomology groups are you considering? For projective varieties there are many choices. – Moishe Kohan Jan 31 '17 at 1:44
• @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. – 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.
• 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 ! – user378546 Mar 13 '17 at 17:34