# Cohomology of the toric variety $X_\Sigma=\mathbb C^2\sqcup \mathbb C^2\big/\left((x,y)_1\sim(x^{-1},y^{-1})_2\right)_{x,y\neq 0}$

I'm writing a thesis on Chow rings of toric varieties and am looking for a reference on the singular cohomology ring of the blowup of $\mathbb C^2$ at xy=0, i.e. at the coordinate axes. The topological description of this blow up is $$\mathbb C^2\sqcup \mathbb C^2\big/\left((x,y)_1\sim(x^{-1},y^{-1})_2\right)_{x,y\neq 0},$$ where $(\cdot,\cdot)_1$ and $(\cdot,\cdot)_2$ denote points in the first and second copy of $\mathbb C^2$, respectively.

If there's a way to calculate the cohomology ring from this topological description, I'd be interested in it as well!

Background: This blow up is obtained as the toric variety $X_\Sigma$ for the fan $\Sigma$ in $\mathbb R^2$ with 2-dimensional cones $\operatorname{Cone}(e_1, e_2)$ and $\operatorname{Cone}(-e_1,-e_2)$. I'd like to compare the combinatorial Chow ring associated to this fan (which is isomorphic to $\mathbb Z[x,y]/\langle x^2,y^2,xy\rangle$) to the chomology ring of $X_\Sigma$.

Edit: The blow-up description I gave above seems to be wrong. Instead, we have $$X_\Sigma \cong \mathbb C\mathbb P^1\times\mathbb C\mathbb P^1\setminus \left\{ ([1:0],[0:1]), ([0:1],[1:0])\right\},$$ by mapping $(x,y)_1\mapsto ([1:x],[1:y])$ and $(x,y)_2\mapsto ([x:1],[y:1])$.

• Usually you blow up subschemes of codimension $\geq 2$. The coordinate axes are codimension one - I must be missing something. Another thing: for nice enough toric varieties, Chow ring and singular cohomology rings are isomorphic via the cycle class map, see Fulton's book on intersection theory (or the one on toric varieties). – Matthias Wendt Aug 18 '14 at 15:13
• @MatthiasWendt This variety is smooth but not complete. The isomorphisms discussed by Fulton go back to Danilov, all assuming completeness. This fan is not complete, but still $n$-pure. On another note, I'm looking at the Chow ring directly associated to $\Sigma$ as $\mathbb Z[X_1, \dots, X_4]/(I+J)$, where $I$ is the Stanley-Reisner ideal and $J$ depends on the coordinates of the rays of $\Sigma$. – Christoph Aug 18 '14 at 15:16
• But how do you blow up something that is already codimension one? – Matthias Wendt Aug 18 '14 at 15:21
• @MatthiasWendt To be honest, I'm not very familiar with blowups. I'm really interested in the toric variety $X_\Sigma$ described in the second paragraph. It has the topological description of gluing $\mathbb C^2\sqcup\mathbb C^2$ along the given relation. I'm not absolutely sure it is a blow up, but I've read that somewhere and thought it would help attract people knowing about this kind of variety... ;-) – Christoph Aug 18 '14 at 15:24
• To compute the cohomology ring, you could use the explicit description you gave: the variety has a covering by two contractible subsets, which are identified along inclusions of (up to homotopy) a 2-torus. The homotopy type of the variety should be the suspension of the 2-torus, so I guess that the singular cohomology ring is isomorphic to the Chow ring after all. – Matthias Wendt Aug 18 '14 at 15:27

Here is a different way of computing the cohomology ring (which starts directly from the fan). The space is constructed by gluing to copies of $\mathbb{C}^2$ along an inclusion of $(\mathbb{C}^\times)^2$. The Mayer-Vietoris sequence associated to that would look like $$\cdots\to H^i(X_\Sigma)\to H^i(\operatorname{pt})\oplus H^i(\operatorname{pt})\to H^i((\mathbb{C}^\times)^2)\to H^{i+1}(X_\Sigma)\to\cdots$$ and therefore induces isomorphisms $H^{i+1}(X_\Sigma)\cong H^i((\mathbb{C}^\times)^2)$ for $i>0$. What you get from the long exact sequence is a priori only additive information, but in this case there isn't much choice for the ring structure.

But there is more information: the pushout $X_\Sigma\cong\operatorname{colim}\left(\mathbb{C}^2\leftarrow(\mathbb{C}^\times)^2\rightarrow \mathbb{C}^2\right)$ is in fact a homotopy pushout. The homotopy type of $X_\Sigma$ can then be identified as $\Sigma(\mathbb{C}^\times)^2\simeq S^2\vee S^2\vee S^3$ because generally $\Sigma(X\times Y)\simeq \Sigma X\vee\Sigma X\vee \Sigma(X\wedge Y)$ - I apologize for the unfortunate notation but in this sentence $\Sigma$ is the suspension of topological spaces. In this description, the ring structure is the one from $S^2\vee S^2\vee S^3$.

There is a motivic generalization of this. First of all, motives are supposed to explain the general patterns behind various sorts of cohomology theories in algebraic geometry. For smooth projective varieties, one has Chow motives and the motivic decompositions for smooth projective toric varieties are the reason behind the cycle class map being an isomorphism (as discussed in Fulton's book on intersection theory, see also the Chow ring formula in the toric varieties book). For smooth but not necessarily projective varieties, we will have to use Voevodsky's construction of the "derived category of mixed motives". If you are interested in studying smooth toric varieties and the relations between Chow rings and singular cohomology rings, I would strongly suggest to have a look at the motives literature - smooth toric varieties are linear, so their motives are mixed Tate and you get strong relations between Chow rings and singular cohomology (or $\ell$-adic cohomology and many others) from this.

Ok, so much for the general advertisement. The argument I gave above for computing the homotopy type of the variety works in what is called $\mathbb{A}^1$-homotopy theory. The covering of $X_\Sigma$ by two copies of affine space shows that the $\mathbb{A}^1$-homotopy type of $X_\Sigma$ is $\Sigma^{\mathbb{A}^1}\mathbb{G}_m^2\simeq\mathbb{P}^1\vee\mathbb{P}^1\vee\mathbb{A}^2\setminus\{0\}$. The motive $M(X_\Sigma)$ is then $\mathbb{Z}\oplus\mathbb{Z}(1)\oplus\mathbb{Z}(1)\oplus\mathbb{Z}(2)$; the first three summands are what you see in Chow groups, the last summand can only be seen in Bloch's higher Chow groups. Applying the Hodge realization to the motive also gives you the Hodge structures, where the first three summands have the right weights (like they come from a smooth projective variety), but the last summand doesn't.

Finally, there are alternatives for doing the computations in motives. There is a localization (or Gysin) sequence for motives (or higher Chow groups) as well, so that the argument of abx can also be done in that setting.

I apologize that the motives part is maybe not very accessible if you have never seen motives before; I will see if I can give better explanations but full details would certainly be beyond the scope of this answer. Anyway, if you are interested in Chow groups of smooth toric varieties, I would say that motives are the way to go.

Once you know that your surface is $S\smallsetminus Z$, with $S=\mathbb{P}^1\times \mathbb{P}^1$ and $Z$ consists of two points, the answer is immediate from the Gysin exact sequence : $$H^p(S,\mathbb{Z})\rightarrow H^p(S\smallsetminus Z,\mathbb{Z})\rightarrow H^{p-3}(Z,\mathbb{Z})\rightarrow H^{p+1}(S,\mathbb{Z})\rightarrow \ldots$$ You get $H^p(S\smallsetminus Z,\mathbb{Z})\cong H^p(S,\mathbb{Z})$ for $p\leq 2$, $H^3(S\smallsetminus Z,\mathbb{Z})\cong \mathbb{Z}$ and $H^4(S\smallsetminus Z,\mathbb{Z})=0$.

• That looks very clean and easy. Do you have a reference for Gysin exact sequences? I've never heard about that! – Christoph Aug 18 '14 at 20:46
• There must be many, but one fairly general is Proposition 3.2.11 in A. Dimca, Sheaves in Topology (Springer Universitext). – abx Aug 19 '14 at 5:09