How to calculate the equivariant cohomology ring of $P^2$?

Question

It is well known that Kirwan's injection theorem gives an ring injection from $H^{\ast}_T(M)$ to $H^{\ast}_T(M^T)$ which is induced by the inclusion $M^T \to M$, where $T$ is a torus acting on manifold $M$ and $M^T$ is the fixed point set of this torus action.

I came across a problem when my professor tried to use Kirwan's injection theorem to explore the ring structure of $\mathbb{CP}^2$. Here $\mathbb{S}^1\times\mathbb{S}^1$ acts on $\mathbb{CP}^2$. The professor just regards $\mathbb{CP}^2$ as a triangle with edges $\mathbb{CP}^1$, with orthogonal axis $u$ and $v$. Then he said on each vertex there is a polynomial since $H^{\ast}_T(M^T)=H^{\ast}(M^T)\otimes\mathbb{C}[u,v]$. Suppose the triangle is put with two orthogonal edges parallel to the axis $u$ and $v$. Then for the two vertex on the edge of $u$ direction, set $u=0$ to obtain the relations between coefficients. For the case $\ast=2$, each vertex has a polynomial of the form $au+bv$. So there would be 6 unknowns with 3 equations, which gives the rank of $H^2_T(M^T)$ to be 3, same for $H^2_T(M)$.

Now my questions are: Firstly, how should I understand the view of $\mathbb{CP}^2$ as a triangle sitting in the orthogonal coordinate system, and why the $u$ and $v$ here coincident with the coordinate axis? Secondly, what is the intepretation of setting $u=0$ when we are trying to find the structure of the cohomology ring? Hope someone can help me with those questions.

Answer 1

It sounds like you're talking about GKM (=Goresky–Kottwitz–MacPherson) theory, in which case it's better to think of tori as complex tori, i.e. as products of copies of $\mathbb C^\times$ and not of $S^1$. The triangle to which you're referring is the so-called moment graph of $\mathbb{CP}^2 = SL_3(\mathbb C)/P$, where $$ P = \begin{pmatrix} \ast & \ast & \ast \\ 0 & \ast & \ast \\ 0 & \ast & \ast \end{pmatrix} $$ and $T=\mathbb C^\times \times \mathbb C^\times$ is the diagonal subgroup of $SL_3$ acting by left multiplication. The vertices of the moment graph are the $T$-fixed points, of which there are three in this case. Two vertices are connected by an edge if and only if there's a one-dimensional $T$-orbit whose closure contains the corresponding fixed points. The closure of such an orbit is a copy of $\mathbb{CP}^1$, so that might explain why your professor labeled the edges as such. But anyway, the moment graph already comes with a useful labeling and a direction, though let me not say more about this here.

GKM theory provides a combinatorial description of $H_T^\ast(M)$ in terms of the moment graph of $M$, and it appears that this is what your professor was using. (Here $M$ refers to a projective variety on which a complex torus $T$ is acting in some "nice" fashion. If $M=G/P$ is a generalized flag variety then the action of a maximal torus $T\subset P$ of $G$ is "nice" enough for GKM theory.) One is also provided with an isomorphism $$ H^\ast(M) = \frac{H_T^\ast(M)}{\mathfrak{m} H_T^\ast(M)}, $$ where $\mathfrak m$ is the augmentation ideal $(x_1,\ldots,x_n)$ in $H_T^\ast({\text pt}) \cong S(\mathfrak{t}^\ast) \cong \mathbb C[x_1,\ldots,x_n]$.

For more on this, I recommend Julianna Tymoczko nice survey article. Be sure to check out example 4.1, where she computes $H_T^\ast(\mathbb{CP}^2)$ for our $T$ above.