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 $\mathbb{P}^2$. \mathbb{CP}^2$. Here$\mathbb{S}^1\times\mathbb{S}^1$acts on$\mathbb{P}^2$. \mathbb{CP}^2$. The professor just regards $\mathbb{P}^2$ \mathbb{CP}^2$as a triangle with edges$\mathbb{P}^1$, \mathbb{CP}^1$, 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 $H^2_T(M)$.