Compute the weights of a $(\mathbb C^*)^{m+1}$-action on $H^0(\mathbb P^m, \mathcal O_{\mathbb P^m}(1))$

Let $\mathbb T:=(\mathbb C^{*})^{m+1}$ be the complex torus and suppose $\mathbb T$ acts on $\mathbb C^{m+1}$ diagonally as follows $$ (t_0, \cdots, t_m) : (x_0, \cdots, x_m) \mapsto (t_0x_0, \cdots, t_mx_m)$$

This action can induces a natural $\mathbb T$-action on $\mathcal O_{\mathbb P^m}(1)$ and also on $X:=H^0(\mathbb P^m, \mathcal O_{\mathbb P^m}(1))$, so the vector space $X$ becomes a $\mathbb T$-representation.

Question How to compute the weights of this $\mathbb T$-action explicitly? I will appreciate it if you can provide details.

(PS:Someone told me that the weights are related to some Chern classes. Is this true? )

Solutions Collecting From Web of "Compute the weights of a $(\mathbb C^*)^{m+1}$-action on $H^0(\mathbb P^m, \mathcal O_{\mathbb P^m}(1))$"

A possible solution (take it with some grains of salts, I am not perfectly familiar with the subject) relies on the following two facts:

  1. The $0$’th sheaf cohomology $H^0\left(\mathbb P^m, \mathcal O_{\mathbb P^m}(1)\right)$ is the space of global sections of the bundle $\mathcal O_{\mathbb P^m}(1)$. (c.f. [Wiki])
  2. The space of global sections of the bundle $\mathcal O_{\mathbb P^m}(1)$ can be identified with the space of homogeneous degree $1$ polynomials in the homogeneous coordinates $\{x_0, \cdots, x_m\}$. (c.f. [Vakil]: Proposition, §1.2.)

Let us define some notation:
\begin{align}
\pi: \mathcal O_{\mathbb P^m}(1) \to \mathbb P^m \; :&\; \mbox{the canonical projection of the bundle to the base} \\
\{U_i\}_{i=0}^m \;:&\; \mbox{standard open cover of $\mathbb P^m$} \\
\{x_i\}_{i=0}^m \;:&\; \mbox{a basis of polynomials of homobeneous degree 1}
\end{align}
Now, a basis of global sections of $\mathcal O_{\mathbb P^m}(1)$ can be defined by (c.f. [Vakil]: Proposition, §1.2.):
$$ g_i^{(j)} : U_j \to \pi^{-1}(U_j) \to \mathbb C\,, \qquad g_i^{(j)}:[x_0:\cdots :x_m] \mapsto \frac{x_i}{x_j}‌\,. $$
The action of $(\mathbb C^*)^{m+1}$ on $g_i^{(j)}$ is defined by imposing:
$$ (t_0, \cdots, t_m): g_i^{(j)} \mapsto \tilde g_i^{(j)}\,, \qquad \tilde g_i^{(j)}([t_0 x_0: \cdots : t_m x_m]) := g_i^{(j)}([x_0: \cdots : x_m])\,, $$
which implies:
$$ \tilde g_i^{(j)}([x_0: \cdots : x_m]) = \frac{t_j}{t_i} \frac{x_i}{x_j} = \frac{t_j}{t_i} g_i^{(j)}([x_0: \cdots : x_m])\,. $$
We can write this action of $(\mathbb C^*)^{m+1}$ on the global sections (in the chart $U_j$) as follows:
$$ (t_0, \cdots, t_m): \left(g_0^{(j)}, \cdots, g_m^{(j)}\right) \mapsto \left(\frac{t_j}{t_0} g_0^{(j)}, \cdots, \frac{t_j}{t_m} g_m^{(j)}\right)\,, $$
so if we denote the $(m+1)$ $(m+1)$-tuples of weights (the weights are $(m+1)$-tuples because the group is an $(m+1)$-dimensional torus, and there are $(m+1)$ of these tuples because the module is $(m+1)$-dimensional) in the chart $U_j$ by $w_a^{(j)} \in \mathbb C^{m+1}$ where $a \in \{0, \cdots, m\}$, then $w_a^{(j)}$ has $1$ at the $j$’th position, $-1$ at the $a$’th position and $0$ everywhere else, for example, $w_1^{(3)}$ looks like $(0,-1,0,1,0, 0, \cdots, 0)$.