Let $\mathbb{C}P^1 = \mathbb{C}^2 / \mathbb{C}_{*}$, where $\mathbb{C}_{*} = \mathbb{C} \backslash \{0\}$. Denote by $[z_0, z_1]$ the equivalence class of $(z_0, z_1) \neq (0, 0)$.
How can I show that the tangent space $T_{[z_0, z_1]}\mathbb{C}P^1$ can be identified with the quotient vector space $\mathbb{C^2} / \mathbb{C}(z_0, z_1)$ ?
I guess I should find an isomorphism between them. Moreover, if if the image of $(\tilde{z_0}, \tilde{z_1})$ in this quotient is denoted by $[[\tilde{z_0},\tilde{z_1}]]$ and $J:\mathbb{C}^2 \rightarrow \mathbb{C}^2$ is a map defined by $J[[\tilde{z_0},\tilde{z_1}]] = [[i\tilde{z_0}, i\tilde{z_1}]],$
how can I prove that $J$ is an almost complex structure on $T_{[z_0, z_1]}\mathbb{C}P^1$ or even better a complex structure on $\mathbb{C}P^1$ ?
So I think I should check that $J$ is a $(1, 1)$-tensor on $T_{[z_0, z_1]}\mathbb{C}P^1$ satisfying $J^2 = -Id$. To prove that $J$ is a complex structure I need to verify that it comes from a holomorphic structure.
Can someone help ? Any hint will be much appreciated.
Many thanks.