This question is motivated by this answer here.
Let $\mathbb{C}P^{n}$ be a complex projective space. Let $X\in\Gamma(T\mathbb{C}P^{n})$, be a vector field. It seems, by the answer I got in the mentioned link, that the zeros of $X$ are the points $[z]\in\mathbb{C}P^{n}$ such that $X([z])=\lambda[z]$, for some $\lambda$. Can anyone please make this clear for me ? In another way : What does it mean to be a zero of a vector field in a complex projective space ? Thanks!
Let's try a different approach. We can think of $\Bbb CP^n$ as the usual quotient of $\Bbb C^{n+1}-\{0\}$ with the projection map $\pi(z) = [z]$. Let's think of a vector field $X$ on $U\subset\Bbb CP^n$ as being pushed down from an appropriate vector field $\tilde X$ on $\pi^{-1}(U)$ with the property that $\pi_{*z}\tilde X(z) = \pi_{*\lambda z}\tilde X(\lambda z)$ for all nonzero (functions) $\lambda$ on $U$. Now, we can visualize a zero $[z]$ of $X$ in terms of saying that $\pi_{*z}\tilde X(z) = 0$, which means that $\tilde X(z)$ is a multiple of $z\in\Bbb C^{n+1}$.
Along these lines, also see my answer to this question about the Euler sequence.