About the holomorphic tangent bundle of $\mathbb{CP}^n$

Question

Are there global sections of $T{\mathbb{CP}^n}$, which vanish only in a finite number of points? If yes, how many? Here $T{\mathbb{CP}^n}$ is the holomorphic bundle. Any help would mean a lot, thanks.

Answer 1

Hint: If $\Lambda$ is an $(n+1) \times (n+1)$ complex matrix whose eigenspaces are one-dimensional and span $\mathbf{C}^{n+1}$, then every element of the one-parameter group $\exp(t\Lambda)$ induces a projective automorphism that fixes the images of the eigenspaces and (except for $t = 0$) no other points.