Is $T\mathbb{C}\mathbb{P}^n$ globally generated?

Question

A vector bundle $E\to X$ is globally generated if there exists global holomorphic sections $s_1,\dots,s_n$ such that $E_x$ is spanned by $s_1(x),\dots,s_n(x)$ for all $x\in X$.

Consider the projection map $p:\mathbb{C}^{n+1}\to\mathbb{P}^n$. It can be shown that $p$ is a smooth submersion. Furthermore, given $n+1$ linear functions $l_i:\mathbb{C}^{n+1}\to\mathbb{C}$, with $i=0,\dots,n$, and taking the coordinates in $\mathbb{C}^{n+1}$ to be $(z_0,\dots,z_n)$, it can be shown that the global vector field $$ \sum_{i=0}^nl(z)\frac{\partial}{\partial z_i} $$ in $\mathbb{C}^{n+1}$ can be pushed forward to a global vector field of $\mathbb{P}^n$. Motivated by Ted Shifrin's answer in this question I believe that all vector fields in $\mathbb{P}^n$ must be of this form, implying that $T\mathbb{P}^n$ is globally generated.

Explicitely, (I believe, a confirmation of the following would be nice) the pushforward of this vector field is: $$ p_*\left(\sum_{i=0}^nl(z)\frac{\partial}{\partial z_i}\right)= \sum_{i=0}^nl(z)p_*\left(\frac{\partial}{\partial z_i}\right), $$ where, for instance in the coordinate chart $U_0$ of $\mathbb{P}^n$ with coordinates $w_k=z_k/z_0$, $$ p_*\left(\frac{\partial}{\partial z_0}\right)=-\sum_{i=1}^n \frac{z_i}{z_0^2}\frac{\partial}{\partial w_i}, $$ and $$ p_*\left(\frac{\partial}{\partial z_k}\right)=\frac{1}{z_0}\frac{\partial}{\partial w_k}. $$ In particular this calculation in coordinates shows the need of multiplying by a linear function to get a well defined vector field.

My problem is I don't see why every vector field in projective space must be of this form, locally and/or globally.

Answer 1

$\newcommand{\Cpx}{\mathbf{C}}\newcommand{\Proj}{\mathbf{P}}$Yes, $T\Cpx\Proj^{n}$ is globally generated. Note that it doesn't matter whether or not all global holomorphic vector fields on $\Cpx\Proj^{n}$ are of the stated form; what matters is:

1. At some point $p$ (and hence every point, by homogeneity) there exist vector fields of the stated form spanning the tangent space at $p$.

2. The zero set of a generic vector field is a finite set (of $(n+1)$ points, if it matters).

(Incidentally, it appears you're using $n$ to denote two distinct integers: The complex dimension of the projective space, and the number of holomorphic vector fields needed to span the tangent bundle globally.)