Trivial bundle on projective space?

Question

On $P^n(\mathbb R)$ I consider the open sets $U_j$ (with $x_j \neq 0$) and the transiction functions of a linear vector bundle $E_d$, $f_{hk}:p\to(x_k /x_h)^d$. I have to demonstrate that, if $d$ is even, $E_d$ is trivial. How can I find the global section?

Answer 1

You can show that every homogeneous polynomial of degree $d$ gives you a global section of $E_d$. Indeed, if $G = \sum a_{i_0,\dotsc,i_n} x_0^{i_0} \dotsm x_n^{i_n}$ is a homogeneous polynomial of degree $d$ and $g_j$ is the dehomogenised polynomial on $U_j$ for every $j = 0,\dotsc,n$, then $$ g_h = \frac{x_j^d}{x_h^d} g_j $$ so $\{U_j,g_j\}$ is a global section of $E_d$.

In particular, a homogeneous polynomial with no (non-trivial) root corresponds to a nowhere $0$ global section. Now, suppose that $d = 2b$ is even and consider the homogeneous polynomial of degree $d$ $$ G = (x_0^2 + \dotsb + x_n^2)^b $$ If $G$ has a root $(a_0 : \dotsc : a_n) \in \Bbb{P}^n(\Bbb{R})$ we may assume without loss of generality that $a_n \neq 0$. Then $a_0$ is a real root of the polynomial $$ x_0^2 + (a_1^2 + \dotsc + a_n^2) $$ which is absurd because $x_0^2 + c$ is irreducible in $\Bbb{R}[x_0]$ for every $c>0$.