I'm asked to provide an example of a section of the tautological bundle over $\Bbb C P^1$ such that it has just a zero. This is a step necessary to prove that the first Chern class is $-1\cdot [\Bbb C P^1]$.
Problem is that I'm unable to provide such section. The obvious candidate would be $[u:v] \mapsto ([u:v],(u/v,1))$ if $v\neq 0$ and $[u:v] \mapsto ([u:v],(1, v/u))$ if $u\neq 0$ but it's not well defined on the overlapping of the two charts of $\Bbb C P^1$.
I'm not looking for a solution, just some hints
As a hint, it is a fact (that follows from the Chern class, for example) that there is no global holomorphic section. So perhaps you'd like to look for a smooth section that is, say, anti-holomorphic.