This comes from a problem I'm working on, but I would like to phrase it in generality.
I consider a vector bundle $E$ on the complex projective space $\mathbb{P}^2$. Is there some criterion to decide whether $E$ admits a global section or not, for example by looking at chern classes of $E$, or the determinant of $E$...
For sure it's easy to answer when the vector bundle splits, $E=\oplus_i \mathcal{O}(d_i)$, and I know the $d_i$'s, but I'm unable to find an analogy for a non-splitting vector bundle. Hirzebruch-Riemann-Roch may help, that's why I'm thinking about chern classes: if $\chi(E)$ is positive, I know that either $E$ or $(E^*\otimes \mathcal{O}_{\mathbb{P}^2}(-3))$ is effective, but then I don't know how I can choose which one is effective.
On $\mathbb{P}^1$ everything is easy, since any vector bundle splits on it, but this is already not true for the projective plane. Maybe some classification result could help?
Thank you!