Complex Geometry/Complex and holomorphic vector bundles

Exercises

1. Let $M:=\mathbb {C} P^{n}$ and define the tautological line bundle over $\mathbb {C} P^{n}$ to be the vector bundle $\tau :=\{([z],x)\in \mathbb {C} P^{n}\times \mathbb {C} |x\in [z]\}$ with the projection $\pi :([z],x)\mapsto [z]$. Find natural local trivialisations of $\tau$ that turn $\tau$ into a holomorphic line bundle.