Why is the obstruction, to lifting a projective bundle to a (complex) vector bundle on a space $X$, given by an element $\alpha \in H^3(X, \mathbb{Z})$?
Vector bundles are classified by $H^1(X, U(1))$ (e.g their transition functions give cocycles in the first Cech groups), and projective bundles by $H^1(X, PU(1))$. We have an exact sequence $$1 \to U(1) \to U(n) \to PU(n)\to 1$$ So the obstruction lies in $H^2(X, U(1))\simeq H^2(X, \mathcal{O}_X^*)$ (I think these are isomorphic). Then maybe use the exponential sequence sequence $0 \to \mathbb{Z} \to \mathcal{O}_X \to \mathcal{O}_X^* \to 1$ to get a map $$H^2(X, \mathcal{O}_X^*) \to H^3(X, \mathbb{Z})$$ I don't see why this map is an isomorphism. Or maybe I am doing this all wrong.