Divisors and higher cohomologies

Question

Given a divisor $D$ on a smooth, complex manifold there is an associated holomorphic line bundle $\mathcal{O}(D)$. The space of holomorphic sections $H^0(\mathcal{O}(D))$ is directly related to the complete linear system $|D|$ of the divisor, $$ |D| = \mathbb{P}\big(H^0(\mathcal{O}(D))\big) \,. $$ Given this relation, it seems natural to ask: is there also a relatively simple relation between the higher cohomologies $H^{i>0}(\mathcal{O}(D))$ and objects determined by the divisor $D$?