Andreotti–Grauert theorem

In mathematics, the Andreotti–Grauert theorem, introduced by, gives conditions for cohomology groups of coherent sheaves over complex manifolds to vanish or to be finite-dimensional.

statement
Let $X$ be a (not necessarily reduced) complex analytic space, and $$\mathcal{F}$$ a coherent analytic sheaf over X. Then,


 * $$\rm{dim}_{\mathbb{C}} \; H^i (X, \mathcal{F}) < \infty$$ for $$i \geq q$$ (resp. $$i < \rm{codh} \; (\mathcal{F}) - q$$), if $X$ is q-pseudoconvex (resp. q-pseudoconcave). (finiteness)
 * $$H^i (X, \mathcal{F}) = 0$$ for $$i \geq q$$, if $X$ is q-complete. (vanish)