Andreotti–Grauert theorem

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

• ${\displaystyle {\rm {{dim}_{\mathbb {C} }\;H^{i}(X,{\mathcal {F}})<\infty }}}$ for ${\displaystyle i\geq q}$ (resp. ${\displaystyle i<{\rm {{codh}\;({\mathcal {F}})-q}}}$), if X is q-pseudoconvex (resp. q-pseudoconcave). (finiteness)^[1][2]
• ${\displaystyle H^{i}(X,{\mathcal {F}})=0}$ for ${\displaystyle i\geq q}$, if X is q-complete. (vanish)^[3][2]