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  be a (not necessarily reduced) complex analytic space, and  a coherent analytic sheaf over X. Then,

  for  (resp. ), if  is q-pseudoconvex (resp. q-pseudoconcave). (finiteness)

  for , if  is q-complete. (vanish)

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Complex manifolds
Theorems in abstract algebra