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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