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Andreotti–Grauert theorem

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In mathematics, the Andreotti–Grauert theorem, introduced by Andreotti and Grauert (1962), 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 a coherent analytic sheaf over X. Then,

  • for (resp. ), if X is q-pseudoconvex (resp. q-pseudoconcave). (finiteness)[1][2]
  • for , if X is q-complete. (vanish)[3][2]
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