Andreotti–Grauert_theorem

Andreotti–Grauert theorem

Andreotti–Grauert theorem

Theorem

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 ${\mathcal {F}}$ 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)^[1]^[2]
$H^{i}(X,{\mathcal {F}})=0$ for $i\geq q$ , if X is q-complete. (vanish)^[3]^[2]

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