Andreotti–Grauert theorem explained

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

l{F}

a coherent analytic sheaf over X. Then,

\rm{dim}C Hi(X,l{F})<infty

for

i\geqq

(resp.

i<\rm{codh} (l{F})-q

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

Hi(X,l{F})=0

for

i\geqq

, if is q-complete. (vanish)

References

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Andreotti–Grauert theorem".

Except where otherwise indicated, Everything.Explained.Today is © Copyright 2009-2024, A B Cryer, All Rights Reserved. Cookie policy.