Andreotti–Grauert theorem

Template:Short description In mathematics, the Andreotti–Grauert theorem, introduced by Template:Harvs, gives conditions for cohomology groups of coherent sheaves over complex manifolds to vanish or to be finite-dimensional.

Statement

Let Template:Mvar be a (not necessarily reduced) complex analytic space, and $ℱ$ a coherent analytic sheaf over X. Then,

• ${\mathrm{d}\mathrm{i}\mathrm{m}}_{ℂ}{\mathrm{H}}^{\mathrm{i}}\mathrm{(}\mathrm{X},ℱ\mathrm{)}<\mathrm{\infty }$ for $i\ge q$ (resp. $i<\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{h}\mathrm{(}ℱ\mathrm{)}\mathrm{-}\mathrm{q}$), if Template:Mvar is q-pseudoconvex (resp. q-pseudoconcave). (finiteness)
• $H^i(X,ℱ)=0$ for $i\ge q$, if Template:Mvar is q-complete. (vanish)Template:R

References

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External links

Template:Abstract-algebra-stub