In mathematics, the Andreotti–Grauert theorem, introduced by , gives conditions for

cohomology group In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

s of

coherent sheaves In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...

over

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...

s to vanish or to be finite-dimensional.

Statement

Let be a (not necessarily

reduced Reduction, reduced, or reduce may refer to: Science and technology Chemistry * Reduction (chemistry), part of a reduction-oxidation (redox) reaction in which atoms have their oxidation state changed. ** Organic redox reaction, a redox reacti ...

)

complex analytic space In mathematics, particularly differential geometry and complex geometry, a complex analytic varietyComplex analytic variety (or just variety) is sometimes required to be irreducible and (or) Reduced ring, reduced or complex analytic space is a g ...

, and

\mathcal

a coherent analytic sheaf over X. Then, *

\rm_ \; H^i (X, \mathcal) < \infty

for

i \geq q

(resp.

i < \rm \; (\mathcal) - q

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

H^i (X, \mathcal) = 0

for

i \geq q

, if is q-complete. (vanish)

Citations

References

* * * * * * * * *

External links

Complex manifolds Theorems in abstract algebra {{abstract-algebra-stub