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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
F
{\displaystyle {\mathcal {F}}}
a coherent analytic sheaf over X. Then,
d
i
m
C
H
i
(
X
,
F
)
<
∞
{\displaystyle {\rm {{dim}_{\mathbb {C} }\;H^{i}(X,{\mathcal {F}})<\infty }}}
for
i
≥
q
{\displaystyle i\geq q}
(resp.
i
<
c
o
d
h
(
F
)
−
q
{\displaystyle i<{\rm {{codh}\;({\mathcal {F}})-q}}}
), if X is q-pseudoconvex (resp. q-pseudoconcave). (finiteness)
[1]
[2]
H
i
(
X
,
F
)
=
0
{\displaystyle H^{i}(X,{\mathcal {F}})=0}
for
i
≥
q
{\displaystyle i\geq q}
, if X is q-complete. (vanish)
[3]
[2]
Citations
References
Andreotti, Aldo; Grauert, Hans (1962),
"Théorème de finitude pour la cohomologie des espaces complexes" , Bulletin de la Société Mathématique de France , 90 : 193–259,
doi :
10.24033/bsmf.1581 ,
ISSN
0037-9484 ,
MR
0150342
Demailly, Jean-Pierre (1990).
"Cohomology of q-convex Spaces in Top Degrees" . Mathematische Zeitschrift . 204 (2): 283–296.
doi :
10.1007/BF02570874 .
S2CID
15197568 .
Demailly, Jean-Pierre; Peternell, Thomas; Schneider, Michael (1996). "Holomorphic line bundles with partially vanishing cohomology". Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry . Israel mathematical conference proceedings; vol. 9.
OCLC
33806479 .
S2CID
19030117 .
Demailly, Jean-Pierre (2011).
"A converse to the Andreotti-Grauert theorem" . Annales de la Faculté des Sciences de Toulouse: Mathématiques . 20 : 123–135.
arXiv :
1011.3635 .
doi :
10.5802/afst.1308 .
S2CID
18656051 .
Henkin, Gennadi M.; Leiterer, Jürgen (1988). "The Cauchy-Riemann Equation on q-Convex Manifolds".
Andreotti-Grauert Theory by Integral Formulas . Progress in Mathematics. Vol. 74. pp. 77–116.
doi :
10.1007/978-1-4899-6724-4_3 .
ISBN
978-0-8176-3413-1 .
Henkin, Gennadi M.; Leiterer, Jürgen (1988). "The Cauchy-Riemann Equation on q-Concave Manifolds".
Andreotti-Grauert Theory by Integral Formulas . Progress in Mathematics. Vol. 74. pp. 117–196.
doi :
10.1007/978-1-4899-6724-4_4 .
ISBN
978-0-8176-3413-1 .
Ohsawa, Takeo (1984).
"Completeness of noncompact analytic spaces" . Publications of the Research Institute for Mathematical Sciences . 20 (3): 683–692.
doi :
10.2977/PRIMS/1195181418 .
Ohsawa, Takeo; Pawlaschyk, Thomas (2022). "Q-Convexity and q-Cycle Spaces".
Analytic Continuation and q-Convexity . SpringerBriefs in Mathematics. pp. 37–47.
doi :
10.1007/978-981-19-1239-9_4 .
ISBN
978-981-19-1238-2 .
Ramis, J. P. (1973).
"Théorèmes de séparation et de finitude pour l'homologie et la cohomologie des espaces (p,q)-convexes-concaves" . Annali della Scuola Normale Superiore di Pisa - Classe di Scienze . 27 (4): 933–997.
External links
Parshin, A.N. (2001) [1994],
"Finiteness theorems" ,
Encyclopedia of Mathematics ,
EMS Press