Andreotti–Grauert theorem

(Redirected from Andreotti–Grauert vanishing 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)
$H^{i}(X,{\mathcal {F}})=0$ for $i\geq q$ , if X is q-complete. (vanish)

Citations

(Andreotti & Grauert 1962, THÉORÈME 14.)
^ (Ohsawa1984)
(Andreotti & Grauert 1962, COROLLAIRE.)

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) , "Finiteness theorems", Encyclopedia of Mathematics, EMS Press

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