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 ${\displaystyle {\mathcal {F}}}$ a coherent analytic sheaf over X. Then,

• ${\displaystyle {\rm {{dim}_{\mathbb {C} }\;H^{i}(X,{\mathcal {F}})<\infty }}}$ for ${\displaystyle i\geq q}$ (resp. ${\displaystyle i<{\rm {{codh}\;({\mathcal {F}})-q}}}$), if X is q-pseudoconvex (resp. q-pseudoconcave). (finiteness)^[1][2]
• ${\displaystyle H^{i}(X,{\mathcal {F}})=0}$ for ${\displaystyle i\geq q}$, 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