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Andreotti–Grauert theorem

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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

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Let X be a (not necessarily reduced) complex analytic space, and a coherent analytic sheaf over X. Then,

  • for (resp. ), if X is q-pseudoconvex (resp. q-pseudoconcave). (finiteness)[1][2]
  • for , if X is q-complete. (vanish)[3][2]

Citations

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  1. ^ (Andreotti & Grauert 1962, THÉORÈME 14.)
  2. ^ a b (Ohsawa1984)
  3. ^ (Andreotti & Grauert 1962, COROLLAIRE.)

References

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  • 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.
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Parshin, A.N. (2001) [1994], "Finiteness theorems", Encyclopedia of Mathematics, EMS Press