Andreotti–Grauert theorem

Short description: 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 ${ℱ}$ a coherent analytic sheaf over X. Then,

• ${\mathrm{d}\mathrm{i}\mathrm{m}}_{\mathbb{ℂ}}{\mathrm{H}}^{\mathrm{i}}\mathrm{(}\mathrm{X},{ℱ}\mathrm{)}<\mathrm{\infty }$ for $i\ge q$ (resp. $i<\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{h}\mathrm{(}{ℱ}\mathrm{)}\mathrm{-}\mathrm{q}$), if X is q-pseudoconvex (resp. q-pseudoconcave). (finiteness)
• $H^i(X,{ℱ})=0$ for $i\ge q$, if X is q-complete. (vanish)

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, http://www.numdam.org/item?id=BSMF_1962__90__193_0 
• Demailly, Jean-Pierre (1990). "Cohomology of q-convex Spaces in Top Degrees". Mathematische Zeitschrift 204 (2): 283–296. doi:10.1007/BF02570874. https://eudml.org/doc/183771. 
• 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. 
• Demailly, Jean-Pierre (2011). "A converse to the Andreotti-Grauert theorem". Annales de la Faculté des Sciences de Toulouse: Mathématiques 20: 123–135. doi:10.5802/afst.1308. 
• Henkin, Gennadi M.; Leiterer, Jürgen (1988). "The Cauchy-Riemann Equation on q-Convex Manifolds". Andreotti-Grauert Theory by Integral Formulas. Progress in Mathematics. 74. pp. 77–116. doi:10.1007/978-1-4899-6724-4_3. ISBN 978-0-8176-3413-1. https://books.google.com/books?id=AJWnEAAAQBAJ&pg=PA116. 
• Henkin, Gennadi M.; Leiterer, Jürgen (1988). "The Cauchy-Riemann Equation on q-Concave Manifolds". Andreotti-Grauert Theory by Integral Formulas. Progress in Mathematics. 74. pp. 117–196. doi:10.1007/978-1-4899-6724-4_4. ISBN 978-0-8176-3413-1. https://books.google.com/books?id=AJWnEAAAQBAJ&pg=PA143. 
• 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. https://books.google.com/books?id=aBZzEAAAQBAJ&pg=PA38. 
• 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. https://eudml.org/doc/83667. 

External links

Hazewinkel, Michiel, ed. (2001), "Finiteness theorems", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=44303