logo AFST
Oka manifolds: From Oka to Stein and back
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 22 (2013) no. 4, pp. 747-809.

Oka theory has its roots in the classical Oka-Grauert principle whose main result is Grauert’s classification of principal holomorphic fiber bundles over Stein spaces. Modern Oka theory concerns holomorphic maps from Stein manifolds and Stein spaces to Oka manifolds. It has emerged as a subfield of complex geometry in its own right since the appearance of a seminal paper of M. Gromov in 1989.

In this expository paper we discuss Oka manifolds and Oka maps. We describe equivalent characterizations of Oka manifolds, the functorial properties of this class, and geometric sufficient conditions for being Oka, the most important of which is Gromov’s ellipticity. We survey the current status of the theory in terms of known examples of Oka manifolds, mention open problems and outline the proofs of the main results. In the appendix by F. Lárusson it is explained how Oka manifolds and Oka maps, along with Stein manifolds, fit into an abstract homotopy-theoretic framework.

The article is an expanded version of lectures given by the author at Winter School KAWA 4 in Toulouse, France, in January 2013. A comprehensive exposition of Oka theory is available in the monograph [32].

La théorie d’Oka tire ses origines du principe classique d’Oka-Grauert, dont la principale application est la classification par Grauert des fibrés holomorphes principaux sur les espaces de Stein. La théorie d’Oka moderne traite des applications holomorphes depuis des variétés ou des espaces de Stein vers des variétés d’Oka. Elle est devenue un sous-domaine à part entière de la géométrie complexe depuis la parution d’un article fondateur de M. Gromov en 1989.

Nous présentons ici les variétés et les applications d’Oka. Nous décrivons les caractérisations équivalentes des variétés d’Oka, les propriétés fonctorielles de cette classe, et des conditions suffisantes géométriques pour qu’une variété soit d’Oka, dont la plus importante est l’ellipticité de Gromov. Nous donnons un panorama de l’état actuel de la théorie en ce qui concerne les exemples connus de variétés d’Oka, mentionnons les problèmes ouverts et esquissons les démonstrations des résultats principaux. Dans l’appendice, dû à F. Lárusson, on explique comment les variétés d’Oka, et les applications d’Oka, s’inscrivent dans le cadre d’une théorie homotopique abstraite.

Le présent article est une version augmentée des exposés de l’auteur lors de l’Ecole d’Hiver KAWA 4 à Toulouse, France, en janvier 2013. On trouvera une présentation exhaustive de la théorie d’Oka dans la monographie [32].

Published online:
DOI: 10.5802/afst.1388
@article{AFST_2013_6_22_4_747_0,
     author = {Franc Forstneri\v{c}},
     title = {Oka manifolds: {From} {Oka} to {Stein} and back},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {747--809},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 22},
     number = {4},
     year = {2013},
     doi = {10.5802/afst.1388},
     zbl = {06250447},
     mrnumber = {3137250},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1388/}
}
TY  - JOUR
TI  - Oka manifolds: From Oka to Stein and back
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2013
DA  - 2013///
SP  - 747
EP  - 809
VL  - Ser. 6, 22
IS  - 4
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1388/
UR  - https://zbmath.org/?q=an%3A06250447
UR  - https://www.ams.org/mathscinet-getitem?mr=3137250
UR  - https://doi.org/10.5802/afst.1388
DO  - 10.5802/afst.1388
LA  - en
ID  - AFST_2013_6_22_4_747_0
ER  - 
%0 Journal Article
%T Oka manifolds: From Oka to Stein and back
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2013
%P 747-809
%V Ser. 6, 22
%N 4
%I Université Paul Sabatier, Toulouse
%U https://doi.org/10.5802/afst.1388
%R 10.5802/afst.1388
%G en
%F AFST_2013_6_22_4_747_0
Franc Forstnerič. Oka manifolds: From Oka to Stein and back. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 22 (2013) no. 4, pp. 747-809. doi : 10.5802/afst.1388. https://afst.centre-mersenne.org/articles/10.5802/afst.1388/

[1] Alarcón (A.), Forstnerič (F.).— Null curves and directed immersions of open Riemann surfaces, Inventiones Math., in press. arXiv:1210.5617 http://link.springer.com/article/10.1007/s00222-013-0478-8

[2] Andrist (R.B.), Wold (E.F.).— The complement of the closed unit ball in 3 is not subelliptic, arXiv:1303.1804

[3] Arzhantsev (I.V.), Flenner (H.), Kaliman (S.), Kutzschebauch (F.), Zaidenberg (M.).— Flexible varieties and automorphism groups, Duke Math. J. 162, p. 767-823 (2013). | MR | Zbl

[4] Arzhantsev (I.V.), Kuyumzhiyan (K.G.), Zaidenberg (M.G.).— Flag varieties, toric varieties, and suspensions: three instances of infinite transitivity, Mat. Sb. 203, 3-30 (2012); English translation: Sb. Math. 203, p. 923-949 (2012). | MR | Zbl

[5] Barth (W.), Hulek (K.), Peters (C.A.M.), Van de Ven (A.).— Compact Complex Surfaces. 2nd Ed, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 4. Springer-Verlag, Berlin (2004). | MR | Zbl

[6] Behnke (H.), Stein (K.).— Entwicklung analytischer Funktionen auf Riemannschen Flächen, Math. Ann. 120, p. 430-461 (1948). | MR | Zbl

[7] Bishop (E.).— Mappings of partially analytic spaces, Amer. J. Math. 83, p. 209-242 (1961). | MR | Zbl

[8] Brody (R.).— Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc. 235, p. 213-219 (1978). | MR | Zbl

[9] Buzzard (G.T.).— Tame sets, dominating maps, and complex tori, Trans. Amer. Math. Soc. 355, p. 2557-2568 (2002). | MR | Zbl

[10] Buzzard (G.), Lu (S.S.Y.).— Algebraic surfaces holomorphically dominable by 2 , Invent. Math. 139, p. 617-659 (2000). | MR | Zbl

[11] Campana (F.).— Orbifolds, special varieties and classification theory, Ann. Inst. Fourier 54, 499-630 (2004). | Numdam | MR | Zbl

[12] Campana (F.).— Orbifolds, special varieties and classification theory: an appendix, Ann. Inst. Fourier 54, p. 631-665 (2004). | Numdam | MR | Zbl

[13] Campana (F.), Winkelmann (J.).— On h-principle and specialness for complex projective manifolds, arxiv.org/abs/1210.7369

[14] Cartan (H.).— Espaces fibrés analytiques. 1958 Symposium internacional de topología algebraica, pp. 97-121, Universidad Nacional Autónoma de México and UNESCO, Mexico City (1958). | Numdam | MR | Zbl

[15] Cox (D.), Little (J.), Schenck (H.).— Toric varieties, Graduate Studies in Mathematics, vol. 124. Amer. Math. Soc., Providence (2011). | MR | Zbl

[16] Drnovšek (B.), Forstnerič (F.).— Holomorphic curves in complex spaces, Duke Math. J. 139, p. 203-254 (2007). | Zbl

[17] Drnovšek (B.), Forstnerič (F.).— Approximation of holomorphic mappings on strongly pseudoconvex domains, Forum Math. 20, p. 817-840 (2008). | MR | Zbl

[18] Dwyer (W.G.), Spaliński (J.).— Homotopy theories and model categories, Handbook of algebraic topology, pp. 73-126. North-Holland, Amsterdam (1995). | MR | Zbl

[19] Eisenman (D.A.).— Intrinsic measures on complex manifolds and holomorphic mappings, Memoirs of the Amer. Math. Soc., 96. Amer. Math. Soc., Providence (1970). | MR | Zbl

[20] Eliashberg (Y.).— Topological characterization of Stein manifolds of dimension >2, Internat. J. Math. 1, p. 29-46 (1990). | MR | Zbl

[21] Eliashberg (Y.), Gromov (M.).— Nonsingular mappings of Stein manifolds, Funkcional. Anal. i Priložen. 5, p. 82-83 (1971). | MR | Zbl

[22] Eliashberg (Y.), Gromov (M.).— Embeddings of Stein manifolds of dimension n into the affine space of dimension 3n/2+1, Ann. Math. (2) 136, p. 123-135 (1992). | MR | Zbl

[23] Forster (O.).— Plongements des variétés de Stein, Comment. Math. Helv. 45, p. 170-184 (1970). | MR | Zbl

[24] Forstnerič (F.).— The Oka principle for sections of subelliptic submersions, Math. Z. 241, p. 527-551 (2002). | MR | Zbl

[25] Forstnerič (F.).— Extending holomorphic mappings from subvarieties in Stein manifolds, Ann. Inst. Fourier 55, p. 733-751 (2005). | Numdam | MR | Zbl

[26] Forstnerič (F.).— Runge approximation on convex sets implies the Oka property, Ann. Math. (2) 163, p. 689-707 (2006). | MR | Zbl

[27] Forstnerič (F.).— Manifolds of holomorphic mappings from strongly pseudoconvex domains, Asian J. Math. 11, p. 113-126 (2007). | MR | Zbl

[28] Forstnerič (F.).— Oka manifolds, C. R. Acad. Sci. Paris, Ser. I, 347, p. 1017-1020 (2009). | MR | Zbl

[29] Forstnerič (F.).— Oka maps, C. R. Acad. Sci. Paris, Ser. I, 348, p. 145-148 (2010). | MR | Zbl

[30] Forstnerič (F.).— The Oka principle for sections of stratified fiber bundles, Pure Appl. Math. Q. 6, p. 843-874 (2010). | MR | Zbl

[31] Forstnerič (F.).— Invariance of the parametric Oka property, Complex analysis, p. 125-144, Trends Math., Birkhäuser/Springer Basel AG, Basel (2010). | MR | Zbl

[32] Forstnerič (F.).— Stein Manifolds and Holomorphic Mappings (The Homotopy Principle in Complex Analysis), Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 56. Springer-Verlag, Berlin-Heidelberg (2011). | MR | Zbl

[33] Forstnerič (F.), Lárusson (F.).— Survey of Oka theory, New York J. Math. 17a, p. 1-28 (2011). | MR | Zbl

[34] Forstnerič (F.), Lárusson (F.).— Holomorphic flexibility properties of compact complex surfaces, Int. Math. Res. Notices IMRN (2013), http://dx.doi.org/10.1093/imrn/rnt044

[35] Forstnerič (F.), Prezelj (J.).— Oka’s principle for holomorphic fiber bundles with sprays, Math. Ann. 317, p. 117-154 (2000). | MR | Zbl

[36] Forstnerič (F.), Prezelj (J.).— Extending holomorphic sections from complex subvarieties, Math. Z. 236, p. 43-68 (2001). | MR | Zbl

[37] Forstnerič (F.), Prezelj (J.).— Oka’s principle for holomorphic submersions with sprays, Math. Ann. 322, p. 633-666 (2002). | MR | Zbl

[38] Forstnerič (F.), Ritter (T.).— Oka properties of ball complements. arXiv:1303.2239

[39] Forstnerič (F.), Slapar (M.).— Stein structures and holomorphic mappings, Math. Z. 256, p. 615-646 (2007). | MR | Zbl

[40] Forstnerič (F.), Wold (E.F.).— Bordered Riemann surfaces in 2 , J. Math. Pures Appl. 91, p. 100-114 (2009). | MR | Zbl

[41] Forstnerič (F.), Wold (E.F.).— Fibrations and Stein neighborhoods, Proc. Amer. Math. Soc. 138, p. 2037-2042 (2010). | MR | Zbl

[42] Forstnerič (F.), Wold (E.F.).— Embeddings of infinitely connected planar domains into 2 , Anal. PDE, in press. arXiv:1110.5354 | MR | Zbl

[43] Goerss (P.G.), Jardine (J.F.).— Simplicial homotopy theory, Progress in Mathematics, 174. Birkhäuser, Basel (1999). | MR | Zbl

[44] Gompf (R.E.).— Handlebody construction of Stein surfaces, Ann. Math. (2) 148, p. 619-693 (1998). | MR | Zbl

[45] Gompf (R.E.).— Stein surfaces as open subsets of 2 , J. Symplectic Geom. 3, p. 565-587 (2005). | MR | Zbl

[46] Grauert (H.).— Approximationssätze für holomorphe Funktionen mit Werten in komplexen Räumen, Math. Ann. 133, p. 139-159 (1957). | MR | Zbl

[47] Grauert (H.).— Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen, Math. Ann. 133, p. 450-472 (1957). | MR | Zbl

[48] Grauert (H.).— Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann. 135, p. 263-273 (1958). | MR | Zbl

[49] Grauert (H.).— On Levi’s problem and the embedding of real-analytic manifolds, Ann. Math. (2) 68, p. 460-472 (1958). | MR | Zbl

[50] Grauert (H.), Remmert (R.).— Theory of Stein spaces, Translated from the German by Alan Huckleberry. Reprint of the 1979 translation, Classics in Mathematics. Springer-Verlag, Berlin (2004). | MR | Zbl

[51] Green (M.).— Holomorphic maps into complex projective spaces omitting hyperplanes, Trans. Amer. Math. Soc. 169, p. 89-103 (1972). | MR | Zbl

[52] Gromov (M.).— Oka’s principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2, p. 851-897 (1989). | MR | Zbl

[53] Gunning (R.C.), Rossi (H.).— Analytic functions of several complex variables, Prentice-Hall, Englewood Cliffs (1965); AMS Chelsea Publishing, Providence (2009). | MR | Zbl

[54] Hanysz (A.).— Oka properties of some hypersurface complements, Proc. Amer. Math. Soc., in press. arXiv:1111.6655

[55] Hanysz (A.).— Holomorphic flexibility properties of the space of cubic rational maps, arXiv:1211.0765

[56] Henkin (G.M.), Leiterer ( J.).— Theory of Functions on Complex Manifolds, Akademie-Verlag, Berlin (1984). | MR | Zbl

[57] Henkin (G.M.), Leiterer (J.).— The Oka-Grauert principle without induction over the basis dimension, Math. Ann. 311, p. 71-93 (1998). | MR | Zbl

[58] Hirschhorn (P.S.).— Model categories and their localizations, Mathematical Surveys and Monographs, vol. 99. Amer. Math. Soc., Providence (2003). | MR | Zbl

[59] Hörmander (L.).— L 2 estimates and existence theorems for the ¯ operator, Acta Math. 113, p. 89-152 (1965). | MR | Zbl

[60] Hörmander (L.).— An introduction to complex analysis in several variables, Third edn. North-Holland Mathematical Library, 7, North Holland Publishing Co., Amsterdam (1990). | MR | Zbl

[61] Hovey (M.).— Model categories. Mathematical Surveys and Monographs, 63, Amer. Math. Soc., Providence (1999). | MR | Zbl

[62] Ivarsson (B.), Kutzschebauch (F.).— Holomorphic factorization of mappings into SL n (), Ann. of Math. (2) 75, p. 45-69 (2012). | MR | Zbl

[63] Jardine (J.F.).— Intermediate model structures for simplicial presheaves, Canad. Math. Bull. 49, p. 407-413 (2006). | MR | Zbl

[64] Kaliman (S.), Kutzschebauch (F.).— On the present state of the Andersén-Lempert theory, In: Affine algebraic geometry, CRM Proc. Lecture Notes, vol. 54, p. 85-122. Am. Math. Soc., Providence (2011). | MR | Zbl

[65] Kobayashi (S.).— Hyperbolic Manifolds and Holomorphic Mappings, Marcel Dekker, New York (1970), Second edn.: World Scientific Publishing Co. Pte. Ltd., Hackensack (2005). | MR | Zbl

[66] Kobayashi (S.).— Hyperbolic complex spaces. Grundlehren der Mathematischen Wissenschaften, 318, Springer-Verlag, Berlin (1998). | MR | Zbl

[67] Kobayashi (S.), Ochiai (T.).— Meromorphic mappings onto compact complex spaces of general type, Invent. Math. 31, p. 7-16 (1975). | MR | Zbl

[68] Lárusson (F.).— Excision for simplicial sheaves on the Stein site and Gromov’s Oka principle, Internat. J. Math. 14, p. 191-209 (2003). | MR | Zbl

[69] Lárusson (F.).— Model structures and the Oka principle, J. Pure Appl. Algebra 192, p. 203-223 (2004). | MR | Zbl

[70] Lárusson (F.).— Mapping cylinders and the Oka principle, Indiana Univ. Math. J. 54, p. 1145-1159 (2005). | MR | Zbl

[71] Lárusson (F.).— Affine simplices in Oka manifolds, Documenta Math. 14, p. 691-697 (2009). | MR | Zbl

[72] Lárusson (F.).— Applications of a parametric Oka principle for liftings, In: Ebenfelt, P., Hungerbuehler, N., Kohn, J.J., Mok, N., Straube, E.J. (eds.) Complex Analysis, Trends in Mathematics, p. 205-212. Birkhäuser, Basel (2010). | MR | Zbl

[73] Lárusson (F.).— Deformations of Oka manifolds, Math. Z. 272, p. 1051-1058 (2012). | MR | Zbl

[74] Lárusson (F.).— Smooth toric varieties are Oka, arXiv:1107.3604

[75] Lárusson (F.), Ritter (T.).— Proper holomorphic immersions in homotopy classes of maps from finitely connected planar domains into × * , arXiv:1209.4430

[76] Luna (D.).— Slices étales, Sur les groupes algébriques, Bull. Soc. Math. France, Mémoire 33, p. 81-105 (1973). | Numdam | MR | Zbl

[77] Majcen (I.).— Embedding certain infinitely connected subsets of bordered Riemann surfaces properly into 2 , J. Geom. Anal. 19, p. 695-707 (2009). | MR | Zbl

[78] May (J.P.).— Simplicial objects in algebraic topology. Reprint of the 1967 original. Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL (1992). | MR | Zbl

[79] May (J.P.), Ponto (K.).— More concise algebraic topology. Localization, completion, and model categories, Chicago Lectures in Mathematics. University of Chicago Press, Chicago (2012). | MR | Zbl

[80] Narasimhan (R.).— Imbedding of holomorphically complete complex spaces, Amer. J. Math. 82, p. 917-934 (1960). | MR | Zbl

[81] Narasimhan (R.).— The Levi problem for complex spaces, II. Math. Ann. 146, p. 195-216 (1962). | MR | Zbl

[82] Oka (K.).— Sur les fonctions des plusieurs variables, III: Deuxième problème de Cousin, J. Sc. Hiroshima Univ. 9, p. 7-19 (1939).

[83] Quillen (D.).— Homotopical algebra, Lecture Notes in Mathematics, 43, Springer-Verlag, Berlin-New York (1967). | MR | Zbl

[84] Range (M.), Siu (Y.-T.).— Uniform estimates for the ¯-equation on domains with piecewise smooth strictly pseudoconvex boundary, Math. Ann. 206, p. 325-354 (1973). | MR | Zbl

[85] Ritter (T.).— A strong Oka principle for embeddings of some planar domains into × * , J. Geom. Anal. 23, p. 571-597 (2013). | MR | Zbl

[86] Ritter (T.).— Acyclic embeddings of open Riemann surfaces into new examples of elliptic manifolds, Proc. Amer. Math. Soc. 141, p. 597-603 (2013). | MR | Zbl

[87] Rosay (J.-P.), Rudin (W.).— Holomorphic maps from n to n , Trans. Amer. Math. Soc. 310, p. 47-86 (1988). | MR | Zbl

[88] Schürmann (J.).— Embeddings of Stein spaces into affine spaces of minimal dimension, Math. Ann. 307, p. 381-399 (1997). | MR | Zbl

[89] Siu (Y.-T.).— Techniques of extension of analytic objects, Lecture Notes in Pure and Applied Mathematics, 8, Marcel Dekker, Inc., New York (1974). | MR | Zbl

[90] Siu (Y.-T.).— Every Stein subvariety admits a Stein neighborhood, Invent. Math. 38, p. 89-100 (1976). | MR | Zbl

[91] Siu (Y.-T.).— Hyperbolicity of generic high-degree hypersurfaces in complex projective spaces, arXiv:1209.2723

[92] Siu (Y.-T.), Yeung (S.-K.).— Hyperbolicity of the complement of a generic smooth curve of high degree in the complex projective plane, Invent. Math. 124, p. 573-618 (1996). | MR | Zbl

[93] Snow (D.M.).— Reductive group actions on Stein spaces, Math. Ann. 259, p. 79-97 (1982). | MR | Zbl

[94] Stein (K.).— Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem, Math. Ann. 123, p. 201-222 (1951). | MR | Zbl

[95] Teleman (A.).— Donaldson theory on non-Kählerian surfaces and class VII surfaces with b 2 =1, Invent. Math. 162, p. 493-521 (2005). | MR | Zbl

[96] Toën (B.), Vezzosi (G.).— Homotopical algebraic geometry, I. Topos theory, Adv. Math. 193, p. 257-372 (2005). | MR | Zbl

[97] Voevodsky (V.).— A 1 -homotopy theory, Proceedings of the International Congress of Mathematicians, vol. I (Berlin, 1998), Documenta Math., extra vol. I, p. 579-604 (1998). | MR | Zbl

[98] Winkelmann (J.).— The Oka-principle for mappings between Riemann surfaces, Enseign. Math. 39, p. 143-151 (1993). | MR | Zbl

Cited by Sources: