Oka manifolds: From Oka to Stein and back

Franc Forstnerič

Annales de la faculté des sciences de Toulouse Mathématiques (2013)

  • Volume: 22, Issue: 4, page 747-809
  • ISSN: 0240-2963

Abstract

top
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].

How to cite

top

Forstnerič, Franc. "Oka manifolds: From Oka to Stein and back." Annales de la faculté des sciences de Toulouse Mathématiques 22.4 (2013): 747-809. <http://eudml.org/doc/275300>.

@article{Forstnerič2013,
abstract = {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].},
author = {Forstnerič, Franc},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {Oka-Grauert principle; Stein manifolds; Oka manifolds; Gromov's ellipticity},
language = {eng},
month = {6},
number = {4},
pages = {747-809},
publisher = {Université Paul Sabatier, Toulouse},
title = {Oka manifolds: From Oka to Stein and back},
url = {http://eudml.org/doc/275300},
volume = {22},
year = {2013},
}

TY - JOUR
AU - Forstnerič, Franc
TI - Oka manifolds: From Oka to Stein and back
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2013/6//
PB - Université Paul Sabatier, Toulouse
VL - 22
IS - 4
SP - 747
EP - 809
AB - 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 - eng
KW - Oka-Grauert principle; Stein manifolds; Oka manifolds; Gromov's ellipticity
UR - http://eudml.org/doc/275300
ER -

References

top
  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). Zbl1295.14057MR3039680
  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). Zbl1311.14059MR2986429
  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). Zbl0718.14023MR2030225
  6. Behnke (H.), Stein (K.).— Entwicklung analytischer Funktionen auf Riemannschen Flächen, Math. Ann. 120, p. 430-461 (1948). Zbl0038.23502MR29997
  7. Bishop (E.).— Mappings of partially analytic spaces, Amer. J. Math. 83, p. 209-242 (1961). Zbl0118.07701MR123732
  8. Brody (R.).— Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc. 235, p. 213-219 (1978). Zbl0416.32013MR470252
  9. Buzzard (G.T.).— Tame sets, dominating maps, and complex tori, Trans. Amer. Math. Soc. 355, p. 2557-2568 (2002). Zbl1026.32029MR1974003
  10. Buzzard (G.), Lu (S.S.Y.).— Algebraic surfaces holomorphically dominable by 2 , Invent. Math. 139, p. 617-659 (2000). Zbl0967.14025MR1738063
  11. Campana (F.).— Orbifolds, special varieties and classification theory, Ann. Inst. Fourier 54, 499-630 (2004). Zbl1062.14014MR2097416
  12. Campana (F.).— Orbifolds, special varieties and classification theory: an appendix, Ann. Inst. Fourier 54, p. 631-665 (2004). Zbl1062.14015MR2097417
  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). Zbl0121.30503MR98196
  15. Cox (D.), Little (J.), Schenck (H.).— Toric varieties, Graduate Studies in Mathematics, vol. 124. Amer. Math. Soc., Providence (2011). Zbl1223.14001MR2810322
  16. Drnovšek (B.), Forstnerič (F.).— Holomorphic curves in complex spaces, Duke Math. J. 139, p. 203-254 (2007). Zbl1133.32002
  17. Drnovšek (B.), Forstnerič (F.).— Approximation of holomorphic mappings on strongly pseudoconvex domains, Forum Math. 20, p. 817-840 (2008). Zbl1155.32008MR2445119
  18. Dwyer (W.G.), Spaliński (J.).— Homotopy theories and model categories, Handbook of algebraic topology, pp. 73-126. North-Holland, Amsterdam (1995). Zbl0869.55018MR1361887
  19. Eisenman (D.A.).— Intrinsic measures on complex manifolds and holomorphic mappings, Memoirs of the Amer. Math. Soc., 96. Amer. Math. Soc., Providence (1970). Zbl0197.05901MR259165
  20. Eliashberg (Y.).— Topological characterization of Stein manifolds of dimension &gt; 2 , Internat. J. Math. 1, p. 29-46 (1990). Zbl0699.58002MR1044658
  21. Eliashberg (Y.), Gromov (M.).— Nonsingular mappings of Stein manifolds, Funkcional. Anal. i Priložen. 5, p. 82-83 (1971). Zbl0234.32011MR301236
  22. Eliashberg (Y.), Gromov (M.).— Embeddings of Stein manifolds of dimension n into the affine space of dimension 3 n / 2 + 1 , Ann. Math. (2) 136, p. 123-135 (1992). Zbl0758.32012MR1173927
  23. Forster (O.).— Plongements des variétés de Stein, Comment. Math. Helv. 45, p. 170-184 (1970). Zbl0184.31403MR269880
  24. Forstnerič (F.).— The Oka principle for sections of subelliptic submersions, Math. Z. 241, p. 527-551 (2002). Zbl1023.32008MR1938703
  25. Forstnerič (F.).— Extending holomorphic mappings from subvarieties in Stein manifolds, Ann. Inst. Fourier 55, p. 733-751 (2005). Zbl1076.32003MR2149401
  26. Forstnerič (F.).— Runge approximation on convex sets implies the Oka property, Ann. Math. (2) 163, p. 689-707 (2006). Zbl1103.32004MR2199229
  27. Forstnerič (F.).— Manifolds of holomorphic mappings from strongly pseudoconvex domains, Asian J. Math. 11, p. 113-126 (2007). Zbl1131.58007MR2304585
  28. Forstnerič (F.).— Oka manifolds, C. R. Acad. Sci. Paris, Ser. I, 347, p. 1017-1020 (2009). Zbl1175.32005MR2554568
  29. Forstnerič (F.).— Oka maps, C. R. Acad. Sci. Paris, Ser. I, 348, p. 145-148 (2010). Zbl1201.32013MR2600066
  30. Forstnerič (F.).— The Oka principle for sections of stratified fiber bundles, Pure Appl. Math. Q. 6, p. 843-874 (2010). Zbl1216.32005MR2677316
  31. Forstnerič (F.).— Invariance of the parametric Oka property, Complex analysis, p. 125-144, Trends Math., Birkhäuser/Springer Basel AG, Basel (2010). Zbl1208.32012MR2885112
  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). Zbl1247.32001MR2975791
  33. Forstnerič (F.), Lárusson (F.).— Survey of Oka theory, New York J. Math. 17a, p. 1-28 (2011). Zbl1225.32019MR2782726
  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 Zbl1310.32013
  35. Forstnerič (F.), Prezelj (J.).— Oka’s principle for holomorphic fiber bundles with sprays, Math. Ann. 317, p. 117-154 (2000). Zbl0964.32017MR1760671
  36. Forstnerič (F.), Prezelj (J.).— Extending holomorphic sections from complex subvarieties, Math. Z. 236, p. 43-68 (2001). Zbl0968.32005MR1812449
  37. Forstnerič (F.), Prezelj (J.).— Oka’s principle for holomorphic submersions with sprays, Math. Ann. 322, p. 633-666 (2002). Zbl1011.32006MR1905108
  38. Forstnerič (F.), Ritter (T.).— Oka properties of ball complements. arXiv:1303.2239 Zbl1297.32010
  39. Forstnerič (F.), Slapar (M.).— Stein structures and holomorphic mappings, Math. Z. 256, p. 615-646 (2007). Zbl1129.32013MR2299574
  40. Forstnerič (F.), Wold (E.F.).— Bordered Riemann surfaces in 2 , J. Math. Pures Appl. 91, p. 100-114 (2009). Zbl1157.32010MR2487902
  41. Forstnerič (F.), Wold (E.F.).— Fibrations and Stein neighborhoods, Proc. Amer. Math. Soc. 138, p. 2037-2042 (2010). Zbl1192.32008MR2596039
  42. Forstnerič (F.), Wold (E.F.).— Embeddings of infinitely connected planar domains into 2 , Anal. PDE, in press. arXiv:1110.5354 Zbl1277.32009MR3071396
  43. Goerss (P.G.), Jardine (J.F.).— Simplicial homotopy theory, Progress in Mathematics, 174. Birkhäuser, Basel (1999). Zbl0949.55001MR1711612
  44. Gompf (R.E.).— Handlebody construction of Stein surfaces, Ann. Math. (2) 148, p. 619-693 (1998). Zbl0919.57012MR1668563
  45. Gompf (R.E.).— Stein surfaces as open subsets of 2 , J. Symplectic Geom. 3, p. 565-587 (2005). Zbl1118.32011MR2235855
  46. Grauert (H.).— Approximationssätze für holomorphe Funktionen mit Werten in komplexen Räumen, Math. Ann. 133, p. 139-159 (1957). Zbl0080.29201MR98197
  47. Grauert (H.).— Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen, Math. Ann. 133, p. 450-472 (1957). Zbl0080.29202MR98198
  48. Grauert (H.).— Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann. 135, p. 263-273 (1958). Zbl0081.07401MR98199
  49. Grauert (H.).— On Levi’s problem and the embedding of real-analytic manifolds, Ann. Math. (2) 68, p. 460-472 (1958). Zbl0108.07804MR98847
  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). Zbl1137.32001MR2029201
  51. Green (M.).— Holomorphic maps into complex projective spaces omitting hyperplanes, Trans. Amer. Math. Soc. 169, p. 89-103 (1972). Zbl0256.32015MR308433
  52. Gromov (M.).— Oka’s principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2, p. 851-897 (1989). Zbl0686.32012MR1001851
  53. Gunning (R.C.), Rossi (H.).— Analytic functions of several complex variables, Prentice-Hall, Englewood Cliffs (1965); AMS Chelsea Publishing, Providence (2009). Zbl1204.01045MR2568219
  54. Hanysz (A.).— Oka properties of some hypersurface complements, Proc. Amer. Math. Soc., in press. arXiv:1111.6655 Zbl1297.32015
  55. Hanysz (A.).— Holomorphic flexibility properties of the space of cubic rational maps, arXiv:1211.0765 Zbl06476824
  56. Henkin (G.M.), Leiterer ( J.).— Theory of Functions on Complex Manifolds, Akademie-Verlag, Berlin (1984). Zbl0573.32001MR774049
  57. Henkin (G.M.), Leiterer (J.).— The Oka-Grauert principle without induction over the basis dimension, Math. Ann. 311, p. 71-93 (1998). Zbl0955.32019MR1624267
  58. Hirschhorn (P.S.).— Model categories and their localizations, Mathematical Surveys and Monographs, vol. 99. Amer. Math. Soc., Providence (2003). Zbl1017.55001MR1944041
  59. Hörmander (L.).— L 2 estimates and existence theorems for the ¯ operator, Acta Math. 113, p. 89-152 (1965). Zbl0158.11002MR179443
  60. Hörmander (L.).— An introduction to complex analysis in several variables, Third edn. North-Holland Mathematical Library, 7, North Holland Publishing Co., Amsterdam (1990). Zbl0271.32001MR1045639
  61. Hovey (M.).— Model categories. Mathematical Surveys and Monographs, 63, Amer. Math. Soc., Providence (1999). Zbl0909.55001MR1650134
  62. Ivarsson (B.), Kutzschebauch (F.).— Holomorphic factorization of mappings into S L n ( ) , Ann. of Math. (2) 75, p. 45-69 (2012). Zbl1243.32007MR2874639
  63. Jardine (J.F.).— Intermediate model structures for simplicial presheaves, Canad. Math. Bull. 49, p. 407-413 (2006). Zbl1107.18007MR2252262
  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). Zbl1266.32028MR2768636
  65. Kobayashi (S.).— Hyperbolic Manifolds and Holomorphic Mappings, Marcel Dekker, New York (1970), Second edn.: World Scientific Publishing Co. Pte. Ltd., Hackensack (2005). Zbl1084.32018MR277770
  66. Kobayashi (S.).— Hyperbolic complex spaces. Grundlehren der Mathematischen Wissenschaften, 318, Springer-Verlag, Berlin (1998). Zbl0917.32019MR1635983
  67. Kobayashi (S.), Ochiai (T.).— Meromorphic mappings onto compact complex spaces of general type, Invent. Math. 31, p. 7-16 (1975). Zbl0331.32020MR402127
  68. Lárusson (F.).— Excision for simplicial sheaves on the Stein site and Gromov’s Oka principle, Internat. J. Math. 14, p. 191-209 (2003). Zbl1078.32017MR1966772
  69. Lárusson (F.).— Model structures and the Oka principle, J. Pure Appl. Algebra 192, p. 203-223 (2004). Zbl1052.32020MR2067196
  70. Lárusson (F.).— Mapping cylinders and the Oka principle, Indiana Univ. Math. J. 54, p. 1145-1159 (2005). Zbl1085.32011MR2164421
  71. Lárusson (F.).— Affine simplices in Oka manifolds, Documenta Math. 14, p. 691-697 (2009). Zbl1200.32016MR2578806
  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). Zbl1208.32025MR2885117
  73. Lárusson (F.).— Deformations of Oka manifolds, Math. Z. 272, p. 1051-1058 (2012). Zbl1262.32015MR2995155
  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). Zbl0286.14014MR342523
  77. Majcen (I.).— Embedding certain infinitely connected subsets of bordered Riemann surfaces properly into 2 , J. Geom. Anal. 19, p. 695-707 (2009). Zbl1172.32300MR2496573
  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). Zbl0769.55001MR1206474
  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). Zbl1249.55001MR2884233
  80. Narasimhan (R.).— Imbedding of holomorphically complete complex spaces, Amer. J. Math. 82, p. 917-934 (1960). Zbl0104.05402MR148942
  81. Narasimhan (R.).— The Levi problem for complex spaces, II. Math. Ann. 146, p. 195-216 (1962). Zbl0131.30801MR182747
  82. Oka (K.).— Sur les fonctions des plusieurs variables, III: Deuxième problème de Cousin, J. Sc. Hiroshima Univ. 9, p. 7-19 (1939). Zbl0020.24002
  83. Quillen (D.).— Homotopical algebra, Lecture Notes in Mathematics, 43, Springer-Verlag, Berlin-New York (1967). Zbl0168.20903MR223432
  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). Zbl0248.32015MR338450
  85. Ritter (T.).— A strong Oka principle for embeddings of some planar domains into × * , J. Geom. Anal. 23, p. 571-597 (2013). Zbl1269.32012MR3023850
  86. Ritter (T.).— Acyclic embeddings of open Riemann surfaces into new examples of elliptic manifolds, Proc. Amer. Math. Soc. 141, p. 597-603 (2013). Zbl1278.32015MR2996964
  87. Rosay (J.-P.), Rudin (W.).— Holomorphic maps from n to n , Trans. Amer. Math. Soc. 310, p. 47-86 (1988). Zbl0708.58003MR929658
  88. Schürmann (J.).— Embeddings of Stein spaces into affine spaces of minimal dimension, Math. Ann. 307, p. 381-399 (1997). Zbl0881.32007MR1437045
  89. Siu (Y.-T.).— Techniques of extension of analytic objects, Lecture Notes in Pure and Applied Mathematics, 8, Marcel Dekker, Inc., New York (1974). Zbl0294.32007MR361154
  90. Siu (Y.-T.).— Every Stein subvariety admits a Stein neighborhood, Invent. Math. 38, p. 89-100 (1976). Zbl0343.32014MR435447
  91. Siu (Y.-T.).— Hyperbolicity of generic high-degree hypersurfaces in complex projective spaces, arXiv:1209.2723 Zbl1333.32020
  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). Zbl0856.32017MR1369429
  93. Snow (D.M.).— Reductive group actions on Stein spaces, Math. Ann. 259, p. 79-97 (1982). Zbl0509.32021MR656653
  94. Stein (K.).— Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem, Math. Ann. 123, p. 201-222 (1951). Zbl0042.08703MR43219
  95. Teleman (A.).— Donaldson theory on non-Kählerian surfaces and class VII surfaces with b 2 = 1 , Invent. Math. 162, p. 493-521 (2005). Zbl1093.32006MR2198220
  96. Toën (B.), Vezzosi (G.).— Homotopical algebraic geometry, I. Topos theory, Adv. Math. 193, p. 257-372 (2005). Zbl1120.14012MR2137288
  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). Zbl0907.19002MR1648048
  98. Winkelmann (J.).— The Oka-principle for mappings between Riemann surfaces, Enseign. Math. 39, p. 143-151 (1993). Zbl0783.30031MR1225261

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.