Abstract
In this preliminary chapter we review the basic notions concerning complex manifolds, complex spaces with singularities, holomorphic mappings, holomorphic fibre bundles and especially vector bundles, the tangent and cotangent bundle of a complex manifold, differential forms, de Rham and Dolbeault cohomology, plurisubharmonic functions, the Levi form, vector fields and their flows. We also recall a basic form of Gromov’s homotopy principle for first order partial differential relations that are ample in the coordinate directions.
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Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications, 2nd edn. Applied Mathematical Sciences, vol. 75. Springer, New York (1988)
Adams, J.F.: Lectures on Lie Groups. W. A. Benjamin, New York (1969)
Ahern, P., Flores, M., Rosay, J.-P.: On \({\mathbb {R}}^{+}\) and ℂ complete holomorphic vector fields. Proc. Am. Math. Soc. 128(10), 3107–3113 (2000)
Baouendi, M.S., Ebenfelt, P., Rothschild, L.P.: Real Submanifolds in Complex Space and Their Mappings. Princeton Mathematical Series, vol. 47. Princeton University Press, Princeton (1999)
Behnke, H., Stein, K.: Modifikation komplexer Mannigfaltigkeiten und Riemannscher Gebiete. Math. Ann. 124, 1–16 (1951)
Bröcker, T., tom Dieck, T.: Representations of Compact Lie Groups. Graduate Texts in Mathematics, vol. 98. Springer, New York (1995). Corrected reprint of the 1985 translation
Brody, R.: Compact manifolds and hyperbolicity. Trans. Am. Math. Soc. 235, 213–219 (1978)
Cartan, H.: Sur les fonctions de plusieurs variables complexes: les espaces analytiques. In: Proc. Internat. Congress Math., vol. 1958, pp. 33–52. Cambridge Univ. Press, New York (1960)
Chirka, E.: Complex Analytic Sets. Kluwer Academic Publishers, Dordrecht (1989). Translated from the Russian by R.A.M. Hoksbergen
Demailly, J.-P.: Complex analytic and differential geometry. https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf
Donaldson, S.: Riemann Surfaces. Oxford Graduate Texts in Mathematics, vol. 22. Oxford University Press, Oxford (2011)
Eisenman, D.A.: Intrinsic Measures on Complex Manifolds and Holomorphic Mappings. Memoirs of the Am. Math. Soc., vol. 96. Am. Math. Soc., Providence (1970)
Eliashberg, Y., Mishachev, N.: Introduction to the \(h\)-Principle. Graduate Studies in Mathematics, vol. 48. Am. Math. Soc., Providence (2002)
Fischer, G.: Complex Analytic Geometry. Lecture Notes in Math., vol. 538. Springer, Berlin (1976)
Fischer, W., Grauert, H.: Lokal-triviale Familien kompakter komplexer Mannigfaltigkeiten. Nachr. Akad. Wiss. Gött. Math.-Phys. Kl., 2B 1965, 89–94 (1965)
Forster, O.: Lectures on Riemann Surfaces. Graduate Texts in Mathematics, vol. 81. Springer, New York (1991). Translated from the 1977 German original by Bruce Gilligan. Reprint of the 1981 English translation
Forstnerič, F.: Actions of \((\mathbb {R},+)\) and \((\mathbb {C},+)\) on complex manifolds. Math. Z. 223(1), 123–153 (1996)
Golubitsky, M., Guillemin, V.: Stable Mappings and Their Singularities. Graduate Texts in Mathematics, vol. 14. Springer, New York (1973)
Grauert, H.: Charakterisierung der holomorph vollständigen komplexen Räume. Math. Ann. 129, 233–259 (1955)
Grauert, H.: On Levi’s problem and the imbedding of real-analytic manifolds. Ann. Math. (2) 68, 460–472 (1958)
Grauert, H., Remmert, R.: Komplexe Räume. Math. Ann. 136, 245–318 (1958)
Grauert, H., Remmert, R.: Theory of Stein Spaces. Grundlehren Math. Wiss., vol. 236. Springer, Berlin (1979). Translated from the German by Alan Huckleberry
Grauert, H., Remmert, R.: Coherent Analytic Sheaves. Grundlehren Math. Wiss., vol. 265. Springer, Berlin (1984)
Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley Classics Library. John Wiley & Sons, New York (1994). Reprint of the 1978 original
Gromov, M.: Partial Differential Relations. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 9. Springer, Berlin (1986)
Gromov, M.L.: Stable mappings of foliations into manifolds. Izv. Akad. Nauk SSSR, Ser. Mat. 33, 707–734 (1969)
Gromov, M.L.: Convex integration of differential relations. I. Izv. Akad. Nauk SSSR, Ser. Mat. 37, 329–343 (1973)
Gunning, R.C., Rossi, H.: Analytic Functions of Several Complex Variables. AMS Chelsea Publishing, Providence (2009). Reprint of the 1965 original
Hirsch, M.W.: Immersions of manifolds. Trans. Am. Math. Soc. 93, 242–276 (1959)
Hörmander, L.: An Introduction to Complex Analysis in Several Variables, 3rd edn. North-Holland Mathematical Library, vol. 7. North-Holland Publishing Co., Amsterdam (1990)
Hörmander, L.: Notions of Convexity. Progress in Mathematics, vol. 127. Birkhäuser Boston, Boston (1994)
Josefson, B.: On the equivalence between locally polar and globally polar sets for plurisubharmonic functions on \({\mathbb {C}}^{n}\). Ark. Mat. 16(1), 109–115 (1978)
Jouanolou, J.P.: Une suite exacte de Mayer-Vietoris en \(K\)-théorie algébrique. In: Algebraic \(K\)-Theory, I: Higher \(K\)-Theories, Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972. Lecture Notes in Math., vol. 341, pp. 293–316. Springer, Berlin (1973)
Kaliman, S.: Some facts about Eisenman intrinsic measures. Complex Var. Theory Appl. 27(2), 163–173 (1995)
Kelley, J.L.: General Topology. Graduate Texts in Mathematics, vol. 27. Springer, New York (1975). Reprint of the 1955 edition, Van Nostrand, Toronto, Ont.
Klimek, M.: Pluripotential Theory. London Mathematical Society Monographs. New Series, vol. 6. The Clarendon Press/Oxford University Press, New York (1991)
Kobayashi, S.: Hyperbolic Manifolds and Holomorphic Mappings. Pure and Applied Mathematics, vol. 2. Marcel Dekker, New York (1970)
Kobayashi, S.: Intrinsic distances, measures and geometric function theory. Bull. Am. Math. Soc. 82(3), 357–416 (1976)
Kobayashi, S., Ochiai, T.: Meromorphic mappings onto compact complex spaces of general type. Invent. Math. 31(1), 7–16 (1975)
Koebe, P.: Über die Uniformisierung beliebiger analytischer Kurven. Zweite Mitteilung. Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1907, 633–669 (1907)
Lelong, P.: Définition des fonctions plurisousharmoniques. C. R. Acad. Sci. Paris 215, 398–400 (1942)
Lelong, P.: Fonctions plurisousharmoniques et formes différentielles positives. Gordon & Breach, Paris (1968)
McDuff, D., Salamon, D.: \(J\)-Holomorphic Curves and Symplectic Topology, 2nd edn. Amer. Math. Soc. Colloquium Publications, vol. 52. Am. Math. Soc., Providence (2012)
Narasimhan, R.: Analysis on Real and Complex Manifolds. North-Holland Mathematical Library, vol. 35. North-Holland Publishing Co., Amsterdam (1985). Reprint of the 1973 edition
Newlander, A., Nirenberg, L.: Complex analytic coordinates in almost complex manifolds. Ann. Math. (2) 65, 391–404 (1957)
Oka, K.: Collected Papers. Springer, Berlin (1984). Translated from the French by R. Narasimhan. With commentaries by H. Cartan. Edited by R. Remmert
Phillips, A.: Submersions of open manifolds. Topology 6, 171–206 (1967)
Poincaré, H.: Sur l’uniformisation des fonctions analytiques. Acta Math. 31, 1–64 (1907)
Remmert, R.: Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes. C. R. Acad. Sci. Paris 243, 118–121 (1956)
Remmert, R.: Holomorphe und meromorphe Abbildungen komplexer Räume. Math. Ann. 133, 328–370 (1957)
Rudin, W.: Function Theory in the Unit Ball of \(\mathbb {C}^{n}\). Classics in Mathematics. Springer, Berlin (2008). Reprint of the 1980 edition
Serre, J-P.: Géométrie algébrique et géométrie analytique. Ann. Inst. Fourier 6, 1–42 (1955/1956)
Smale, S.: The classification of immersions of spheres in Euclidean spaces. Ann. Math. (2) 69, 327–344 (1959)
Spring, D.: Convex Integration Theory. Solutions to the \(h\)-Principle in Geometry and Topology. Monographs in Mathematics, vol. 92. Birkhäuser, Basel (1998)
Stein, K.: Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem. Math. Ann. 123, 201–222 (1951)
Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics, vol. 94. Springer, New York (1983). Corrected reprint of the 1971 edition
Wells, R.O. Jr.: Differential Analysis on Complex Manifolds, 3rd edn. Graduate Texts in Mathematics, vol. 65. Springer, New York (2008)
Whitney, H.: Local properties of analytic varieties. In: Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), pp. 205–244. Princeton University Press, Princeton (1965)
Whitney, H., Bruhat, F.: Quelques propriétés fondamentales des ensembles analytiques-réels. Comment. Math. Helv. 33, 132–160 (1959)
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Forstnerič, F. (2017). Preliminaries. In: Stein Manifolds and Holomorphic Mappings. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 56. Springer, Cham. https://doi.org/10.1007/978-3-319-61058-0_1
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