Riemann Surfaces
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Bulletin De La S
BULLETIN DE LA S. M. F. NGAIMING MOK An embedding theorem of complete Kähler manifolds of positive bisectional curvature onto affine algebraic varieties Bulletin de la S. M. F., tome 112 (1984), p. 197-258 <http://www.numdam.org/item?id=BSMF_1984__112__197_0> © Bulletin de la S. M. F., 1984, tous droits réservés. L’accès aux archives de la revue « Bulletin de la S. M. F. » (http: //smf.emath.fr/Publications/Bulletin/Presentation.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/ conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Bull. Soc. math. France, 112, 1984, p. 197-258. AN EMBEDDING THEOREM OF COMPLETE KAHLER MANIFOLDS OF POSITIVE BISECTIONAL CURVATURE ONTO AFFINE ALGEBRAIC VARIETIES BY NGAIMING MOK (*) R£SUM£. — Nous prouvons qu'une variete complete kahleriennc non compacte X de courbure biscctionnclle positive satisfaisant qudques conditions quantitatives geometriques est biholomorphiqucment isomorphe a une varictc affine algebrique. Si X est une surface complcxe de courbure riemannienne positive satisfaisant les memes conditions quantitatives, nous demontrons que X est en fait biholomorphiquement isomorphe a C2. ABSTRACT. - We prove that a complete noncompact Kahler manifold X of positive bisectional curvature satisfying suitable growth conditions can be biholomorphicaUy embed- ded onto an affine algebraic variety. In case X is a complex surface of positive Riemannian sectional curvature satisfying the same growth conditions, we show that X is biholomorphic toC2. -
Oka Manifolds: from Oka to Stein and Back
ANNALES DE LA FACULTÉ DES SCIENCES Mathématiques FRANC FORSTNERICˇ Oka manifolds: From Oka to Stein and back Tome XXII, no 4 (2013), p. 747-809. <http://afst.cedram.org/item?id=AFST_2013_6_22_4_747_0> © Université Paul Sabatier, Toulouse, 2013, tous droits réservés. L’accès aux articles de la revue « Annales de la faculté des sci- ences de Toulouse Mathématiques » (http://afst.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://afst.cedram. org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement personnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Annales de la Facult´e des Sciences de Toulouse Vol. XXII, n◦ 4, 2013 pp. 747–809 Oka manifolds: From Oka to Stein and back Franc Forstnericˇ(1) ABSTRACT. — Oka theory has its roots in the classical Oka-Grauert prin- ciple whose main result is Grauert’s classification of principal holomorphic fiber bundles over Stein spaces. Modern Oka theory concerns holomor- phic 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 de- scribe equivalent characterizations of Oka manifolds, the functorial prop- erties of this class, and geometric sufficient conditions for being Oka, the most important of which is Gromov’s ellipticity. -
![Arxiv:1003.6025V1 [Math.HO] 31 Mar 2010 Karl Stein (1913-2000)](https://docslib.b-cdn.net/cover/1257/arxiv-1003-6025v1-math-ho-31-mar-2010-karl-stein-1913-2000-651257.webp)
Arxiv:1003.6025V1 [Math.HO] 31 Mar 2010 Karl Stein (1913-2000)
Karl Stein (1913-2000) Alan Huckleberry Karl Stein was born on the first of January 1913 in Hamm in Westfalen, grew up there, received his Abitur in 1932 and immediately thereafter be- gan his studies in M¨unster. Just four years later, under the guidance of Heinrich Behnke, he passed his Staatsexam, received his promotion and became Behnke’s assistant. Throughout his life, complex analysis, primarily in higher dimensions (“mehrere Ver¨anderliche”), was the leitmotif of Stein’s mathematics. As a fresh Ph.D. in M¨unster in 1936, under the leadership of the master Behnke, he had already been exposed to the fascinating developments in this area. The brilliant young Peter Thullen was proving fundamental theorems, Henri Cartan had visited M¨unster, and Behnke and Thullen had just writ- ten the book on the subject. It must have been clear to Stein that this was the way to go. Indeed it was! The amazing phenomenon of analytic continuation in higher dimensions had already been exemplified more than 20 years be- fore in the works of Hartogs and E. E. Levi. Thullen’s recent work had gone much further. In the opposite direction, Cartan and Thullen had arXiv:1003.6025v1 [math.HO] 31 Mar 2010 proved their characterization of domains in Cn which admit a holomor- phic function which can not be continued any further. Behnke himself was also an active participant in mathematics research, always bringing new ideas to M¨unster. This was indeed an exciting time for the young researcher, Karl Stein. Even though the pest of the Third Reich was already invading academia, Behnke kept things going for as long as possible. -
From Riemann Surfaces to Complex Spaces Reinhold Remmert∗
From Riemann Surfaces to Complex Spaces Reinhold Remmert∗ We must always have old memories and young hopes Abstract This paper analyzes the development of the theory of Riemann surfaces and complex spaces, with emphasis on the work of Rie- mann, Klein and Poincar´e in the nineteenth century and on the work of Behnke-Stein and Cartan-Serre in the middle of this cen- tury. R´esum´e Cet article analyse le d´eveloppement de la th´eorie des surfaces de Riemann et des espaces analytiques complexes, en ´etudiant notamment les travaux de Riemann, Klein et Poincar´eauXIXe si`ecle et ceux de Behnke-Stein et Cartan-Serre au milieu de ce si`ecle. Table of Contents 1. Riemann surfaces from 1851 to 1912 1.1. Georg Friedrich Bernhard Riemann and the covering principle 1.1∗. Riemann’s doctorate 1.2. Christian Felix Klein and the atlas principle 1.3. Karl Theodor Wilhelm Weierstrass and analytic configurations AMS 1991 Mathematics Subject Classification: 01A55, 01A60, 30-03, 32-03 ∗Westf¨alische Wilhelms–Universit¨at, Mathematisches Institut, D–48149 Munster,¨ Deutschland This expos´e is an enlarged version of my lecture given in Nice. Gratias ago to J.-P. Serre for critical comments. A detailed exposition of sections 1 and 2 will appear elsewhere. SOCIET´ EMATH´ EMATIQUE´ DE FRANCE 1998 204 R. REMMERT 1.4. The feud between G¨ottingen and Berlin 1.5. Jules Henri Poincar´e and automorphic functions 1.6. The competition between Klein and Poincar´e 1.7. Georg Ferdinand Ludwig Philipp Cantor and countability of the topology 1.8. -
The Cousin Problems and the Emergence of the Sheaf Concept Author(S): Renaud Chorlay Source: Archive for History of Exact Sciences, Vol
From Problems to Structures: the Cousin Problems and the Emergence of the Sheaf Concept Author(s): Renaud Chorlay Source: Archive for History of Exact Sciences, Vol. 64, No. 1 (January 2010), pp. 1-73 Published by: Springer Stable URL: http://www.jstor.org/stable/41342411 Accessed: 23-05-2017 00:50 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms Springer is collaborating with JSTOR to digitize, preserve and extend access to Archive for History of Exact Sciences This content downloaded from 132.248.9.8 on Tue, 23 May 2017 00:50:12 UTC All use subject to http://about.jstor.org/terms Arch. Hist. Exact Sci. (2010) 64:1-73 DOI 10. 1007/s00407-009-0052-3 From Problems to Structures: the Cousin Problems and the Emergence of the Sheaf Concept Renaud Chorlay Received: 19 November 2008 / Published online: 30 July 2009 © Springer- Verlag 2009 Abstract Historical work on the emergence of sheaf theory has mainly concen- trated on the topological origins of sheaf cohomology in the period from 1945 to 1950 and on subsequent developments. However, a shift of emphasis both in time-scale and disciplinary context can help gain new insight into the emergence of the sheaf concept. -
Removal of Singularites for Stein Manifolds
Removal of Singularities for Stein Manifolds Undergraduate Honors Thesis in Mathematics Luis Kumanduri Abstract We adapt the technique of removal of singularities to the holomorphic setting and prove a general flexibility result for holomorphic vector bundles over Stein manifolds. If D : V ! W is an elliptic differential operator between holomorphic vector bundles over a Stein Manifold, then a q-tuple (θ1; : : : ; θq) of holomorphic sections generating W may be deformed to an exact holomorphic q-tuple (Dφ1; : : : ; Dφq) generating W . We also prove a parametric version of this theorem with holomorphic dependence on a Stein parameter X and obtain a 1-parametric h-principle. The parametric h- principle works relative to closed complex analytic subsets A of X. As corollaries we will obtain h-principles for holomorphic immersions and free maps of Stein Manifolds. Contents 1 Introduction 2 2 Jets and the h-principle 4 2.1 Jet Bundles and Transversality . .4 2.2 Differential Relations and the h-principle . .6 3 Geometry of Several Complex Variables 9 3.1 Basics of Several Complex Variables . .9 3.2 Stein Manifolds . 12 3.3 Cartan's Theorems . 16 4 Holomorphic Transversality 20 5 Main Results 24 5.1 Proof of Main Theorem . 24 5.2 Parametric h-principle . 29 5.3 Applications . 31 1 1 Introduction Problems in smooth geometry are often subject to a partial differential inequality or rela- tion that satisfies an h-principle. For any differential relation, there is a notion of a formal solution where the derivatives are replaced with algebraic relations. In situations where formal solutions can be deformed into actual solutions, we say that a problem satisfies an h-principle. -
Complex Analysis and Complex Geometry
Complex Analysis and Complex Geometry Dan Coman (Syracuse University) Finnur L´arusson (University of Adelaide) Stefan Nemirovski (Steklov Institute) Rasul Shafikov (University of Western Ontario) May 31 – June 5, 2009 1 Overview of the Field Complex analysis and complex geometry can be viewed as two aspects of the same subject. The two are inseparable, as most work in the area involves interplay between analysis and geometry. The fundamental objects of the theory are complex manifolds and, more generally, complex spaces, holomorphic functions on them, and holomorphic maps between them. Holomorphic functions can be defined in three equivalent ways as complex-differentiable functions, as sums of complex power series, and as solutions of the homogeneous Cauchy-Riemann equation. The threefold nature of differentiability over the complex numbers gives complex analysis its distinctive character and is the ultimate reason why it is linked to so many areas of mathematics. Plurisubharmonic functions are not as well known to nonexperts as holomorphic functions. They were first explicitly defined in the 1940s, but they had already appeared in attempts to geometrically describe domains of holomorphy at the very beginning of several complex variables in the first decade of the 20th century. Since the 1960s, one of their most important roles has been as weights in a priori estimates for solving the Cauchy-Riemann equation. They are intimately related to the complex Monge-Amp`ere equation, the second partial differential equation of complex analysis. There is also a potential-theoretic aspect to plurisubharmonic functions, which is the subject of pluripotential theory. In the early decades of the modern era of the subject, from the 1940s into the 1970s, the notion of a complex space took shape and the geometry of analytic varieties and holomorphic maps was developed. -
Complex Geometry
Complex Geometry Weiyi Zhang Mathematics Institute, University of Warwick March 13, 2020 2 Contents 1 Course Description 5 2 Structures 7 2.1 Complex manifolds . .7 2.1.1 Examples of Complex manifolds . .8 2.2 Vector bundles and the tangent bundle . 10 2.2.1 Holomorphic vector bundles . 15 2.3 Almost complex structure and integrability . 17 2.4 K¨ahlermanifolds . 23 2.4.1 Examples. 24 2.4.2 Blowups . 25 3 Geometry 27 3.1 Hermitian Vector Bundles . 27 3.2 (Almost) K¨ahleridentities . 31 3.3 Hodge theorem . 37 3.3.1 @@¯-Lemma . 42 3.3.2 Proof of Hodge theorem . 44 3.4 Divisors and line bundles . 48 3.5 Lefschetz hyperplane theorem . 52 3.6 Kodaira embedding theorem . 55 3.6.1 Proof of Newlander-Nirenberg theorem . 61 3.7 Kodaira dimension and classification . 62 3.7.1 Complex dimension one . 63 3.7.2 Complex Surfaces . 64 3.7.3 BMY line . 66 3.8 Hirzebruch-Riemann-Roch Theorem . 66 3.9 K¨ahler-Einsteinmetrics . 68 3 4 CONTENTS Chapter 1 Course Description Instructor: Weiyi Zhang Email: [email protected] Webpage: http://homepages.warwick.ac.uk/staff/Weiyi.Zhang/ Lecture time/room: Wednesday 9am - 10am MS.B3.03 Friday 9am - 11am MA B3.01 Reference books: • P. Griffiths, J. Harris: Principles of Algebraic Geometry, Wiley, 1978. • D. Huybrechts: Complex geometry: An Introduction, Universitext, Springer, 2005. • K. Kodaira: Complex manifolds and deformation of complex struc- tures, Springer, 1986. • R.O. Wells: Differential Analysis on Complex Manifolds, Springer- Verlag, 1980. • C. Voisin: Hodge Theory and Complex Algebraic Geometry I/II, Cam- bridge University Press, 2002. -
Riemann Surfaces Lecture Notes
Introduction to Riemann Surfaces Lecture Notes Armin Rainer 0 0 1 dim H (X; L−D) − dim H (X; LD) = 1 − g − deg D June 17, 2018 PREFACE i Preface These are lecture notes for the course Riemann surfaces held in Vienna in Spring 2018 (three semester hours). The presentation is primarily based on the book [4] which is followed quite closely. Also [13] had some influence. Apart from some familiarity with basic complex analysis, general topology, and basic algebra no other prerequisites are demanded. All necessary tools will be developed when needed. Riemann surfaces were originally conceived in complex analysis in order to deal with multivalued functions. The analytic continuation of a given holomorphic function element along different paths leads in general to different branches of that function. Riemann replaced the domain of the function by a multiple-sheeted covering of the complex plane to get a single valued function on the covering space. Abstract Riemann surfaces are by definition connected complex one- dimensional manifolds. They are the natural domains of definitions of holomorphic functions in one variable. In chapter 1 we introduce Riemann surfaces and discuss basic properties. We develop the fundamentals of the theory of topological covering spaces including the fundamental group, the universal covering, and deck transformations. It will turn out that non-constant holomorphic maps between Riemann surfaces are covering maps, possibly with branch points. In chapter 2 we get acquainted with the language of sheaves. It proves very useful in the construction of Riemann surfaces which arise from the analytic con- tinuation of germs of holomorphic functions. -
Some Recent Results in Complex Manifold Theory Related to Vanishing Theorems for the Semipositive Case
Some Recent Results in Complex Manifold Theory Related to Vanishing Theorems for the Semipositive Case Yum-Tong Siu Department of Mathematics Harvard University Cambridge, MA 02138, U.S.A. To put this survey in the proper perspective, let me first make some rather general remarks. To study complex manifolds or in general complex spaces, one works with holomorphic objects like holomorphic maps, holomorphic functions, holomorphic vector bundles and their holomorphic sections. One has to construct such objects. For example, to prove that a complex manifold is biholomorphic to £n one tries to produce n suitable holomorphic functions. To prove that a complex manifold is biholomorphic to Pn' one tries to produce n+l good holomorphic sections of a suitable holomorphic line bundle. To prove that two complex manifolds are biholomorphic, one tries to produce a biholomorphic map. How does one produce such holomorphic objects? So far we have mainly the following methods: (i) The method of constructing harmonic objects first and then getting holomorphic objects from them. An example is the use of the Dirichlet principle to construct harmonic functions on open Riemann surfaces and then obtaining holomorphic functions from them. Examples of the construction of harmonic objects are the results of Eel ls-Sampson [i0] and Sacns-Uhlenbeck [40] on the existence of harmonic maps. However, unlike the one-dimensional case, in the higher-dimensional case the gap between a harmonic object and a holomorphic object is very wide and, except for some special cases, is impossible to bridge. (2) The method of using the vanishing theorem of Kodaira to construct holomorphic sections of high powers of positive line bundles [24]. -
A Beginner's Guide to Holomorphic Manifolds
This is page Printer: Opaque this A Beginner’s Guide to Holomorphic Manifolds Andrew D. Hwang This is page i Printer: Opaque this Preface This book constitutes notes from a one-semester graduate course called “Complex Manifolds and Hermitian Differential Geometry” given during the Spring Term, 1997, at the University of Toronto. Its aim is not to give a thorough treatment of the algebraic and differential geometry of holomorphic manifolds, but to introduce material of current interest as quickly and concretely as possible with a minimum of prerequisites. There are several excellent references available for the reader who wishes to see subjects in more depth. The coverage includes standard introductory analytic material on holo- morphic manifolds, sheaf cohomology and deformation theory, differential geometry of vector bundles (Hodge theory, and Chern classes via curva- ture), and some applications to the topology and projective embeddability of K¨ahlerian manifolds. The final chapter is a short survey of extremal K¨ahler metrics and related topics, emphasizing the geometric and “soft” analytic aspects. There is a large number of exercises, particularly for a book at this level. The exercises introduce several specific but colorful ex- amples scattered through “folklore” and “the literature.” Because there are recurrent themes and varying viewpoints in the subject, some of the exercises overlap considerably. The course attendees were mostly advanced graduate students in math- ematics, but it is hoped that these notes will reach a wider audience, in- cluding theoretical physicists. The “ideal” reader would be familiar with smooth manifolds (charts, forms, flows, Lie groups, vector bundles), differ- ential geometry (metrics, connections, and curvature), and basic algebraic topology (simplicial and singular cohomology, the long exact sequence, and ii the fundamental group), but in reality the prerequisites are less strenuous, though a good reference for each subject should be kept at hand. -
L2 Extension of Adjoint Line Bundle Sections D
L2 extension of adjoint line bundle sections D. Kim REPORT No. 20, 2007/2008, spring ISSN 1103-467X ISRN IML-R- -20-07/08- -SE+spring L2 extension of adjoint line bundle sections Dano Kim ∗ Abstract We prove an L2 extension theorem of Ohsawa-Takegoshi type for extending holo- morphic sections of line bundles from a subvariety which is given as a maximal log-canonical center of a pair and is of general codimension in a projective variety. Our method of proof indicates that such a setting is the most natural one in a sense, for general L2 extension of line bundle sections. Contents 1 Introduction 1 2 Preliminaries 6 2.1 Singular hermitian metrics . 6 2.1.1 The first kind . 6 2.1.2 The second kind and the adjoint norm . 8 2.2 Plurisubharmonic functions . 10 2.3 Stein manifolds . 12 2.4 ∂ operators on the Hilbert spaces of (p, q) forms . 15 3 Kawamata metric on a log-canonical center 17 3.1 A refined log-resolution and the Kawamata metric . 17 3.2 Appendix . 22 4 L2 extension 23 4.1 Statement of the main theorem . 23 4.2 Proof of the main theorem . 25 4.2.1 Setup of the ∂ equation . 26 4.2.2 Two main inequalities and the extension . 29 5 Pluriadjoint extension 34 1 Introduction The purpose of this paper is to prove an L2 extension theorem (Theorem 4.2) of Ohsawa-Takegoshi type to lift line bundle sections from a closed subvariety of general codimension of a projective variety. For the moment, let Z X be a complex sub- ⊂ manifold of a complex manifold.