Some Recent Results in Complex Manifold Theory Related to Vanishing Theorems for the Semipositive Case
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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. -
Informal Lecture Notes for Complex Analysis
Informal lecture notes for complex analysis Robert Neel I'll assume you're familiar with the review of complex numbers and their algebra as contained in Appendix G of Stewart's book, so we'll pick up where that leaves off. 1 Elementary complex functions In one-variable real calculus, we have a collection of basic functions, like poly- nomials, rational functions, the exponential and log functions, and the trig functions, which we understand well and which serve as the building blocks for more general functions. The same is true in one complex variable; in fact, the real functions we just listed can be extended to complex functions. 1.1 Polynomials and rational functions We start with polynomials and rational functions. We know how to multiply and add complex numbers, and thus we understand polynomial functions. To be specific, a degree n polynomial, for some non-negative integer n, is a function of the form n n−1 f(z) = cnz + cn−1z + ··· + c1z + c0; 3 where the ci are complex numbers with cn 6= 0. For example, f(z) = 2z + (1 − i)z + 2i is a degree three (complex) polynomial. Polynomials are clearly defined on all of C. A rational function is the quotient of two polynomials, and it is defined everywhere where the denominator is non-zero. z2+1 Example: The function f(z) = z2−1 is a rational function. The denomina- tor will be zero precisely when z2 = 1. We know that every non-zero complex number has n distinct nth roots, and thus there will be two points at which the denominator is zero. -
Bounded Holomorphic Functions on Finite Riemann Surfaces
BOUNDED HOLOMORPHIC FUNCTIONS ON FINITE RIEMANN SURFACES BY E. L. STOUT(i) 1. Introduction. This paper is devoted to the study of some problems concerning bounded holomorphic functions on finite Riemann surfaces. Our work has its origin in a pair of theorems due to Lennart Carleson. The first of the theorems of Carleson we shall be concerned with is the following [7] : Theorem 1.1. Let fy,---,f„ be bounded holomorphic functions on U, the open unit disc, such that \fy(z) + — + \f„(z) | su <5> 0 holds for some ô and all z in U. Then there exist bounded holomorphic function gy,---,g„ on U such that ftEi + - +/A-1. In §2, we use this theorem to establish an analogous result in the setting of finite open Riemann surfaces. §§3 and 4 consider certain questions which arise naturally in the course of the proof of this generalization. We mention that the chief result of §2, Theorem 2.6, has been obtained independently by N. L. Ailing [3] who has used methods more highly algebraic than ours. The second matter we shall be concerned with is that of interpolation. If R is a Riemann surface and if £ is a subset of R, call £ an interpolation set for R if for every bounded complex-valued function a on £, there is a bounded holo- morphic function f on R such that /1 £ = a. Carleson [6] has characterized interpolation sets in the unit disc: Theorem 1.2. The set {zt}™=1of points in U is an interpolation set for U if and only if there exists ô > 0 such that for all n ** n00 k = l;ktn 1 — Z.Zl An alternative proof is to be found in [12, p. -
Complex-Differentiability
Complex-Differentiability Sébastien Boisgérault, Mines ParisTech, under CC BY-NC-SA 4.0 April 25, 2017 Contents Core Definitions 1 Derivative and Complex-Differential 3 Calculus 5 Cauchy-Riemann Equations 7 Appendix – Terminology and Notation 10 References 11 Core Definitions Definition – Complex-Differentiability & Derivative. Let f : A ⊂ C → C. The function f is complex-differentiable at an interior point z of A if the derivative of f at z, defined as the limit of the difference quotient f(z + h) − f(z) f 0(z) = lim h→0 h exists in C. Remark – Why Interior Points? The point z is an interior point of A if ∃ r > 0, ∀ h ∈ C, |h| < r → z + h ∈ A. In the definition above, this assumption ensures that f(z + h) – and therefore the difference quotient – are well defined when |h| is (nonzero and) small enough. Therefore, the derivative of f at z is defined as the limit in “all directions at once” of the difference quotient of f at z. To question the existence of the derivative of f : A ⊂ C → C at every point of its domain, we therefore require that every point of A is an interior point, or in other words, that A is open. 1 Definition – Holomorphic Function. Let Ω be an open subset of C. A function f :Ω → C is complex-differentiable – or holomorphic – in Ω if it is complex-differentiable at every point z ∈ Ω. If additionally Ω = C, the function is entire. Examples – Elementary Functions. 1. Every constant function f : z ∈ C 7→ λ ∈ C is holomorphic as 0 λ − λ ∀ z ∈ C, f (z) = lim = 0. -
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. -
Complex Manifolds
Complex Manifolds Lecture notes based on the course by Lambertus van Geemen A.A. 2012/2013 Author: Michele Ferrari. For any improvement suggestion, please email me at: [email protected] Contents n 1 Some preliminaries about C 3 2 Basic theory of complex manifolds 6 2.1 Complex charts and atlases . 6 2.2 Holomorphic functions . 8 2.3 The complex tangent space and cotangent space . 10 2.4 Differential forms . 12 2.5 Complex submanifolds . 14 n 2.6 Submanifolds of P ............................... 16 2.6.1 Complete intersections . 18 2 3 The Weierstrass }-function; complex tori and cubics in P 21 3.1 Complex tori . 21 3.2 Elliptic functions . 22 3.3 The Weierstrass }-function . 24 3.4 Tori and cubic curves . 26 3.4.1 Addition law on cubic curves . 28 3.4.2 Isomorphisms between tori . 30 2 Chapter 1 n Some preliminaries about C We assume that the reader has some familiarity with the notion of a holomorphic function in one complex variable. We extend that notion with the following n n Definition 1.1. Let f : C ! C, U ⊆ C open with a 2 U, and let z = (z1; : : : ; zn) be n the coordinates in C . f is holomorphic in a = (a1; : : : ; an) 2 U if f has a convergent power series expansion: +1 X k1 kn f(z) = ak1;:::;kn (z1 − a1) ··· (zn − an) k1;:::;kn=0 This means, in particular, that f is holomorphic in each variable. Moreover, we define OCn (U) := ff : U ! C j f is holomorphicg m A map F = (F1;:::;Fm): U ! C is holomorphic if each Fj is holomorphic. -
Introduction to Holomorphic Functions of Several Variables, by Robert C
BOOK REVIEWS 205 BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 25, Number 1, July 1991 ©1991 American Mathematical Society 0273-0979/91 $1.00+ $.25 per page Introduction to holomorphic functions of several variables, by Robert C. Gunning. Wadsworth & Brooks/Cole, vol. 1, Function the ory, 202 pp. ISBN 0-534-13308-8; vol. 2, Local theory, 215 pp. ISBN 0-534-13309-6; vol. 3, Homological theory, 194 pp. ISBN 0-534-13310-X What is several complex variables? The answer depends on whom you ask. Some will speak of coherent analytic sheaves, commutative al gebra, and Cousin problems (see for instance [GRR1, GRR2]). Some will speak of Kàhler manifolds and uniformization—how many topologically trivial complex manifolds are there of strictly negative (bounded from zero) curvature (see [KOW])? (In one complex variable the important answer is one,) Some will speak of positive line bundles (see [GRA]). Some will speak of partial differential equations, subelliptic es timates, and noncoercive boundary value problems (see [FOK]). Some will speak of intersection theory, orders of contact of va rieties with real manifolds, and algebraic geometry (see [JPDA]). Some will speak of geometric function theory (see [KRA]). Some will speak of integral operators and harmonic analysis (see [STE1, STE2, KRA]). Some will speak of questions of hard analysis inspired by results in one complex variable (see [RUD1, RUD2]). As with any lively and diverse field, there are many points of view, and many different tools that may be used to obtain useful results. Several complex variables, perhaps more than most fields, seems to be a meeting ground for an especially rich array of tech niques. -
Lecture Note for Math 220A Complex Analysis of One Variable
Lecture Note for Math 220A Complex Analysis of One Variable Song-Ying Li University of California, Irvine Contents 1 Complex numbers and geometry 2 1.1 Complex number field . 2 1.2 Geometry of the complex numbers . 3 1.2.1 Euler's Formula . 3 1.3 Holomorphic linear factional maps . 6 1.3.1 Self-maps of unit circle and the unit disc. 6 1.3.2 Maps from line to circle and upper half plane to disc. 7 2 Smooth functions on domains in C 8 2.1 Notation and definitions . 8 2.2 Polynomial of degree n ...................... 9 2.3 Rules of differentiations . 11 3 Holomorphic, harmonic functions 14 3.1 Holomorphic functions and C-R equations . 14 3.2 Harmonic functions . 15 3.3 Translation formula for Laplacian . 17 4 Line integral and cohomology group 18 4.1 Line integrals . 18 4.2 Cohomology group . 19 4.3 Harmonic conjugate . 21 1 5 Complex line integrals 23 5.1 Definition and examples . 23 5.2 Green's theorem for complex line integral . 25 6 Complex differentiation 26 6.1 Definition of complex differentiation . 26 6.2 Properties of complex derivatives . 26 6.3 Complex anti-derivative . 27 7 Cauchy's theorem and Morera's theorem 31 7.1 Cauchy's theorems . 31 7.2 Morera's theorem . 33 8 Cauchy integral formula 34 8.1 Integral formula for C1 and holomorphic functions . 34 8.2 Examples of evaluating line integrals . 35 8.3 Cauchy integral for kth derivative f (k)(z) . 36 9 Application of the Cauchy integral formula 36 9.1 Mean value properties . -
1 Holomorphic Functions
MM Vercelli. L2: Holomorphic and analytic functions. Contents: ² Holomorphic functions. ² Cauchy-Riemann conditions. ² Complex di®erentials and complex partial derivatives. ² Analytic functions. ² The exponential function. 1 Holomorphic functions A complex function f(x; y) = u(x; y) + i v(x; y) can be also regarded as a function of z = x + i y and its conjugated z because we can write z + z z ¡ z z + z z ¡ z z + z z ¡ z f( ; ) = u( ; ) + i v( ; ) 2 2 i 2 2 i 2 2 i Classically a holomorphic function f was de¯ned as a function f such that the expres- z+z z¡z z+z z¡z sion u( 2 ; 2 i )+i v( 2 ; 2 i ) does not contain, after simpli¯cation, the letter z or what is the same f does not depend upon z. It is standard to denote O(D) the set of holomorphic functions on D. From calculus we know that a given function f do not depends on a variable, say w @f if the partial derivative @w is identically zero. So a simpler way of saying that a function f does not depend on z is as follows: @f = 0 (1) @z The problem is that we have no de¯nition of the partial derivative with respect to z. n n¡1 Example 1.1. Any polynomial f(z) = anz + an¡1z + ¢ ¢ ¢ + a0 gives an holomorphic P1 k function. Moreover, a convergent power series f(z) = k=0 ckz gives a holomorphic function in his disc of convergence. The function f(x; y) = x2 + y2 is not holomorphic. -
Math 372: Fall 2015: Solutions to Homework
Math 372: Fall 2015: Solutions to Homework Steven Miller December 7, 2015 Abstract Below are detailed solutions to the homeworkproblemsfrom Math 372 Complex Analysis (Williams College, Fall 2015, Professor Steven J. Miller, [email protected]). The course homepage is http://www.williams.edu/Mathematics/sjmiller/public_html/372Fa15 and the textbook is Complex Analysis by Stein and Shakarchi (ISBN13: 978-0-691-11385-2). Note to students: it’s nice to include the statement of the problems, but I leave that up to you. I am only skimming the solutions. I will occasionally add some comments or mention alternate solutions. If you find an error in these notes, let me know for extra credit. Contents 1 Math 372: Homework #1: Yuzhong (Jeff) Meng and Liyang Zhang (2010) 3 1.1 Problems for HW#1: Due September 21, 2015 . ................ 3 1.2 SolutionsforHW#1: ............................... ........... 3 2 Math 372: Homework #2: Solutions by Nick Arnosti and Thomas Crawford (2010) 8 3 Math 372: Homework #3: Carlos Dominguez, Carson Eisenach, David Gold 12 4 Math 372: Homework #4: Due Friday, October 12, 2015: Pham, Jensen, Kolo˘glu 16 4.1 Chapter3,Exercise1 .............................. ............ 16 4.2 Chapter3,Exercise2 .............................. ............ 17 4.3 Chapter3,Exercise5 .............................. ............ 19 4.4 Chapter3Exercise15d ..... ..... ...... ..... ...... .. ............ 22 4.5 Chapter3Exercise17a ..... ..... ...... ..... ...... .. ............ 22 4.6 AdditionalProblem1 .............................. ............ 22 5 Math 372: Homework #5: Due Monday October 26: Pegado, Vu 24 6 Math 372: Homework #6: Kung, Lin, Waters 34 7 Math 372: Homework #7: Due Monday, November 9: Thompson, Schrock, Tosteson 42 7.1 Problems. ....................................... ......... 46 1 8 Math 372: Homework #8: Thompson, Schrock, Tosteson 47 8.1 Problems. -
Zeroes of Holomorphic Functions
LECTURE-19 : ZEROES OF HOLOMORPHIC FUNCTIONS VED V. DATAR∗ A complex number a is called a zero of a holomorphic function f :Ω ! C if f(a) = 0. A basic fact is that zeroes of holomorphic functions are isolated. This follows from the following theorem. Theorem 0.1. Let f :Ω ! C be a holomorphic function that is not identi- cally zero, and let a 2 Ω be a zero of f. Then there exists a disc D around a, a non vanishing holomorphic function g : D ! C (that is, g(z) 6= 0 for all z 2 D), and a unique positive integer n such that f(z) = (z − a)ng(z): The positive integer n is called the order of the zero at a. The main tool that we need in order to prove this is the strong principle of analytic continuation. Lemma 0.1. Let Ω ⊂ C be a connected domain, and f be holomorphic on Ω. If there is a p 2 Ω such that f k(p) = 0 for all k = 0; 1; 2 ··· , then f(z) = 0 for all z 2 Ω. Proof. Let E ⊂ Ω defined by E = fz 2 Ω j f (n)(z) = 0; 8 n = 0; 1; 2; · · · g: Then by hypothesis p 2 E, and hence E is non-empty. It is enough to show that E = Ω. The proof is completed by proving three claims. Claim 1. E is an open subset. To see this, let a 2 E. Since Ω is open, there exists a disc D"(a) such that D"(a) ⊂ Ω. -
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.