DOCSLIB.ORG

Tasty Bits of Several Complex Variables

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Tasty Bits of Several Complex Variables Tasty Bits of Several Complex Variables A whirlwind tour of the subject Jiří Lebl December 23, 2020 (version 3.4) 2 Typeset in LATEX. Copyright ©2014–2020 Jiří Lebl License: This work is dual licensed under the Creative Commons Attribution-Noncommercial-Share Alike 4.0 International License and the Creative Commons Attribution-Share Alike 4.0 International License. To view a copy of these licenses, visit https://creativecommons.org/licenses/ by-nc-sa/4.0/ or https://creativecommons.org/licenses/by-sa/4.0/ or send a letter to Creative Commons PO Box 1866, Mountain View, CA 94042, USA. You can use, print, duplicate, share this book as much as you want. You can base your own notes on it and reuse parts if you keep the license the same. You can assume the license is either the CC-BY-NC-SA or CC-BY-SA, whichever is compatible with what you wish to do, your derivative works must use at least one of the licenses. Acknowledgments: I would like to thank Debraj Chakrabarti, Anirban Dawn, Alekzander Malcom, John Treuer, Jianou Zhang, Liz Vivas, Trevor Fancher, Nicholas Lawson McLean, Alan Sola, Achinta Nandi, Sivaguru Ravisankar, Richard Lärkäng, Elizabeth Wulcan, Tomas Rodriguez, and students in my classes for pointing out typos/errors and helpful suggestions. During the writing of this book, the author was in part supported by NSF grant DMS-1362337. More information: See https://www.jirka.org/scv/ for more information (including contact information). The LATEX source for the book is available for possible modification and customization at github: https://github.com/jirilebl/scv Contents Introduction 5 0.1 Motivation, single variable, and Cauchy’s formula ..................5 1 Holomorphic functions in several variables 11 1.1 Onto several variables ................................. 11 1.2 Power series representation .............................. 16 1.3 Derivatives ...................................... 24 1.4 Inequivalence of ball and polydisc .......................... 28 1.5 Cartan’s uniqueness theorem ............................. 32 1.6 Riemann extension theorem, zero sets, and injective maps .............. 35 2 Convexity and pseudoconvexity 40 2.1 Domains of holomorphy and holomorphic extensions ................ 40 2.2 Tangent vectors, the Hessian, and convexity ..................... 44 2.3 Holomorphic vectors, the Levi form, and pseudoconvexity ............. 51 2.4 Harmonic, subharmonic, and plurisubharmonic functions .............. 63 2.5 Hartogs pseudoconvexity ............................... 73 2.6 Holomorphic convexity ................................ 80 3 CR functions 86 3.1 Real-analytic functions and complexification ..................... 86 3.2 CR functions ..................................... 92 3.3 Approximation of CR functions ........................... 98 3.4 Extension of CR functions .............................. 105 ¯ 4 The m-problem 109 4.1 The generalized Cauchy integral formula ....................... 109 ¯ 4.2 Simple case of the m-problem ............................. 111 4.3 The general Hartogs phenomenon .......................... 113 5 Integral kernels 116 5.1 The Bochner–Martinelli kernel ............................ 116 5.2 The Bergman kernel ................................. 119 5.3 The Szegö kernel ................................... 123 4 CONTENTS 6 Complex analytic varieties 125 6.1 The ring of germs ................................... 125 6.2 Weierstrass preparation and division theorems .................... 127 6.3 The dependence of zeros on parameters ....................... 132 6.4 Properties of the ring of germs ............................ 135 6.5 Varieties ........................................ 137 6.6 Hypervarieties ..................................... 140 6.7 Irreducibility, local parametrization, and Puiseux theorem .............. 143 6.8 Segre varieties and CR geometry ........................... 146 A Basic notation and terminology 151 B Results from one complex variable 152 C Differential forms and Stokes’ theorem 162 D Basic terminology and results from algebra 170 Further reading 173 Index 174 List of Notation 179 Introduction This book is a polished version of my course notes for Math 6283, Several Complex Variables, given in Spring 2014, Spring 2016, and Spring 2019 semesters at Oklahoma State University. It is meant for a semester-long course. Quite a few exercises of various difficulty are sprinkled throughout the text, and I hope a reader is at least attempting all of them. Many are required later in the text. The reader should attempt exercises in sequence as earlier exercises can help or even be required to solve later exercises. The prerequisites are a decent knowledge of vector calculus, basic real analysis, and a working knowledge of complex analysis in one variable. Measure theory (Lebesgue integral and its con- vergence theorems) is useful, but it is not essential except in a couple of places later in the book. The first two chapters and most of the third is accessible to beginning graduate students after one semester of a standard single-variable complex analysis graduate course. From time to time (e.g. proof of Baouendi–Trèves in chapter 3, and most of chapter 4, and chapter 5), basic knowledge of differential forms is useful, and in chapter 6 we use some basic ring theory from algebra. By design, it can replace the second semester of complex analysis. This book is not meant as an exhaustive reference. It is simply a whirlwind tour of several complex variables with slightly more material than can be covered within a semester. See the end of the book for a list of books for reference and further reading. There are also appendices for a list of one-variable results, an overview of differential forms, and some basic algebra. See appendix B , appendix C, and appendix D . 0.1 Motivation, single variable, and Cauchy’s formula Let us start with some standard notation. We use C for the complexp numbers, R for real numbers, Z for integers, N = f1, 2, 3,...g for natural numbers, 8 = −1. Throughout this book, we use the standard terminology of domain to mean connected open set. We will try to avoid using it if connectedness is not needed, but sometimes we use it just for simplicity. p As complex analysis deals with the complexp numbers, perhaps we should begin with −1. We start with the real numbers, R, and we add −1 into our field. We call this square root 8, and we write the complex numbers, C, by identifying C with R2 using I = G ¸ 8H, where I 2 C, and ¹G, Hº 2 R2. A subtle philosophical issue is that there are two square roots of −1. Two chickens are running around in our yard, and because we like to know which is which, we catch one and write “8” on it. If we happened to have caught the other chicken, we would have gotten an exactly equivalent theory, which we could not tell apart from the original. 6 INTRODUCTION Given a complex number I, its “opposite” is the complex conjugate of I and is defined as def I¯ = G − 8H. The size of I is measured by the so-called modulus, which is just the Euclidean distance: def p q jIj = II¯ = G2 ¸ H2. If I = G ¸ 8H 2 C for G, H 2 R, then G is called the real part and H is called the imaginary part. We write I ¸ I¯ I − I¯ Re I = Re¹G ¸ 8Hº = = G, Im I = Im¹G ¸ 8Hº = = H. 2 28 A function 5 : * ⊂ R= ! C for an open set * is said to be continuously differentiable, or 퐶1 if the first (real) partial derivatives exist and are continuous. Similarly, it is 퐶: or 퐶: -smooth if the first : partial derivatives all exist and are differentiable. Finally, a function is said to be 퐶1 or ∗ : simply smooth if it is infinitely differentiable, or in other words, if it is 퐶 for all : 2 N. Complex analysis is the study of holomorphic (or complex-analytic) functions. Holomorphic functions are a generalization of polynomials, and to get there one leaves the land of algebra to arrive in the realm of analysis. One can do an awful lot with polynomials, but sometimes they are just not enough. For example, there is no nonzero polynomial function that solves the simplest of differential equations, 5 0 = 5 . We need the exponential function, which is holomorphic. Let us start with polynomials. In one variable, a polynomial in I is an expression of the form 3 ∑︁ 9 %¹Iº = 29 I , 9=0 where 29 2 C and 23 < 0. The number 3 is called the degree of the polynomial %. We can plug in some number I and simply compute %¹Iº, so we have a function % : C ! C. We try to write 1 ∑︁ 9 5 ¹Iº = 29 I 9=0 and all is very fine until we wish to know what 5 ¹Iº is for some number I 2 C. We usually mean 1 3 ∑︁ 9 ∑︁ 9 29 I = lim 29 I . 3!1 9=0 9=0 As long as the limit exists, we have a function. You know all this; it is your one-variable complex analysis. We usually start with the functions and prove that we can expand into series. Let * ⊂ C be open. A function 5 : * ! C is holomorphic (or complex-analytic) if it is complex-differentiable at every point, that is, if 5 ¹I ¸ bº − 5 ¹Iº 5 0¹Iº = lim exists for all I 2 *. b2C!0 b ∗While 퐶1 is the common definition of smooth, not everyone always means the same thing by the word smooth.I have seen it mean differentiable, 퐶1, piecewise-퐶1, 퐶1, holomorphic, . 0.1. MOTIVATION, SINGLE VARIABLE, AND CAUCHY’S FORMULA 7 Importantly, the limit is taken with respect to complex b. Another vantage point is to start with a ∗ continuously differentiable 5 , and say 5 = D ¸8 E is holomorphic if it satisfies the Cauchy–Riemann equations: mD mE mD mE = , = − . mG mH mH mG The so-called Wirtinger operators, m 1 m m m 1 m m def= − 8 , def= ¸ 8 , mI 2 mG mH mI¯ 2 mG mH provide an easier way to understand the Cauchy–Riemann equations.
Load more

Recommended publications
  • Topic 7 Notes 7 Taylor and Laurent Series
    Topic 7 Notes Jeremy Orloff 7 Taylor and Laurent series 7.1 Introduction We originally defined an analytic function as one where the derivative, defined as a limit of ratios, existed. We went on to prove Cauchy's theorem and Cauchy's integral formula. These revealed some deep properties of analytic functions, e.g. the existence of derivatives of all orders. Our goal in this topic is to express analytic functions as infinite power series. This will lead us to Taylor series. When a complex function has an isolated singularity at a point we will replace Taylor series by Laurent series. Not surprisingly we will derive these series from Cauchy's integral formula. Although we come to power series representations after exploring other properties of analytic functions, they will be one of our main tools in understanding and computing with analytic functions. 7.2 Geometric series Having a detailed understanding of geometric series will enable us to use Cauchy's integral formula to understand power series representations of analytic functions. We start with the definition: Definition. A finite geometric series has one of the following (all equivalent) forms. 2 3 n Sn = a(1 + r + r + r + ::: + r ) = a + ar + ar2 + ar3 + ::: + arn n X = arj j=0 n X = a rj j=0 The number r is called the ratio of the geometric series because it is the ratio of consecutive terms of the series. Theorem. The sum of a finite geometric series is given by a(1 − rn+1) S = a(1 + r + r2 + r3 + ::: + rn) = : (1) n 1 − r Proof.
    [Show full text]
  • Complex Analysis Class 24: Wednesday April 2
    Complex Analysis Math 214 Spring 2014 Fowler 307 MWF 3:00pm - 3:55pm c 2014 Ron Buckmire http://faculty.oxy.edu/ron/math/312/14/ Class 24: Wednesday April 2 TITLE Classifying Singularities using Laurent Series CURRENT READING Zill & Shanahan, §6.2-6.3 HOMEWORK Zill & Shanahan, §6.2 3, 15, 20, 24 33*. §6.3 7, 8, 9, 10. SUMMARY We shall be introduced to Laurent Series and learn how to use them to classify different various kinds of singularities (locations where complex functions are no longer analytic). Classifying Singularities There are basically three types of singularities (points where f(z) is not analytic) in the complex plane. Isolated Singularity An isolated singularity of a function f(z) is a point z0 such that f(z) is analytic on the punctured disc 0 < |z − z0| <rbut is undefined at z = z0. We usually call isolated singularities poles. An example is z = i for the function z/(z − i). Removable Singularity A removable singularity is a point z0 where the function f(z0) appears to be undefined but if we assign f(z0) the value w0 with the knowledge that lim f(z)=w0 then we can say that we z→z0 have “removed” the singularity. An example would be the point z = 0 for f(z) = sin(z)/z. Branch Singularity A branch singularity is a point z0 through which all possible branch cuts of a multi-valued function can be drawn to produce a single-valued function. An example of such a point would be the point z = 0 for Log (z).
    [Show full text]
  • Residue Theorem
    Topic 8 Notes Jeremy Orloff 8 Residue Theorem 8.1 Poles and zeros f z z We remind you of the following terminology: Suppose . / is analytic at 0 and f z a z z n a z z n+1 ; . / = n. * 0/ + n+1. * 0/ + § a ≠ f n z n z with n 0. Then we say has a zero of order at 0. If = 1 we say 0 is a simple zero. f z Suppose has an isolated singularity at 0 and Laurent series b b b n n*1 1 f .z/ = + + § + + a + a .z * z / + § z z n z z n*1 z z 0 1 0 . * 0/ . * 0/ * 0 < z z < R b ≠ f n z which converges on 0 * 0 and with n 0. Then we say has a pole of order at 0. n z If = 1 we say 0 is a simple pole. There are several examples in the Topic 7 notes. Here is one more Example 8.1. z + 1 f .z/ = z3.z2 + 1/ has isolated singularities at z = 0; ,i and a zero at z = *1. We will show that z = 0 is a pole of order 3, z = ,i are poles of order 1 and z = *1 is a zero of order 1. The style of argument is the same in each case. At z = 0: 1 z + 1 f .z/ = ⋅ : z3 z2 + 1 Call the second factor g.z/. Since g.z/ is analytic at z = 0 and g.0/ = 1, it has a Taylor series z + 1 g.z/ = = 1 + a z + a z2 + § z2 + 1 1 2 Therefore 1 a a f .z/ = + 1 +2 + § : z3 z2 z This shows z = 0 is a pole of order 3.
    [Show full text]
  • Chapter 2 Complex Analysis
    Chapter 2 Complex Analysis In this part of the course we will study some basic complex analysis. This is an extremely useful and beautiful part of mathematics and forms the basis of many techniques employed in many branches of mathematics and physics. We will extend the notions of derivatives and integrals, familiar from calculus, to the case of complex functions of a complex variable. In so doing we will come across analytic functions, which form the centerpiece of this part of the course. In fact, to a large extent complex analysis is the study of analytic functions. After a brief review of complex numbers as points in the complex plane, we will ¯rst discuss analyticity and give plenty of examples of analytic functions. We will then discuss complex integration, culminating with the generalised Cauchy Integral Formula, and some of its applications. We then go on to discuss the power series representations of analytic functions and the residue calculus, which will allow us to compute many real integrals and in¯nite sums very easily via complex integration. 2.1 Analytic functions In this section we will study complex functions of a complex variable. We will see that di®erentiability of such a function is a non-trivial property, giving rise to the concept of an analytic function. We will then study many examples of analytic functions. In fact, the construction of analytic functions will form a basic leitmotif for this part of the course. 2.1.1 The complex plane We already discussed complex numbers briefly in Section 1.3.5.
    [Show full text]
  • Compex Analysis
    ( ! ) 1. The amplitu.e of a comple2 number isisis a) 0 b) 89: c) 8 .) :8 2.The amplitu.e of the quotient of two comple2 numbers is a) the sum of their amplitu.es b) the .ifference of their amplitu.es c) the pro.uct of their amplitu.es .) the quotient of their amplitu.es 3. The ? = A B then ?C = : : : : : : a) AB b) A B c) DB .) imaginary J ? K ?: J 4. If ? I ?: are any two comple2 numbers, then J ?.?: J isisis J ? K ?: J J ? K ?: J J ? K ?: J J J J J a)a)a) J ?: J b) J ?. J c) J ?: JJ? J .) J ? J J ?: J : : 555.5...MNO?P:Q: A B R = a) S b) DS c) T .) U 666.The6.The equation J? A UJ A J? D UJ = W represents a) Circle b) ellipse c) parparabolaabola .) hyperbola 777.The7.The equation J? A ZJ = J? D ZJ represents a) Circle b) ellipse c) parparabolaabola .) Real a2is 888.The8.The equation J? A J = ]:J? D J represents a) Circle b) ellipse c) parparabolaabola .) hyperbola 999.The9.The equation J?A:J A J?D:J = U represents a) Circle b) a st. line c) parabola .) hyperbohyperbolalalala 101010.The10.The equation J:? D J = J?D:J represents a) Circle b) a st. line c) parabola .) hyperbola : : 111111.The11.The equation J? D J A J?D:J = U represents a) Circle b) a st. line c) parabola .) hyperbola ?a 121212.The12.The equation !_ `?a b = c represents a) Circle b) a st.
    [Show full text]
  • Extremal Positive Pluriharmonic Functions on Euclidean Balls
    EXTREMAL POSITIVE PLURIHARMONIC FUNCTIONS ON EUCLIDEAN BALLS By Farhad Jafari and Mihai Putinar IMA Preprint Series # 2180 ( November 2007 ) INSTITUTE FOR MATHEMATICS AND ITS APPLICATIONS UNIVERSITY OF MINNESOTA 400 Lind Hall 207 Church Street S.E. Minneapolis, Minnesota 55455–0436 Phone: 612-624-6066 Fax: 612-626-7370 URL: http://www.ima.umn.edu EXTREMAL POSITIVE PLURIHARMONIC FUNCTIONS ON EUCLIDEAN BALLS FARHAD JAFARI AND MIHAI PUTINAR To Professor J. J. Kohn on the occasion of his seventy-fifth birthday Abstract. Contrary to the well understood structure of positive har- monic functions in the unit disk, most of the properties of positive pluri- n harmonic functions in symmetric domains of C , in particular the unit ball, remain mysterious. In particular, in spite of efforts spread over quite a few decades, no characterization of the extremal rays in the n cone of positive pluriharmonic functions in the unit ball of C is known. We investigate this question by a geometric tomography technique, and provide some new classes of examples of such extremal functions. 1. Introduction. According to a classical theorem due to Herglotz and F. Riesz, every non- negative harmonic function in the open unit disk D is the Poisson integral of a positive Borel measure on the unit circle T. This correspondence readily characterizes the extreme points of the convex cone of positive harmonic functions h in D, normalized by the condition h(0) = 1, as the Poisson integrals of extremal probability measures on the unit circle, namely the Dirac measures on the unit circle. Thus these extremal objects are simply the one-parameter class of functions uζ (z) = P (z, ζ), ζ ∈ T, where P (z, ζ) is the classical Poisson kernel in D.
    [Show full text]
  • COMPLEX ANALYSIS Notes Lent 2006 T. K. Carne
    Department of Pure Mathematics and Mathematical Statistics University of Cambridge COMPLEX ANALYSIS Notes Lent 2006 T. K. Carne. [email protected] c Copyright. Not for distribution outside Cambridge University. CONTENTS 1. ANALYTIC FUNCTIONS 1 Domains 1 Analytic Functions 1 Cauchy – Riemann Equations 1 2. POWER SERIES 3 Proposition 2.1 Radius of convergence 3 Proposition 2.2 Power series are differentiable. 3 Corollary 2.3 Power series are infinitely differentiable 4 The Exponential Function 4 Proposition 2.4 Products of exponentials 5 Corollary 2.5 Properties of the exponential 5 Logarithms 6 Branches of the logarithm 6 Logarithmic singularity 6 Powers 7 Branches of powers 7 Branch point 7 Conformal Maps 8 3. INTEGRATION ALONG CURVES 9 Definition of curves 9 Integral along a curve 9 Integral with respect to arc length 9 Proposition 3.1 10 Proposition 3.2 Fundamental Theorem of Calculus 11 Closed curves 11 Winding Numbers 11 Definition of the winding number 11 Lemma 3.3 12 Proposition 3.4 Winding numbers under perturbation 13 Proposition 3.5 Winding number constant on each component 13 Homotopy 13 Definition of homotopy 13 Definition of simply-connected 14 Definition of star domains 14 Proposition 3.6 Winding number and homotopy 14 Chains and Cycles 14 4 CAUCHY’S THEOREM 15 Proposition 4.1 Cauchy’s theorem for triangles 15 Theorem 4.2 Cauchy’s theorem for a star domain 16 Proposition 4.10 Cauchy’s theorem for triangles 17 Theorem 4.20 Cauchy’s theorem for a star domain 18 Theorem 4.3 Cauchy’s Representation Formula 18 Theorem 4.4 Liouville’s theorem 19 Corollary 4.5 The Fundamental Theorem of Algebra 19 Homotopy form of Cauchy’s Theorem.
    [Show full text]
  • SOME PROPERTIES of GRAUERT TYPE SURFACES Contents 1
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Archivio istituzionale della ricerca - Politecnico di Milano SOME PROPERTIES OF GRAUERT TYPE SURFACES SAMUELE MONGODI1, ZBIGNIEW SLODKOWSKI2, AND GIUSEPPE TOMASSINI3 Contents 1. Introduction 1 2. Compact curves in Grauert type surfaces 3 3. Level sets of pluriharmonic functions 6 4. A class of examples 11 4.1. S-antisymmetric bundles and cocycles 12 4.2. Proof of Theorem 4.1 15 References 17 1. Introduction In [5, 6], we addressed the problem of studying a class of weakly complete spaces, namely, complex manifolds of dimension 2 endowed with a real analytic plurisubharmonic exhaustion function. Stein spaces are obviously an instance of weakly complete spaces, where the exhaustion function can be taken to be strictly plurisubhar- monic and we have abundance of holomorhpic functions; more gen- erally, holomorphically convex spaces give a wide range of examples, which admit a proper holomorphic map onto a Stein space, their Cartan- Remmert reduction. A completely different example of weakly complete space is due to Grauert (cfr. [7]). As a generalization of the latter, we say that a com- plex surface X is of Grauert type if it admits a real analytic plurisub- harmonic exhaustion function α, such that the regular level sets of Date: June 1, 2017. 2010 Mathematics Subject Classification. Primary 32E, 32T, 32U; Secondary 32E05, 32T35, 32U10. Key words and phrases. Pseudoconvex domains, Weakly complete spaces, Holo- morphic foliations. The first author was supported by the FIRB2012 grant “Differential Geometry and Geometric Function Theory". 1 2 S.
    [Show full text]
  • MA 528 PRACTICE PROBLEMS 1. Find the Directional Derivative of F(X, Y)
    MA 528 PRACTICE PROBLEMS 1. Find the directional derivative of f(x, y)=5− 4x2 − 3y at (x, y) towards the origin −8x2 − 3y −8x − 3 8x2 +3y a. −8x − 3b.p c. √ d. 8x2 +3y e. p . x2 + y2 64x2 +9 x2 + y2 2. Find a vector pointing in the direction in which f(x, y, z)=3xy − 9xz2 + y increases most rapidly at the point (1, 1, 0). a. 3i +4j b. i + j c. 4i − 3j d. 2i + k e. −i + j. 1 3. Find a vector that is normal to the graph of the equation 2 cos(πxy) = 1 at the point ( 6 , 2). √ a. 6i + j b. − 3 i − j c. 12i + j d. j e. 12i − j. 4. Find an equation of the tangent plane to the surface x2 +2y2 +3z2 = 6 at the point (1, 1, −1). a. −x +2y +3z =2 b. 2x +4y − 6z =6 c. x − 2y +3z = −4 d. 2x +4y − 6z =6 e. x +2y − 3z =6. 5. Find an equation of the plane tangent to the graph of z = π + sin(πx2 +2y)when(x, y)=(2,π). a. 4πx +2y − z =9π b. 4x +2πy − z =10π c. 4πx +2πy + z =10π d. 4x +2πy − z =9π e. 4πx +2y + z =9π. 6. Are the following statements true or false? R 1. The line integral (x3 +2xy)dx +(x2 − y2)dy is independent of path in the xy-plane. R C 3 2 − 2 2. C (x +2xy)dx +(x y )dy = 0 for every closed oriented curve C in the xy-plane.
    [Show full text]
  • On a Monge-Amp`Ere Operator for Plurisubharmonic Functions With
    ON A MONGE-AMPERE` OPERATOR FOR PLURISUBHARMONIC FUNCTIONS WITH ANALYTIC SINGULARITIES MATS ANDERSSON, ZBIGNIEW BLOCKI,ELIZABETH WULCAN Abstract. We study continuity properties of generalized Monge-Amp`ereoperators for plurisubharmonic functions with analytic singularities. In particular, we prove continuity for a natural class of decreasing approximating sequences. We also prove a formula for the total mass of the Monge-Amp`eremeasure of such a function on a compact K¨ahlermanifold. 1. Introduction We say that a plurisubharmonic (psh) function u on a complex manifold X has analytic singularities if locally it can be written in the form (1.1) u = c log jF j + b; where c ≥ 0 is a constant, F = (f1; : : : ; fm) is a tuple of holomorphic functions, and b is bounded. For instance, if fj are holomorphic functions and aj are positive rational numbers, a am then log(jf1j 1 + ··· + jfmj ) has analytic singularities. By the classical Bedford-Taylor theory, [5, 6], if u is of the form (1.1), then in fF 6= 0g, for any k, one can define a positive closed current (ddcu)k recursively as (1.2) (ddcu)k := ddcu(ddcu)k−1: It was shown in [3] that (ddcu)k has locally finite mass near fF = 0g for any k and that the c k−1 natural extension 1fF 6=0g(dd u) across fF = 0g is closed, cf. [3, Eq. (4.8)]. Moreover, by c k−1 [3, Proposition 4.1], u1fF 6=0g(dd u) has locally finite mass as well, and therefore one can define the Monge-Amp`erecurrent c k c c k−1 (1.3) (dd u) := dd u1fF 6=0g(dd u) for any k.
    [Show full text]
  • Traces of Pluriharmonic Functions Compositio Mathematica, Tome 44, No 1-3 (1981), P
    COMPOSITIO MATHEMATICA PAOLO DE BARTOLOMEIS GIUSEPPE TOMASSINI Traces of pluriharmonic functions Compositio Mathematica, tome 44, no 1-3 (1981), p. 29-39 <http://www.numdam.org/item?id=CM_1981__44_1-3_29_0> © Foundation Compositio Mathematica, 1981, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ COMPOSITIO MATHEMATICA, Vol. 44, Fasc. 1-3, 1981, pag. 29-39 @ 1981 Sijthoff &#x26; Noordhoff International Publishers - Alphen aan den Rijn Printed in the Netherlands TRACES OF PLURIHARMONIC FUNCTIONS Paolo de Bartolomeis and Giuseppe Tomassini Introduction Let M be a real oriented hypersurface in a complex manif old X, which divides X into two open sets X+ and X -. In this paper we characterize in terms of tangential linear diff eren- tial operators on M the distributions T on M which are "jumps" or traces (in the sense of currents) of pluriharmonic functions in X+ and X -. The starting point of our investigation is the non-tangential charac- terizing equation ~b~T = 0, which can be deduced from the theory of boundary values of holomorphic forms. If M is not Levi-flat, we construct a second order tangential local linear differential operator wM such that if T is the trace on M of a pluriharmonic function h, then CùM(T) = ah.
    [Show full text]
  • 20. Isolated Singularities Definition 20.1. Let F : U −→ C Be A
    20. Isolated Singularities Definition 20.1. Let f : U −! C be a holomorphic function on a region U. Let a2 = U. We say that a is an isolated singularity of f if U contains a punctured neighbourhood of a. Note that Log z does not have an isolated singularity at 0, since we have to remove all of (−∞; 0] to get a continuous function. By contrast its derivative 1=z is holomorphic except at 0 and so it has an isolated singularity at 0. Suppose that f has an isolated singularity at a. As a punctured neighbourhood of a is a special type of annulus, f has a Laurent ex- pansion centred at a, 1 X k f(z) = ak(z − a) ; k=−∞ valid for 0 < jz − aj < r; for some real r. The behaviour at a is dictated by the negative part of the Laurent expansion. Definition 20.2. If f has an isolated singularity at a and all of the coefficients ak of the Laurent expansion 1 X k f(z) = ak(z − a) ; k=−∞ vanish if k < 0, then we say that f has a removable singularity. If f has a removable singularity then in fact we can extend f to a holomorphic function in a neighbourhood of a. Indeed, the Laurent ex- pansion of f is a power series expansion, and this defines a holomorphic function in a neighbourhood of a. Example 20.3. The function sin z z has a removable singularity at a. 1 Indeed, sin z 1 z3 z3 = z − + + ::: z z 3! 5! z2 z4 = 1 − + + :::; 3! 5! is the Laurent series expansion of sin z=z.
    [Show full text]