D. V. Anosov A. A. Bolibruch The Riemann-Hilbert Problem Asped~f Mathematic~ Edited by Klos Diederich Vol. E 2: M. Knebuseh/M. Koister: Wittrings Vol. E 3: G. Hector/U. Hirsch: Introduction to the Geometry of Foliations, Part B Vol. E 5: P. Stiller: Automorphie Forms and the Pieard Number of an Elliptie Surfaee Vol. E 6: G. Faltings/G. Wüstholz et 01.: Rational Poinls* Vol. E 7: W. StoII: Value Distribution Theory for Meromorphie Mops Vol. E 9: A. Howard/P.-M. Wong (Eds.): Contribution to Several Complex Variables Vol. E 10: A. J Tromba (Ed.): Seminar of New Results in Nonlinear Partial Differential Equations* Vol. E 13: Y. Andre: G-Funetions and Geometry* Vol. E 14: U. Cegrell: Capaeities in Complex Analysis Vol. E 15: J-P. Serre: Lectures on the Mordell-Weil Theorem Vol. E 16: K. Iwasaki/H. Kimura/S. Shimomura/M. Yoshida: From Gauss to Painleve Vol. E 17: K. Diederich (Ed.): Complex Analysis Vol. E 18: W. W. J Hulsbergen: Conjeetures in Arithmetie Aigebraie Geometry Vol. E 19: R. Racke: Leetures on Nonlinear Evolution Equations Vol. E 20: F. Hirzebrueh, Th. Berger, R. Jung: Manifolds and Modular Forms * Vol. E 21: H. Fujimoto: Value Distribution Theory of the Gauss Map of Minimal Surfaees in Rm Vol. E 22: D. V. Anosov/A. A. Bolibrueh: The Riemann-Hilbert Problem Vol. E 23: A. P. Fordy/J C. Wood (Eds.): Harmonie Mops and Integrable Systems Vol. E 24: D. S. Alexander: A History of Complex Dynamies Vol. E 25: A. Tikhomirov/A. Tyurin (Eds.): Algebraie Geometry and its Applieations * A Publicatian of the Max-Planck-Institut für Mathematik, Bonn Volumes of the German-Ianguage subseries "Aspekte der Mathemotik" ore listed at the end of the book. D. V. Anosov A. A. Bolibruch 'he Riemann-Hilbert Problem A Publication from the Steklov Institute of Mathematics Adviser: Armen Sergeev IJ Vleweg Professor D. V. Anosov Professor A. A. Bolibruch Steklov Institute of Mathematics Vavilova 42 117966 Moscow/CIS Russia Die Deutschc Bibliothek - CIP-Einheitsaufnahme v.: Anosov, D. The Riemann Hilbert problem: a publication from the Steklov Institute of Mathematics / D. V. Anosov; A. A. Bolibruch. (Aspects of mathematics: E; VoI. 22) ISBN 978-3-322-92911-2 ISBN 978-3-322-92909-9 (eBook) DOI 10.1007/978-3-322-92909-9 NE: Bolibruch, A. A.:; Aspects of mathematics / E Mathematics Subject Classification: 34A20 AII rights reserved © Springer Fachmedien Wiesbaden 1994 Originally published by Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, BraunschweiglWiesbaden in 1994 Vieweg is a subsidiary company of the Bertelsmann Publishing Group International. No part of this publication may be reproduced, stored in a retrieval system or transmitted, mechanical, photocopying or otherwise without prior permission of the copyright holder. Cover design: Wolfgang Nieger, Wiesbaden Printed on acid-free paper ISSN 0179-2156 ISBN 978-3-322-92911-2 v Preface This book is devoted to Hilbert's 21st problem (the Riemann-Hilbert problem) which belongs to the theory of linear systems of ordinary differential equations in the complex domain. The problem concems the existence of a Fuchsian system with prescribed singularities and monodromy. Hilbert was convinced that such a system always exists. However, this tumed out to be a rare case of a wrong forecast made by hirn. In 1989 the second author (A.B.) discovered a counterexample, thus obtaining a negative solution to Hilbert's 21st problem. 1 After we recognized that some "data" (singularities and monodromy) can be obtai ned from a Fuchsian system and some others cannot, we are enforced to change our point of view. To make the terminology more precise, we shaII caII the foIIowing problem the Riemann-Hilbert problem for such and such data: does there exist a Fuchsian system having these singularities and monodromy? The contemporary version of the 21 st Hilbert problem is to find conditions implying a positive or negative solution to the Riemann-Hilbert problem. In this book we consider only (the contemporary version 00 the cIassical 21st Hilbert's problem and only mention (of course, with due references) various modi fications, generalizations and related problems. We mention aII known results on the cIassical problem, both positive and negative, and prove some of them. We simply do not have enough place to prove aII of them, but the sampies we explain in detail incIude the most important cases and see m to provide a good feeling of the whole picture. The problem under consideration is of global character, but in order to study it one needs so me local theory (a theory describing the behavior of solutions near a singular point). There is a well-known local theory which goes back to Fuchs and Poincare and can be found in such textbooks as those by Coddington-Levinson [CoLe], or Hartman [Ha](we need only a simpler part ofthis theory dealing with the so-called regular singularities). For our purposes this theory has to be supplemented by a new local theory due to Levelt. Our book contains the exposition of both theories inasmuch as we need them. In 1908 Plemelj obtained a positive solution to the problem similar to Hilbert's 1 In the preface we dweIl on the history only inasmuch as it helps us to describe the content of the book. Introductory chapter contains more remarks on the history (which was somewhat fanciful), but complete description of the history was not our goal. VI Preface 21 st problem in its original form, but conceming the so-called regular systems instead of Fuchsian ones. This was a remarkable achievement, although it does not me an a solution to Hilbert's 21st problem, because the cIass of regular systems is broader than the cIass of Fuchsian systems. However, his theorem is useful even if one is interested only in Fuchsian systems - almost all positive resuits on Hilbert's 21st problem are obtained in the following way: one takes a regular system provided by Plemelj and tries to modify it so that it becomes Fuchsian with the same singularities and monodromy. In reality, not only the statement ofPlemelj's theorem is used, but sometimes also some details from its proof - the very proof we give here. This proof is different from PlemeIj's original proof, but goes back to RöhrI (1957) [Rö] and takes into account so me improvements invented later. An essential "ingredient" of this proof is the use of the Birkhoff-Grothendieck theorem about the complex analytic vector bundles over the Riemann sphere. We give (with minor modifications) the elementary proof of the latter theorem developed recently by J.Leiterer [Lei]. This makes our exposition self-contained, modulo more or less standard background. Here follows some information on this background. The reader must be acquainted with standard ("basic") courses on linear algebra (incIuding Jordan normal form of matrices). ordinary differential equations (we need general properties of solutions to linear systems), and the theory of functions of the complex variable. Usually in the basic course of the latter more attention is paid to the "single-valued" functions than to "multi valued" ones, whereas "multi valued" functions are important for our purposes. However, we need not a "deep general theory" of such functions (whatever that means), but rather a good "feeling" of such things as branching of elementary functions, analytic continuation, "complete" analytic function, Riemann surface. Usually this is incIuded in a basic course (although often - in a formal way) and of course can be found in many textbooks. One must know what the universal covering surface and the deck transformations are. We shaII use two functions of matrices - exponential and logarithm. Although the theory of ordinary differential equations in thc complex domain is rather extensive, we need only a few facts of it. It may happen that the reader's knowledge of ordinary differential equations is restricted to the real domain only. We hope that several remarks in the introductory chapter will help such areader to adopt to a complex point of view. At the beginning our exposition is detailed, later it becomes more succinct - we hope that by that time the reader gets some practice in this field and becomes more mature. This book includes without big changes the preprint "An introduction to Hilbert's 21st problem" [An], published by the first author (D.A.) in 1991. This preprint contained: the general introduction (wh ich is extended here); the description of the local theory (which we reproduce here with minor changes); the first counterexample to Hilbert's 21st problem (the exposition in the preprint is somewhat improved VII compared to the original exposition given by the second author); proofs of the theorems of Plemelj and Birkhoff-Grothendieck. The first author wrote also the section on the Bessel equation. All other text was written by the second author. Of course, we planned and discussed it together. The above mentioned preprint was written during the visit of the first author to the Inst. of Math. and Appl., Penn State Univ. and City Univ. of New York. In this preprint the first author already expressed his thanks to a number of persons who helped hirn at his work on this preprint and the entire faculty and staff of the IMA, PSU and CUNY for their hospitality. Big part of this book was written in Moscow where we both work at the Steklov Math. Inst., and the second author finish~d the work on this book during his visit to Max-Planck-Institut für Math. and to the University of Nice. He would like to thank the staff of MPIM and UN for hospitality and excellent working conditions. He is very grateful to Y.A.Golubeva and A. y'Chemavskii , who introduced hirn to the theory of Fuchsian equations and the Riemann-Hilbert problem. D.Y.Anosov A.A.Bolibruch VIII Contents 1 Introduction 2 1.1 Educational notes 2 1.2 Introduction ... 5 2 Counterexample to Hilbert's 21st problem 15 2.1 The first counterexample 15 2.2 Local theory ...... 17 2.3 The second order system 37 2.4 The third order system 44 3 The Plemelj theorem 52 3.1 A weak version of Plemelj's theorem. 52 3.2 Proof of Plemelj's theorem . . . . . . 57 3.3 Proof of the Birkhoff-Grothendieck theorem 64 3.4 Some other known results ......... 74 4 Irreducible representations 78 4.1 Technical preface. . . . ......... 78 4.2 Solution for an irreducible representation 83 5 Miscellaneous topics 90 5.1 Vector bundles and Hilbert's 21st problem 90 5.2 Reducibility and regular systems . 95 5.3 New series of counterexamples . 103 5.4 The cases of positive solvability 109 IX 5.5 On regular systems ............... . 114 5.6 On codimension of "nonfuchsian" representations 129 6 The case p = 3 133 6.1 The complete ans wer for p = 3 133 6.2 Fuchsian weight of a representation 136 6.3 Properties of Fuchsian weight 144 6.4 Instability of Fuchsian weight 151 6.5 The theorem of realization 154 7 Fuchsian equations 158 7.1 The number of apparent singularities 158 7.2 Fuchsian equations and systems 161 7.3 Examples .......... . 169 Bibliography 185 Index 189 1 Introduction 1.1 Educational notes When dealing with the system of ODE in a complex domain ddytj -_ Ji( x,y1, ... ,yP,) J' -- 1 , ... ,p (1.1.1) Ji one assurnes that are holomorphic (i.e. single-valued analytic) in some domain G and asks for analytic solution yl(X), ... , yP(x). The local existence theorem (for brevity, we include here uniqueness and analytic dependence on initial duta and parameters (ifthere are any)) looks like the corresponding theorem in areal domain. The most well-known proof of the latter is obtained by rewriting the system as a system of integral equations which is solved using iterations. A careful analysis of this proof reveals that it works in a complex domain as weil. (We work in a small disk on the x-plane containing the initial value Xo of the independent variable. Integration is performed along linear segments connecting Xo to the "current" x. These integrals are estimated literally in the same way as in the real case). At some points it even becomes simpler. If a sequence of analytic function<; converges uniformly in a domain on the x-plane, then its limit is an analytic function. In fact, one does not even need to assurne apriori that the convergence is uniform; but in our case the proof of the convergence is based on the estimates which imply the uniform convergence in a sufficiently small disko (For brevity, we say "function" instead of "a system of p functions"; one can also have in mind "a vector function"). So (1.1.1) has an analytic solution. Now, as regards to the uniqueness, it is easy to see (differentiating (1.1.1)) that two solutions having the same initial data must have the same derivatives of all orders. Being analytic, they must coincide. As regards to the dependence on initial data and parameters, we have an uniformly convergent sequence of functions holomorphic with respect to x, these data and parameters, so the limit is also holomorphic with respect to them (we again refer to the analyticity of the limit functions, only this time we deal with a function of several complex variables). Compare this easy argument with the situation in the real domain. In the latter case the un iform convergence does not imply any smoothness of the limit function. As regards to its dependence in x, smoothness follows immediately from the integral equation, so this is also easy, but as regards to the dependence on the