Topics on Real Analytic Spaces PDF
Preview Topics on Real Analytic Spaces
Francesco Guaraldo, Patrizia Macri, Alessandro Tancredi Topics on Real Analytic Spaces Advanced Lectures in Mathematics Edited by Gerd Fischer • Jochen Werner Optimization. Theory and Applications Manfred Denker • Asymptotic Distribution Theory in Nonparametric Statistics Klaus Lamotke • Regular Solids and Isolated Singularities Francesco Guaraldo, Patrizia Macri, Alessandro Tancredi Topics on Real Analytic Spaces Ernst Kunz Kahler. Differentials Francesco Guaraldo Patrizia Macri Alessandro Tancredi Topics on Real Analytic Spaces Friedr. Vieweg & Sohn Braunschweig I Wiesbaden AMS Subject Classification: 32B15 - 32C05 - 32C35 - 32F99 - 32K 15 - 32L99 - 58A07 1986 All rights reserved © Friedr. Vieweg & Sohn Verlagsgeselischaft mbH, Braunschweig 1986 No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise; without prior permission of the copyright holder. Produced by Lengericher Handelsdruckerei, Lengerich ISBN 978-3-528-08963-4 ISBN 978-3-322-84243-5 (eBook) DOl 10.1007/978-3-322-84243-5 v TAB L E o F CON TEN T S INTRODUCTION VII Chapter I RINGED SPACES § 1. k-ringed spaces 1 § 2. Coherent sheaves 4 § 3. Embeddings 7 Bibliography 10 Chapter II SPACES AND VARIETIES § 1. General properties 11 § 2. Local properties 18 § 3. Global properties 24 § 4. Antiinvolutions 30 Bibliography 38 Chapter III COMPLEXIFICATION § 1. Complexification of germs 40 § 2. Local complexification 44 § 3. Global complexification 50 Bibliography 59 Chapter IV REAL ANALYTIC VARIETIES § 1. Real part 60 § 2. Analytic subvarieties 63 § 3. Normalization 71 § 4. Desingularization 77 Bibliography 81 Chapter V EMBEDDINGS OF STEIN SPACES § 1. A first relative embedding theorem 83 § 2. A second relative embedding theorem % § 3. (l-invariant embedding theoremti 100 Bibliography 108 VI Chapter VI EMBEDDINGS OF REAL ANALYTIC VARIETIES OR SPACES § 1. Varieties: the general case 109 § 2. Varieties: the pathological case 112 § 3. The non reduced case 118 § 4. Topologies on Cm(X, lRq) 118 Bibliography 127 Chapter VII APPROXIMATIONS § 1. The weak and strong topologies 129 § 2. Approximations 131 Bibliography 147 Chapter VIII FIBRE BUNDLES § 1. Generalities on analytic fibre bundles 149 § 2. A classification theorem 158 Bibliography 160 INDEX 161 VII I N T ROD U C T ION The aim of this book is to present some topics on the global theory of real analytic spaces, on which only fragment ary literature is available: the complexification, the normal ization, the desingularization, the theory of relative ap proximation of differentiable functions by analytic functions, the embedding theorems, the classification of analytic vector bundles. Unlike the complex case, not all the real analytic spaces are coherent; however, if they are, they have properties si milar to those of Stein spaces. Situations which are essen tially new instead exist when working with reduced spaces, in general non coherent. To these we are particularly interested. Al though the coherent spaces have good properties, if they are reduced, however, their category is nei ther parti cularly vast (e.g. it does not contain the algebraic varietie~ nor stable (it can happen that a reduced space is coherent but its singular locus is not). These inconveniences can be eliminated by the introduc tion of non reduced structures, and then of coherent non re duced spaces. Such spaces intervene in many problems in a natural way, and often in an essential way; moreover they are sometimes very useful in the study of the reduced case. For these reasons we shall also examine several of their most im portant properties (see Chapters I, II, III). Now, if X is a real analytic space, from the above it is useful to stress the cases in which it is either reduced or coherent. We shall call X a variety if it is reduced and a space if it is coherent. Not every variety is a space and it is clear that the coherent varieties are precisely the reduced spaces. A great help for the study of the global properties of VIII the spaces (which are coherent, following our terminology) is given by the existence of the complexification and of Stein neighbourhoods. This fact has made it possible to use also in the real case the theory of analytic coherent sheaves which was so rich in results in the complex case. The first to work in this direction was H. Cartan. His work: "VariE~tes analytiques reelles et varietes analytiques complexes" (1957) can be considered the starting point of the theory and contains the principal ideas which have inspired the subsequent research in this field. Subsequently, thanks to the works of F. Bruhat and H. Whitney and of A. Tognoli, by using the solution of the Levi problem given by H. Grauert and R. Narasimhan, the existence of the complexification and of Stein neighbourhoods has been proved in all its generality. Chapter III is devoted to this. In 1958 Grauel't, by exploi ting deep resul ts of the sheaf theory on Stein manifolds, proved that each real analytic manifold with countable topology can be analitically embedded into mq. The complexification and some appropriate adaptations of the techniques used by Narasimhan to embed Stein spaces into n ~ , have subsequently allowed Tognoli e G. Tomassini to give embedding theorems into mq for those real varieties which are the reduction of a space and, in particular, for the coherent varieties (see Chapter VI). In general, however, the use of the complex theory finds great limitations in the non coherent case. In fact the non coherent varieties are not complexifiable; moreover, they can present some pathologies the study of which usually requires non standard techniques. For example, this is the case, point ed out by Cartan, of those non coherent varieties which are not the real part of any complex space (see Chapter IV). IX Nevertheless, a means of analysis which has proved to be useful in several cases is given by some appropriate general izations, due to Tognoli, of Whitney's classical approximation theorem on the differentiable functions. So, by relative approximation theorems (see Chapter VII), interesting global resul ts can be obtained, such as a very general embedding theorem for non coherent varieties (see Theorem 2.7 in Chapter VII) and a classification theorem for analytic vector bundles (see Theorem 2.2 in Chapter VIII). The above approximation results on the differentiable functions have then important applications in the research of analytic structures in the class of differentiable struc tures; but they turn out to be very useful also in the ap proximation problems of differentiable objects by analytic ones. It is interesting to observe that several results along these lines show that, in the study of real analytic manifolds, one finds only topological or differentiable obstructions to carry out certain analytic operations. This is a kind of Oka's principle in the real case. The complex theory is at the basis of the real one; then we presume that the reader has a certain knowledge of the pro perties of the complex analytic spaces, reduced or not. How ever, we give some essential resul ts. In general, for the relati ve proofs one must refer to the bib 1 iography, even if some results particularly used are developed in greater detail. This is the case of the embeddings of Stein spaces in the relative and O-invariant versions (see Chapter V) which will play a fundamental role in the real case. This is also the case when we want to stress the differences existing between the real theory and the complex one. Also for those problems of the real theory which formal ly are not dissimilar to the complex case and which have x already been systematically treated, one must refer to the ample bibliography existent. A reference to Definition 2 of § 1 in Chapter II is written 11.1.2, or 1.2 if it appears in Chapter II. Number in brackets refer to the bibliography given at the end of each chapter.
The list of books you might like
