Table Of ContentFrancesco 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,
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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.
