Table Of ContentStanislaw Lojasiewicz
Introduction to
Complex Analytic Geometry
Translated from the Polish
by Maciej Klimek
1991 Springer Basel AG
Author's addrcss:
Dr. Stanistaw Lojasicwicz
Jagicllonian University
Dcpartmcnt of Mathcmatics
ul. Rcymonta 4
PL-30-05'1 Cracow (Poland)
Originally published as:
Wst<;p do geometrii analitycznej zespoloncj
© PWN-Panstwowe Wydawnictwo Naukowc, Warszawa, 1988
Oeutsche Bibliothek Cataloging-in-Publication Data
Lojasiewicz, Stanistaw:
lntroduction to complex analytic geomctry 1 Stanistaw Lojasicwicz. Transl. from the Polish by
Macicj Klimek.-Basel: Boston: Berlin: Birkhăuscr, 1991
Einheitssacht.: Wstep do geometrii analitycznci zcspolonej <engl.>
ISBN 978-3-0348-7619-3 ISBN 978-3-0348-7617-9 (eBook)
DOI 10.1007/978-3-0348-7617-9
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© 1991 for the English edition: Springer Basel AG
Originally published by Birkhăuser Verlag Basel 1991
Printed from the translator's camera-ready manuscript on acid-free paper
Without optimism one cannot
prove a theorem.
Aldo Andreotti
PREFACE
TO THE POLISH EDITION
The subject of this book is analytic geometry understood as the geometry
of analytic sets (or, more generally, analytic spaces), i.e., sets described locally
by systems of analytic equations C). Except for the last chapter, mostly local
problems are investigated and, throughout the book, only the complex case
is studied. From the purely geometric point of view, the real case is more
natural and more general. But it displays fewer regularities and - by and
large - the corresponding theory is more difficult. The complex structure is
richer. Hence one can expect deeper results. Indeed, some phenomena, such as
analyticity of the set of singular points (see IV. 24) or analyticity of proper
images (Remmert's theorem, see V. 5.1), do not have counterparts in the
real case. More than anything else, the beauty of the interplay between the
geometric and algebraic phenomena constitutes the main attraction of the
"complex" theory e).
This book should be regarded as an introduction. It does not pretend
to reflect the entire theory. Its aim is to familiarize the reader with the basic
range of problems, using means as elementary as possible. They belong to
complex analysis, commutative algebra, and set topology (the methods of al
gebraic topology have not been employed), and are gathered in the first three
preliminary chapters. The author's intention was to provide the reader with
access to complete proofs without the need to rely on so called "well-known"
e)
More or less, until the 50's, the name "analytic geometry" had been associated
with an undergraduate level course dealing with the use of Cartesian coordinates
in the study of linear and quadratic objects.
e)
To an even greater degree this is true in the case of algebraic geometry (i.e.,
in sim pie terms, the geometry of sets defined by systems of polynomial equations).
There, the geometric and algebraic aspects are interconnected in almost every prob
lem.
VI Preface
facts. An elementary acquaintance with topology, algebra, and analysis (in
cluding the notion of a manifold) is sufficient as far as the understanding of
this book is concerned. All the necessary properties and theorems have been
gathered in the preliminary chapters - either with proofs or with references
to standard and elementary textbooks.
The first chapter of the book is devoted to a study of the rings Oa of
holomorphic functions. The notions of analytic sets and germs are introduced
in the second chapter. Its aim is to present elementary properties of these
objects, also in connection with ideals of the rings Oa. The case of principal
germs (§5) and one-dimensional germs (Puiseux theorem, §6) are treated
separately.
The main step towards understanding of the local structure of analytic
sets is Ruckert's descriptive lemma proved in Chapter III. Among its conse
quences is the important Hilbert Nullstellensatz (§4).
In the fourth chapter, a study of local structure (normal triples, §1 ) is
followed by an exposition of the basic properties of analytic sets. The latter
includes theorems on the set of singular points, irreducibility, and decom
position into irreducible branches (§2). The role played by the ring 0 A of
an analytic germ is shown (§4). Then, the Remmert-Stein theorem on re
movable singularities is proved (§6). The last part of the chapter deals with
analytically constructible sets (§7).
The fifth chapter is concerned with holomorphic mappings between ana
lytic sets. To begin with, the theorem on multiplicities and Rouche's theorem
are proved for mappings between manifolds of the same dimension (§2). Next
- theorems concerning dimension of fibres are shown in the general case (§3).
After the introduction of (reduced) analytic spaces (via atlases) and extend
ing properties of analytic sets (onto these spaces), Remmert's proper and
open mapping theorems are proved (§§5 and 6). Finally, the Andreotti-Stoll
theorem on the structure of finite mappings is presented (§7).
Normal spaces and the normalization theorem are the main topic of the
sixth chapter. A necessary tool here is the Cartan-Oka theorem (§1), which
is proved without introducing the full notion of an analytic sheaf.
The seventh chapter is connected with ideas of the well-known paper
"GAGA" by Serre [38] about the phenomenon of the "necessary" algebraic
ity of analytic objects in projective spaces. Serre's theorem (which requires
the use of the theory of analytic sheaves and cohomologies) is not presented
though. After an elementary discussion of the manifold structure on the pro
jective and Grassmann spaces (§§2 and 4), blowings-up (§5) are introduced.
The projective and Grassmann spaces are also investigated later on in §16 as
algebraic manifolds. Next, Chow's theorems on algebraicity of analytic sets
en
(§6) as well as Rudin-Sadullaev criteria of algebraicity in are proved (§7).
Then constructible sets are introduced and Chevalley's theorem is shown.
Preface VII
The algebraic versIOn of Ruckert's lemma (§9) is followed by the Hilbert
Nullstellensatz (§10), the theorem about the degree of algebraic sets (§11),
and Bezout's theorem (§13). One part of the chapter is devoted to a study
of meromorphic and rational functions. After a series of characterizations
of such functions, the Siegel-Thimm theorem, the Hurwitz theorem, and a
theorem of Zariski on constructible graphs are proved. The last theorem is
used in the proof of Serre's algebraic graph theorem (§16). Then, Zariski's
theorem on analytic normality is shown (§16). Subsequently, algebraic spaces
("varietes algebriques" in the sense of Serre) are introduced and Chow's the
orem for such spaces is proved. Hironaka's example of a non-projective al
gebraic variety is given (§17). The purpose of the last part of the chapter
is the exposition of Chow's characterization of biholomorphic mappings of
Grassmann manifolds (§20). Its proof is based on the theorem on biholomor
phic mappings of factorial sets (§lS) and on the Andreotti-Salmon theorem
saying that Grassmann's cone is factorial (§19).
While working on this material, I have used many sources, particularly
the books by Bochner and Martin [11], Herve [24], Narasimhan [33], and
Whitney [44]. Among them, the last one is the closest in character to this
book.
I am grateful to my colleagues for numerous valuable comments. Above
all, I would like to thank Kamil Rusek and Tadeusz Winiarski. Their enthu
siasm, encouragement, and stimulation certainly played an important role
in the writing of this book. Also, I would like to express my gratitude to
Prof. Stanislaw Balcerzyk and Prof. Krzysztof Maurin, as well as to Slawomir
Cynk, Zbigniew Hajto, Marek Jarnicki, Grzegorz Jasinski, Krzysztof Kur
dyka, Wieslaw Pawlucki and Piotr Tworzewski from Cracow, Adam Parusin
ski from Gdansk, Piotr Pragacz from Torun, Fabbrizio Catanese and Ar
turo Vaz Ferreira from Pisa, Hans-Jorg Reiffen from Osnabruck, Jean-Jaques
Risler from Paris, and M. Umemura from Nagoya.
I wish to thank Maria Rataj for her dedicated work in the preparation
of the typescript.
Also, I am most grateful to JaS Ciaptak-Ggsienica and Hania Fronek
Gi}sienica for creating for me a very good environment for the writing of this
book.
Krzeptowki, September 19S3 Stanislaw Lojasiewicz.
PREFACE
TO THE ENGLISH EDITION
The text of the English edition has been slightly extended. Among other
changes, Section 7 of Chapter V (on finite holomorphic mappings) has been
modified and extended by the addition of the Grauert-Remmert formula (the
primitive element theorem) with some of its consequences. Furthermore, the
Grauert-Remmert theorem on uniform convergence (with a simple proof fol
lowing from an observation due to S. Cynk), preceded by Cartan's closedness
theorem, has been included in Section 1 of Chapter VI. Some facts related
to complete intersections have been added in Chapter VI (on normalization);
in particular: Schumacher's, Tsikh's, and Oka-Abhyankar's theorems, as well
as Serre's normality criterion. Naturally, this has required a series of modifi
cations in the preceding chapters.
I would like to convey here my sincere thanks to Professors Edward
Bierstone, Sean Dineen, and Stephen G. Krantz for reading the text and for
their valuable remarks.
I wish to thank very warmly th~ staff of Birkhauser Verlag for their
special effort which they put into this English edition.
Finally, many thanks should go to Dr Maciej Klimek who has undertaken
the translation of the Polish edition with the new supplements.
Krak6w, 21 March 1991 Stanislaw Lojasiewicz.
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Caly swiat obyndzies, nima takiej nika!
CONTENTS
Preface to the Polish Edition V
Preface to the English Edition IX
PRELIMIN ARIES
CHAPTER A. Algebra 1
§ 1. Rings, fields, modules, ideals, vector spaces 1
§ 2. Polynomials 15
§ 3. Polynomial mappings 20
§ 4. Symmetric polynomials. Discriminant 23
§ U. Extensions of fields 26
§ 6. Factorial rings 27
~ 7. Primitive element theorem 29
§ 8. Extensions of rings 30
§ 9. Noetherian rings 34
§10. Local rings 39
§II. Localization 46
§12. Krull's dimension 53
§13. Modules of syzygies and homological dimension 57
§14. The depth of a module 61
§15. Regular rings 66
CHAPTER B. Topology 72
§ 1. Some topological properties of sets and families of sets 72
§ 2. Open, closed and proper mappings 74
§ 3. Local homeomorphisms and coverings 77
§ 4. Germs of sets and functions 80
§ 5. The topology of a finite dimensional vector space
(over C or R) 84
§ 6. The topology of the Grassmann space 87
