The book is a mostly translated reprint of a report on cohomology of groups from the 1950s and 1960s, originally written as background for the Artin-Tate notes on class field theory, following the cohomological approach. This report was first published (in French) by Benjamin. For this new English edition, the author added Tate's local duality, written up from letters which John Tate sent to Lang in 1958 - 1959. Except for this last item, which requires more substantial background in algebraic geometry and especially abelian varieties, the rest of the book is basically elementary, depending only on standard homological algebra at the level of first year graduate students.

This book provides a classification of all three-dimensional complex manifolds for which there exists a transitive action (by biholomorphic transformations) of a real Lie group. This means two homogeneous complex manifolds are considered equivalent if they are isomorphic as complex manifolds. The classification is based on methods from Lie group theory, complex analysis and algebraic geometry. Basic knowledge in these areas is presupposed.

As an interesting object of arithmetic, algebraic and analytic geometry the complex ball was born in a paper of the French Mathematician E. PICARD in 1883. In recent developments the ball finds great interest again in the framework of SHIMURA varieties but also in the theory of diophantine equations (asymptotic FERMAT Problem, see ch. VI). At first glance the original ideas and the advanced theories seem to be rather disconnected. With these lectures I try to build a bridge from the analytic origins to the actual research on effective problems of arithmetic algebraic geometry. The best motivation is HILBERT'S far-reaching program consisting of 23 prob lems (Paris 1900) " . . . one should succeed in finding and discussing those functions which play the part for any algebraic number field corresponding to that of the exponential function in the field of rational numbers and of the elliptic modular functions in the imaginary quadratic number field." This message can be found in the 12-th problem "Extension of KRONECKER'S Theorem on Abelian Fields to Any Algebraic Realm of Rationality" standing in the middle of HILBERTS'S pro gram. It is dedicated to the construction of number fields by means of special value of transcendental functions of several variables. The close connection with three other HILBERT problems will be explained together with corresponding advanced theories, which are necessary to find special effective solutions, namely: 7. Irrationality and Transcendence of Certain Numbers; 21."

Classical algebraic geometry, inseparably connected with the names of Abel, Riemann, Weierstrass, Poincaré, Clebsch, Jacobi and other outstanding mathematicians of the last century, was mainly an analytical theory. In our century the methods and ideas of topology, commutative algebra and Grothendieck's schemes enriched it and seemed to have replaced once and forever the somewhat naive language of classical algebraic geometry. This classic book, written in 1897, covers the whole of algebraic geometry and associated theories. Baker discusses the subject in terms of transcendental functions, and theta functions in particular. Many of the ideas put forward are of continuing relevance today, and some of the most exciting ideas from theoretical physics draw on work presented here.

Preface of the Editors Ce volume prend sa source dans le Colloque en l'honneur de Pierre Dolbeault, organise a l'occasion de son depart a la retraite, a 'initiative des Universites de Paris 6 et de Poitiers. Ce colloque, consacre a l' Analyse Complexe et a la Geometrie Analytique, s'est tenu a Paris, sur le campus de l'Universite Pierreet Marie Curie, du 23 au 26 Juin 1992.11 areuni autour de ces themes une centaine de congressistes, dont de nombreux mathematiciens etrangers (Allemagne, Argentine, Canada, Etats-Unis, Islande, Italie, Pologne, Roumanie, Russie, Suede). Nous avons souhaite prolanger cet hommage par la publication d'un volume dedie a Pierre Dolbeault. Le present recueil d'articles ne constitue pas strictement les actes du Colloque. Nous avons voulu qu'il rassemble uniquement des articles originaux ou synthetiques, qui illustrent l' ceuvre scientifique de Pierre Dolbeault a travers les themes abordes ou la personnalite de leurs auteurs. Nous remercions les conferenciers qui ont bien voulu contribuer a cet ouvrage, et Klas Diederich de l'avoir accueilli dans la collection "Aspects of Mathematics" qu'il dirige. Au nom du Comite d'Organisation du Colloque (C. Laurent-Thiebaut, J. Le Potier, J.B. Poly, J.P. Vigue et nous-memes), nous remercions les institutions qui nous ont apporte leur aide financiere et materielle: les Universites Paris 6 et de Poitiers, la Direction de la Recherche et des Etudes Doctorales, le Centre National de la Recherche Scientifique et le Ministere de la Recherche et de la Technologie.

This book makes a systematic study of the relations between the etale cohomology of a scheme and the orderings of its residue fields. A major result is that in high degrees, etale cohomology is cohomology of the real spectrum. It also contains new contributions in group cohomology and in topos theory. It is of interest to graduate students and researchers who work in algebraic geometry (not only real) and have some familiarity with the basics of etale cohomology and Grothendieck sites. Independently, it is of interest to people working in the cohomology theory of groups or in topos theory.

The equivariant derived category of sheaves is introduced. All usual functors on sheaves are extended to the equivariant situation. Some applications to the equivariant intersection cohomology are given.
The theory may be useful to specialists in representation theory, algebraic geometry or topology.

Tropical geometry is a combinatorial shadow of algebraic geometry, offering new polyhedral tools to compute invariants of algebraic varieties. It is based on tropical algebra, where the sum of two numbers is their minimum and the product is their sum. This turns polynomials into piecewise-linear functions, and their zero sets into polyhedral complexes. These tropical varieties retain a surprising amount of information about their classical counterparts. Tropical geometry is a young subject that has undergone a rapid development since the beginning of the 21st century. While establishing itself as an area in its own right, deep connections have been made to many branches of pure and applied mathematics. This book offers a self-contained introduction to tropical geometry, suitable as a course text for beginning graduate students. Proofs are provided for the main results, such as the Fundamental Theorem and the Structure Theorem. Numerous examples and explicit computations illustrate the main concepts. Each of the six chapters concludes with problems that will help the readers to practice their tropical skills, and to gain access to the research literature.

From the reviews: "The author's book ...] saw its first edition in 1935. ...] Now as before, the original text of the book is an excellent source for an interested reader to study the methods of classical algebraic geometry, and to find the great old results. ...] a timelessly beautiful pearl in the cultural heritage of mathematics as a whole." Zentralblatt MATH

In this book we study Hilbert schemes of zero-dimensional subschemes of smooth varieties and several related parameter varieties of interest in enumerative geometry. The main aim here is to describe their cohomology and Chow rings. Some enumerative applications are also given. The Weil conjectures are used to compute the Betti numbers of many of the varieties considered, thus also illustrating how this powerful tool can be applied. The book is essentially self-contained, assuming only a basic knowledge of algebraic geometry; it is intended both for graduate students and research mathematicians interested in Hilbert schemes, enumertive geometry and moduli spaces.

The aim of this book is to show that Shimura varieties provide a tool to construct certain interesting objects in arithmetic algebraic geometry. These objects are the so-called mixed motives: these are of great arithmetic interest. They can be viewed as quasiprojective algebraic varieties over Q which have some controlled ramification and where we know what we have to add at infinity to compactify them. The existence of certain of these mixed motives is related to zeroes of L-functions attached to certain pure motives. This is the content of the Beilinson-Deligne conjectures which are explained in some detail in the first chapter of the book. The rest of the book is devoted to the description of the general principles of construction (Chapter II) and the discussion of several examples in Chapter II-IV. In an appendix we explain how the (topological) trace formula can be used to get some understanding of the problems discussed in the book. Only some of this material is really proved: the book also contains speculative considerations, which give some hints as to how the problems could be tackled. Hence the book should be viewed as the outline of a programme and it offers some interesting problems which are of importance and can be pursued by the reader. In the widest sense the subject of the paper is number theory and belongs to what is called arithmetic algebraic geometry. Thus the reader should be familiar with some algebraic geometry, number theory, the theory of Liegroups and their arithmetic subgroups. Some problems mentioned require only part of this background knowledge.

In recent years new topological methods, especially the theory of sheaves founded by J. LERAY, have been applied successfully to algebraic geometry and to the theory of functions of several complex variables. H. CARTAN and J. -P. SERRE have shown how fundamental theorems on holomorphically complete manifolds (STEIN manifolds) can be for mulated in terms of sheaf theory. These theorems imply many facts of function theory because the domains of holomorphy are holomorphically complete. They can also be applied to algebraic geometry because the complement of a hyperplane section of an algebraic manifold is holo morphically complete. J. -P. SERRE has obtained important results on algebraic manifolds by these and other methods. Recently many of his results have been proved for algebraic varieties defined over a field of arbitrary characteristic. K. KODAIRA and D. C. SPENCER have also applied sheaf theory to algebraic geometry with great success. Their methods differ from those of SERRE in that they use techniques from differential geometry (harmonic integrals etc. ) but do not make any use of the theory of STEIN manifolds. M. F. ATIYAH and W. V. D. HODGE have dealt successfully with problems on integrals of the second kind on algebraic manifolds with the help of sheaf theory. I was able to work together with K. KODAIRA and D. C. SPENCER during a stay at the Institute for Advanced Study at Princeton from 1952 to 1954."

This book is about the smooth classification of a certain class of algebraicsurfaces, namely regular elliptic surfaces of geometric genus one, i.e. elliptic surfaces with b1 = 0 and b2+ = 3. The authors give a complete classification of these surfaces up to diffeomorphism. They achieve this result by partially computing one of Donalson's polynomial invariants. The computation is carried out using techniques from algebraic geometry. In these computations both thebasic facts about the Donaldson invariants and the relationship of the moduli space of ASD connections with the moduli space of stable bundles are assumed known. Some familiarity with the basic facts of the theory of moduliof sheaves and bundles on a surface is also assumed. This work gives a good and fairly comprehensive indication of how the methods of algebraic geometry can be used to compute Donaldson invariants.

The study of hypersurface quadrilateral singularities can be reduced to the study of elliptic K3 surfaces with a singular fiber of type I * 0 (superscript *, subscript 0), and therefore these notes consider, besides the topics of the title, such K3 surfaces too. The combinations of rational double points that can occur on fibers in the semi-universal deformations of quadrilateral singularities are examined, to show that the possible combinations can be described by a certain law from the viewpoint of Dynkin graphs. This is equivalent to saying that the possible combinations of singular fibers in elliptic K3 surfaces with a singular fiber of type I * 0 (superscript *, subscript 0) can be described by a certain law using classical Dynkin graphs appearing in the theory of semi-simple Lie groups. Further, a similar description for thecombination of singularities on plane sextic curves is given. Standard knowledge of algebraic geometry at the level of graduate students is expected. A new method based on graphs will attract attention of researches.

Etale Cohomology is one of the most important methods in modern Algebraic Geometry and Number Theory. It has, in the last decades, brought fundamental new insights in arithmetic and algebraic geometric problems with many applications and many important results. The book gives a short and easy introduction into the world of Abelian Categories, Derived Functors, Grothendieck Topologies, Sheaves, General Etale Cohomology, and Etale Cohomology of Curves."

This work provides a lucid and rigorous account of the foundations of modern algebraic geometry. The authors have confined themselves to fundamental concepts and geometrical methods, and do not give detailed developments of geometrical properties, but geometrical meaning has been emphasised throughout. This first volume is divided into two parts. The first is devoted to pure algebra; the basic notions, the theory of matrices over a non-commutative ground field and a study of algebraic equations. The second part is concerned with the definitions and basic properties of projective space in n dimensions. It concludes with a purely algebraic account of collineations and correlations. The other two volumes of Hodge and Pedoe's classic work are also available. Together, these books give an insight into algebraic geometry that is unique and unsurpassed.

This volume is based on lectures given at a workshop and conference on symplectic geometry at the University of Warwick in August 1990. The area of symplectic geometry has developed rapidly in the past ten years with major new discoveries that were motivated by and have provided links with many other subjects such as dynamical systems, topology, gauge theory, mathematical physics and singularity theory. The conference brought together a number of leading experts in these areas of mathematics. The contributions to this volume reflect the richness of the subject and include expository papers as well as original research. They will be an essential source for all research mathematicians in symplectic geometry.

Ten years after the first Rennes international meeting on real algebraic geometry, the second one looked at the developments in the subject during the intervening decade - see the 6 survey papers listed below. Further contributions from the participants on recent research covered real algebra and geometry, topology of real algebraic varieties and 16thHilbert problem, classical algebraic geometry, techniques in real algebraic geometry, algorithms in real algebraic geometry, semialgebraic geometry, real analytic geometry. CONTENTS: Survey papers: M. Knebusch: Semialgebraic topology in the last ten years.- R. Parimala: Algebraic and topological invariants of real algebraic varieties.- Polotovskii, G.M.: On the classification of decomposing plane algebraic curves.- Scheiderer, C.: Real algebra and its applications to geometry in the last ten years: some major developments and results.- Shustin, E.L.: Topology of real plane algebraic curves.- Silhol, R.: Moduli problems in real algebraic geometry. Further contributions by: S. Akbulut and H. King; C. Andradas and J. Ruiz; A. Borobia; L. Br-cker; G.W. Brumfield; A. Castilla; Z. Charzynski and P. Skibinski; M. Coste and M. Reguiat; A. Degtyarev; Z. Denkowska; J.-P. Francoise and F. Ronga; J.M. Gamboa and C. Ueno; D. Gondard- Cozette; I.V. Itenberg; P. Jaworski; A. Korchagin; T. Krasinksi and S. Spodzieja; K. Kurdyka; H. Lombardi; M. Marshall and L. Walter; V.F. Mazurovskii; G. Mikhalkin; T. Mostowski and E. Rannou; E.I. Shustin; N. Vorobjov.

About ten years ago, V.D. Goppa found a surprising connection between the theory of algebraic curves over a finite field and error-correcting codes. The aim of the meeting "Algebraic Geometry and Coding Theory" was to give a survey on the present state of research in this field and related topics. The proceedings contain research papers on several aspects of the theory, among them: Codes constructed from special curves and from higher-dimensional varieties, Decoding of algebraic geometric codes, Trace codes, Exponen- tial sums, Fast multiplication in finite fields, Asymptotic number of points on algebraic curves, Sphere packings.

This volume contains three long lecture series by J.L. Colliot-Thelene, Kazuya Kato and P. Vojta. Their topics are respectively the connection between algebraic K-theory and the torsion algebraic cycles on an algebraic variety, a new approach to Iwasawa theory for Hasse-Weil L-function, and the applications of arithemetic geometry to Diophantine approximation. They contain many new results at a very advanced level, but also surveys of the state of the art on the subject with complete, detailed profs and a lot of background. Hence they can be useful to readers with very different background and experience. CONTENTS: J.L. Colliot-Thelene: Cycles algebriques de torsion et K-theorie algebrique.- K. Kato: Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions.- P. Vojta: Applications of arithmetic algebraic geometry to diophantine approximations.

From the very beginning, algebraic topology has developed under the influ- ence of the problems posed by trying to understand the topological properties of complex algebraic varieties (e.g., the pioneering work by Poincare and Lefschetz). Especially in the work of Lefschetz [Lf2], the idea is made explicit that singularities are important in the study of the topology even in the case of smooth varieties. What is known nowadays about the topology of smooth and singular vari- eties is quite impressive. The many existing results may be roughly divided into two classes as follows: (i) very general results or theories, like stratified Morse theory and (mixed) Hodge theory, see, for instance, Goresky-MacPherson [GM], Deligne [Del], and Steenbrink [S6]; and (ii) specific topics of great subtlety and beauty, like the study of the funda- mental group of the complement in [p>2 of a singular plane curve initiated by Zariski or Griffiths' theory relating the rational differential forms to the Hodge filtration on the middle cohomology group of a smooth projec- tive hypersurface. The aim of this book is precisely to introduce the reader to some topics in this latter class. Most of the results to be discussed, as well as the related notions, are at least two decades old, and specialists use them intensively and freely in their work. Nevertheless, it is impossible to find an adequate intro- duction to this subject, which gives a good feeling for its relations with other parts of algebraic geometry and topology.

This book is a general introduction to Higher Algebraic K-groups of rings and algebraic varieties, which were first defined by Quillen at the beginning of the 70's. These K-groups happen to be useful in many different fields, including topology, algebraic geometry, algebra and number theory. The goal of this volume is to provide graduate students, teachers and researchers with basic definitions, concepts and results, and to give a sampling of current directions of research. Written by five specialists of different parts of the subject, each set of lectures reflects the particular perspective ofits author. As such, this volume can serve as a primer (if not as a technical basic textbook) for mathematicians from many different fields of interest.

In this expository paper we sketch some interrelations between several famous conjectures in number theory and algebraic geometry that have intrigued mathematicians for a long period of time. Starting from Fermat's Last Theorem one is naturally led to intro duce L-functions, the main motivation being the calculation of class numbers. In particular, Kummer showed that the class numbers of cyclotomic fields playa decisive role in the corroboration of Fermat's Last Theorem for a large class of exponents. Before Kummer, Dirich let had already successfully applied his L-functions to the proof of the theorem on arithmetic progressions. Another prominent appearance of an L-function is Riemann's paper where the now famous Riemann Hypothesis was stated. In short, nineteenth century number theory showed that much, if not all, of number theory is reflected by proper ties of L-functions. Twentieth century number theory, class field theory and algebraic geometry only strengthen the nineteenth century number theorists's view. We just mention the work of E. Heeke, E. Artin, A. Weil and A. Grothendieck with his collaborators. Heeke generalized Dirichlet's L-functions to obtain results on the distribution of primes in number fields. Artin introduced his L-functions as a non-abelian generaliza tion of Dirichlet's L-functions with a generalization of class field the ory to non-abelian Galois extensions of number fields in mind. Weil introduced his zeta-function for varieties over finite fields in relation to a problem in number theory."

In the Teichm ller theory of Riemann surfaces, besides the classical theory of quasi-conformal mappings, vari- ous approaches from differential geometry and algebraic geometry have merged in recent years. Thus the central subject of "Complex Structure" was a timely choice for the joint meetings in Katata and Kyoto in 1989. The invited participants exchanged ideas on different approaches to related topics in complex geometry and mapped out the prospects for the next few years of research.

The International Conference "Algebraic Geometry and Analytic Geometry, Tokyo 1990" was held at Tokyo Metropolitan University and the Tokyo Training Center of Daihyaku Mutual Life Insurance Co., from August 13 through August 17, 1990, under the co-sponsorship of the Mathematical Society of Japan. It was one of the satellite conferences of ICM90, Kyoto, and approximately 300 participants, including more than 100 from overseas, attended the conference. The academic program was divided into two parts, the morning sessions and the afternoon sessions. The morning sessions were held at Tokyo Metropolitan University, and two one-hour plenary lectures were delivered every day. The afternoon sessions at the Tokyo Training Center, intended for a more specialized audience, consisted of four separate subsessions: Arithemetic Geometry, Algebraic Geometry, Analytic Geometry I and Analytic Geometry II. This book contains papers which grew out of the talks at the conference. The committee in charge of the organization and program consisted of A. Fujiki, K. Kato, T. Katsura, Y. Kawamata, Y. Miyaoka, S. Mori, K. Saito, N. Sasakura, T. Suwa and K. Watanabe. We would like to take this opportunity to thank the many mathematicians and students who cooperated to make the conference possible, especially Professors T. Fukui, S. Ishii, Y. Kitaoka, M. Miyanishi, Y. Namikawa, T. Oda, F. Sakai and T. Shioda for their valuable advice and assistance in organizing this conference. Financial support was mainly provided by personal contributions from Professors M.