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It was the aim of the Erlangen meeting in May 1988 to bring together number theoretists and algebraic geometers to discuss problems of common interest, such as moduli problems, complex tori, integral points, rationality questions, automorphic forms. In recent years such problems, which are simultaneously of arithmetic and geometric interest, have become increasingly important. This proceedings volume contains 12 original research papers. Its main topics are theta functions, modular forms, abelian varieties and algebraic three-folds.
In this book, Miranda takes the approach that algebraic curves are best encountered for the first time over the complex numbers, where the reader's classical intuition about surfaces, integration, and other concepts can be brought into play. Therefore, many examples of algebraic curves are presented in the first chapters. In this way, the book begins as a primer on Riemann surfaces, with complex charts and meromorphic functions taking centre stage. But the main examples come fromprojective curves, and slowly but surely the text moves toward the algebraic category. Proofs of the Riemann-Roch and Serre Dualtiy Theorems are presented in an algebraic manner, via an adaptation of the adelic proof, expressed completely in terms of solving a Mittag-Leffler problem. Sheaves andcohomology are introduced as a unifying device in the later chapters, so that their utility and naturalness are immediately obvious. Requiring a background of one term of complex variable theory and a year of abstract algebra, this is an excellent graduate textbook for a second-term course in complex variables or a year-long course in algebraic geometry.
The articles in this volume were written to commemorate Reinhold Remmert's 60th birthday in June, 1990. They are surveys, meant to facilitate access to some of the many aspects of the theory of complex manifolds, and demonstrate the interplay between complex analysis and many other branches of mathematics, algebraic geometry, differential topology, representations of Lie groups, and mathematical physics being only the most obvious of these branches. Each of these articles should serve not only to describe the particular circle of ideas in complex analysis with which it deals but also as a guide to the many mathematical ideas related to its theme.
In 1961 Smale established the generalized Poincare Conjecture in dimensions greater than or equal to 5 [129] and proceeded to prove the h-cobordism theorem [130]. This result inaugurated a major effort to classify all possible smooth and topological structures on manifolds of dimension at least 5. By the mid 1970's the main outlines of this theory were complete, and explicit answers (especially concerning simply connected manifolds) as well as general qualitative results had been obtained. As an example of such a qualitative result, a closed, simply connected manifold of dimension 2: 5 is determined up to finitely many diffeomorphism possibilities by its homotopy type and its Pontrjagin classes. There are similar results for self-diffeomorphisms, which, at least in the simply connected case, say that the group of self-diffeomorphisms of a closed manifold M of dimension at least 5 is commensurate with an arithmetic subgroup of the linear algebraic group of all automorphisms of its so-called rational minimal model which preserve the Pontrjagin classes [131]. Once the high dimensional theory was in good shape, attention shifted to the remaining, and seemingly exceptional, dimensions 3 and 4. The theory behind the results for manifolds of dimension at least 5 does not carryover to manifolds of these low dimensions, essentially because there is no longer enough room to maneuver. Thus new ideas are necessary to study manifolds of these "low" dimensions.
Recently there has been considerable interest in developing techniques based on number theory to attack problems of 3-manifolds; Contains many examples and lots of problems; Brings together much of the existing literature of Kleinian groups in a clear and concise way; At present no such text exists
The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and clarifications have been made, and the layout has been improved.
This volume presents papers dedicated to Professor Shoshichi Kobayashi, commemorating the occasion of his sixtieth birthday on January 4, 1992.The principal theme of this volume is “Geometry and Analysis on Complex Manifolds”. It emphasizes the wide mathematical influence that Professor Kobayashi has on areas ranging from differential geometry to complex analysis and algebraic geometry. It covers various materials including holomorphic vector bundles on complex manifolds, Kähler metrics and Einstein–Hermitian metrics, geometric function theory in several complex variables, and symplectic or non-Kähler geometry on complex manifolds. These are areas in which Professor Kobayashi has made strong impact and is continuing to make many deep invaluable contributions.
In one guise or another, many mathematicians are familiar with certain arithmetic groups, such as $\mathbf{Z}$ or $\textrm{SL}(n, \mathbf{Z})$. Yet, many applications of arithmetic groups and many connections to other subjects within mathematics are less well known. Indeed, arithmetic groups admit many natural and important generalizations. The purpose of this expository book is to explain, through some brief and informal comments and extensive references, what arithmetic groups and their generalizations are, why they are important to study, and how they can be understood and applied to many fields, such as analysis, geometry, topology, number theory, representation theory, and algebraic geometry. It is hoped that such an overview will shed a light on the important role played by arithmetic groups in modern mathematics. Titles in this series are co-published with International Press, Cambridge, MA.Table of Contents: Introduction; General comments on references; Examples of basic arithmetic groups; General arithmetic subgroups and locally symmetric spaces; Discrete subgroups of Lie groups and arithmeticity of lattices in Lie groups; Different completions of $\mathbb{Q}$ and $S$-arithmetic groups over number fields; Global fields and $S$-arithmetic groups over function fields; Finiteness properties of arithmetic and $S$-arithmetic groups; Symmetric spaces, Bruhat-Tits buildings and their arithmetic quotients; Compactifications of locally symmetric spaces; Rigidity of locally symmetric spaces; Automorphic forms and automorphic representations for general arithmetic groups; Cohomology of arithmetic groups; $K$-groups of rings of integers and $K$-groups of group rings; Locally homogeneous manifolds and period domains; Non-cofinite discrete groups, geometrically finite groups; Large scale geometry of discrete groups; Tree lattices; Hyperbolic groups; Mapping class groups and outer automorphism groups of free groups; Outer automorphism group of free groups and the outer spaces; References; Index. Review from Mathematical Reviews: ...the author deserves credit for having done the tremendous job of encompassing every aspect of arithmetic groups visible in today's mathematics in a systematic manner; the book should be an important guide for some time to come.(AMSIP/43.
This volume presents papers dedicated to Professor Shoshichi Kobayashi, commemorating the occasion of his sixtieth birthday on January 4, 1992.The principal theme of this volume is “Geometry and Analysis on Complex Manifolds”. It emphasizes the wide mathematical influence that Professor Kobayashi has on areas ranging from differential geometry to complex analysis and algebraic geometry. It covers various materials including holomorphic vector bundles on complex manifolds, Kähler metrics and Einstein-Hermitian metrics, geometric function theory in several complex variables, and symplectic or non-Kähler geometry on complex manifolds. These are areas in which Professor Kobayashi has made strong impact and is continuing to make many deep invaluable contributions.