# An Introduction To Riemann Surfaces Algebraic Curves And Moduli Spaces

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## An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces

### An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces by Martin Schlichenmaier Book Summary:

This book gives an introduction to modern geometry. Starting from an elementary level, the author develops deep geometrical concepts that play an important role in contemporary theoretical physics, presenting various techniques and viewpoints along the way. This second edition contains two additional, more advanced geometric techniques: the modern language and modern view of Algebraic Geometry and Mirror Symmetry.

## An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces

### An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces by Martin Schlichenmaier Book Summary:

This lecture is intended as an introduction to the mathematical concepts of algebraic and analytic geometry. It is addressed primarily to theoretical physicists, in particular those working in string theories. The author gives a very clear exposition of the main theorems, introducing the necessary concepts by lucid examples, and shows how to work with the methods of algebraic geometry. As an example he presents the Krichever-Novikov construction of algebras of Virasaro type. The book will be welcomed by many researchers as an overview of an important branch of mathematics, a collection of useful formulae and an excellent guide to the more extensive mathematical literature.

## Riemann Surfaces and Algebraic Curves

### Riemann Surfaces and Algebraic Curves by Renzo Cavalieri,Eric Miles Book Summary:

Classroom-tested and featuring over 100 exercises, this text introduces the key algebraic geometry field of Hurwitz theory.

## Moduli of Riemann Surfaces, Real Algebraic Curves, and Their Superanalogs

### Moduli of Riemann Surfaces, Real Algebraic Curves, and Their Superanalogs by S. M. Natanzon Book Summary:

The space of all Riemann surfaces (the so-called moduli space) plays an important role in algebraic geometry and its applications to quantum field theory. The present book is devoted to the study of topological properties of this space and of similar moduli spaces, such as the space of real algebraic curves, the space of mappings, and also superanalogs of all these spaces. The book can be used by researchers and graduate students working in algebraic geometry, topology, and mathematical physics.

## Algebraic Curves

### Algebraic Curves by Maxim E. Kazaryan,Sergei K. Lando,Victor V. Prasolov Book Summary:

This book offers a concise yet thorough introduction to the notion of moduli spaces of complex algebraic curves. Over the last few decades, this notion has become central not only in algebraic geometry, but in mathematical physics, including string theory, as well. The book begins by studying individual smooth algebraic curves, including the most beautiful ones, before addressing families of curves. Studying families of algebraic curves often proves to be more efficient than studying individual curves: these families and their total spaces can still be smooth, even if there are singular curves among their members. A major discovery of the 20th century, attributed to P. Deligne and D. Mumford, was that curves with only mild singularities form smooth compact moduli spaces. An unexpected byproduct of this discovery was the realization that the analysis of more complex curve singularities is not a necessary step in understanding the geometry of the moduli spaces. The book does not use the sophisticated machinery of modern algebraic geometry, and most classical objects related to curves – such as Jacobian, space of holomorphic differentials, the Riemann-Roch theorem, and Weierstrass points – are treated at a basic level that does not require a profound command of algebraic geometry, but which is sufficient for extending them to vector bundles and other geometric objects associated to moduli spaces. Nevertheless, it offers clear information on the construction of the moduli spaces, and provides readers with tools for practical operations with this notion. Based on several lecture courses given by the authors at the Independent University of Moscow and Higher School of Economics, the book also includes a wealth of problems, making it suitable not only for individual research, but also as a textbook for undergraduate and graduate coursework

## Advances in Moduli Theory

### Advances in Moduli Theory by Kenji Ueno,Yūji Shimizu Book Summary:

The word 'moduli' in the sense of this book first appeared in the epoch-making paper of B. Riemann, Theorie der Abel'schen Funktionen, published in 1857. Riemann defined a Riemann surface of an algebraic function field as a branched covering of a one-dimensional complex projective space, and found out that Riemann surfaces have parameters. This work gave birth to the theory of moduli. However, the viewpoint regarding a Riemann surface as an algebraic curve became the mainstream, and the moduli meant the parameters for the figures (graphs) defined by equations. In 1913, H. Weyl defined a Riemann surface as a complex manifold of dimension one. Moreover, Teichmuller's theory of quasiconformal mappings and Teichmuller spaces made a start for new development of the theory of moduli, making possible a complex analytic approach toward the theory of moduli of Riemann surfaces.This theory was then investigated and made complete by Ahlfors, Bers, Rauch, and others. However, the theory of Teichmuller spaces utilized the special nature of complex dimension one, and it was difficult to generalize it to an arbitrary dimension in a direct way. It was Kodaira-Spencer's deformation theory of complex manifolds that allowed one to study arbitrary dimensional complex manifolds. Initial motivation in Kodaira-Spencer's discussion was the need to clarify what one should mean by number of moduli. Their results, together with further work by Kuranishi, provided this notion with intrinsic meaning. This book begins by presenting the Kodaira-Spencer theory in its original naive form in Chapter 1 and introduces readers to moduli theory from the viewpoint of complex analytic geometry.Chapter 2 briefly outlines the theory of period mapping and Jacobian variety for compact Riemann surfaces, with the Torelli theorem as a goal. The theory of period mappings for compact Riemann surfaces can be generalized to the theory of period mappings in terms of Hodge structures for compact Kahler manifolds. In Chapter 3, the authors state the theory of Hodge structures, focusing briefly on period mappings. Chapter 4 explains conformal field theory as an application of moduli theory. This is the English translation of a book originally published in Japanese. Other books by Kenji Ueno published in this AMS series, ""Translations of Mathematical Monographs"", include ""An Introduction to Algebraic Geometry"", Volume 166, ""Algebraic Geometry 1: From Algebraic Varieties to Schemes"", Volume 185, and ""Algebraic Geometry 2: Sheaves and Cohomology"", Volume 197.

## Moduli Spaces of Riemann Surfaces

### Moduli Spaces of Riemann Surfaces by Benson Farb,Richard Hain,Eduard Looijenga Book Summary:

Mapping class groups and moduli spaces of Riemann surfaces were the topics of the Graduate Summer School at the 2011 IAS/Park City Mathematics Institute. This book presents the nine different lecture series comprising the summer school, covering a selection of topics of current interest. The introductory courses treat mapping class groups and Teichmüller theory. The more advanced courses cover intersection theory on moduli spaces, the dynamics of polygonal billiards and moduli spaces, the stable cohomology of mapping class groups, the structure of Torelli groups, and arithmetic mapping class groups. The courses consist of a set of intensive short lectures offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures do not duplicate standard courses available elsewhere. The book should be a valuable resource for graduate students and researchers interested in the topology, geometry and dynamics of moduli spaces of Riemann surfaces and related topics. Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

## Geometry of Riemann Surfaces

### Geometry of Riemann Surfaces by Frederick P. Gardiner,Gabino González-Diez,Christos Kourouniotis Book Summary:

Riemann surfaces is a thriving area of mathematics with applications to hyperbolic geometry, complex analysis, conformal dynamics, discrete groups, algebraic curves and more. This collection of articles presents original research and expert surveys of important related topics, making the field accessible to research workers, graduate students and teachers.

## The Moduli Space of Curves

### The Moduli Space of Curves by Robert H. Dijkgraaf,Carel Faber,Gerard B.M. van der Geer Book Summary:

The moduli space Mg of curves of fixed genus g – that is, the algebraic variety that parametrizes all curves of genus g – is one of the most intriguing objects of study in algebraic geometry these days. Its appeal results not only from its beautiful mathematical structure but also from recent developments in theoretical physics, in particular in conformal field theory.

## Moduli of Vector Bundles

### Moduli of Vector Bundles by Masaki Maruyama Book Summary:

"Contains papers presented at the 35th Taniguchi International Symposium held recently in Sanda and Kyoto, Japan. Details the latest developments concerning moduli spaces of vector bundles or instantons and their application. Covers a broad array of topics in both differential and algebraic geometry."

## Computational Aspects of Algebraic Curves

### Computational Aspects of Algebraic Curves by N.A Book Summary:

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## An Introduction to Riemann Surfaces

### An Introduction to Riemann Surfaces by Terrence Napier,Mohan Ramachandran Book Summary:

This textbook presents a unified approach to compact and noncompact Riemann surfaces from the point of view of the so-called L2 $\bar{\delta}$-method. This method is a powerful technique from the theory of several complex variables, and provides for a unique approach to the fundamentally different characteristics of compact and noncompact Riemann surfaces. The inclusion of continuing exercises running throughout the book, which lead to generalizations of the main theorems, as well as the exercises included in each chapter make this text ideal for a one- or two-semester graduate course.

## Geometric Galois Actions: Volume 2, The Inverse Galois Problem, Moduli Spaces and Mapping Class Groups

### Geometric Galois Actions: Volume 2, The Inverse Galois Problem, Moduli Spaces and Mapping Class Groups by Leila Schneps,Pierre Lochak,J. W. S. Cassels Book Summary:

This book surveys progress in the domains described in the hitherto unpublished manuscript "Esquisse d'un Programme" (Sketch of a Program) by Alexander Grothendieck. It will be of wide interest amongst workers in algebraic geometry, number theory, algebra and topology.

## Mathematical Reviews

### Mathematical Reviews by N.A Book Summary:

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## Geometry of Algebraic Curves

### Geometry of Algebraic Curves by Enrico Arbarello,Maurizio Cornalba,Phillip Griffiths Book Summary:

The second volume of the Geometry of Algebraic Curves is devoted to the foundations of the theory of moduli of algebraic curves. Its authors are research mathematicians who have actively participated in the development of the Geometry of Algebraic Curves. The subject is an extremely fertile and active one, both within the mathematical community and at the interface with the theoretical physics community. The approach is unique in its blending of algebro-geometric, complex analytic and topological/combinatorial methods. It treats important topics such as Teichmüller theory, the cellular decomposition of moduli and its consequences and the Witten conjecture. The careful and comprehensive presentation of the material is of value to students who wish to learn the subject and to experts as a reference source. The first volume appeared 1985 as vol. 267 of the same series.

## Lectures on K3 Surfaces

### Lectures on K3 Surfaces by Daniel Huybrechts Book Summary:

Simple enough for detailed study, rich enough to show interesting behavior, K3 surfaces illuminate core methods in algebraic geometry.

## Journal of the Korean Mathematical Society

### Journal of the Korean Mathematical Society by N.A Book Summary:

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## Russian Mathematical Surveys

### Russian Mathematical Surveys by N.A Book Summary:

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## The Geometry, Topology And Physics Of Moduli Spaces Of Higgs Bundles

### The Geometry, Topology And Physics Of Moduli Spaces Of Higgs Bundles by Richard Wentworth,Graeme Wilkin Book Summary:

In the 25 years since their introduction, Higgs bundles have seen a surprising number of interactions within different areas of mathematics and physics. There is a recent surge of interest following Ngô Bau Châu's proof of the Fundamental Lemma and the work of Kapustin and Witten on the Geometric Langlands program. The program on The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles, was held at the Institute for Mathematical Sciences at the National University of Singapore during 2014. It hosted a number of lectures on recent topics of importance related to Higgs bundles, and it is the purpose of this volume to collect these lectures in a form accessible to graduate students and young researchers interested in learning more about this field.

## Geometry of Algebraic Curves

### Geometry of Algebraic Curves by Enrico Arbarello,Maurizio Cornalba,Phillip Griffiths,Joseph Daniel Harris Book Summary:

In recent years there has been enormous activity in the theory of algebraic curves. Many long-standing problems have been solved using the general techniques developed in algebraic geometry during the 1950's and 1960's. Additionally, unexpected and deep connections between algebraic curves and differential equations have been uncovered, and these in turn shed light on other classical problems in curve theory. It seems fair to say that the theory of algebraic curves looks completely different now from how it appeared 15 years ago; in particular, our current state of knowledge repre sents a significant advance beyond the legacy left by the classical geometers such as Noether, Castelnuovo, Enriques, and Severi. These books give a presentation of one of the central areas of this recent activity; namely, the study of linear series on both a fixed curve (Volume I) and on a variable curve (Volume II). Our goal is to give a comprehensive and self-contained account of the extrinsic geometry of algebraic curves, which in our opinion constitutes the main geometric core of the recent advances in curve theory. Along the way we shall, of course, discuss appli cations of the theory of linear series to a number of classical topics (e.g., the geometry of the Riemann theta divisor) as well as to some of the current research (e.g., the Kodaira dimension of the moduli space of curves).

## Physics letters : [part B].

### Physics letters : [part B]. by N.A Book Summary:

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## Integrable Geodesic Flows on Two-Dimensional Surfaces

### Integrable Geodesic Flows on Two-Dimensional Surfaces by A.V. Bolsinov,A.T. Fomenko Book Summary:

Presenting a new approach to qualitative analysis of integrable geodesic flows based on the theory of topological classification of integrable Hamiltonian systems, this is the first book to apply this technique systematically to a wide class of integrable systems. The first part of the book provides an introduction to the qualitative theory of integrable Hamiltonian systems and their invariants (symplectic geometry, integrability, the topology of Liouville foliations, the orbital classification theory for integrable nondegenerate Hamiltonian systems with two degrees of freedom, obstructions to integrability, etc). In the second part, the class of integrable geodesic flows on two-dimensional surfaces is discussed both from the classical and contemporary point of view. The authors classify them up to different equivalence relations such as an isometry, the Liouville equivalence, the trajectory equivalence (smooth and continuous), and the geodesic equivalence. A new technique, which provides the possibility to classify integrable geodesic flows up to these kinds of equivalences, is presented together with applications. Together with systematic presentation of wide material on this subject, the book contains previously unpublished new results, and is enhanced with many original illustrations.

## Manifolds on which Analysis Meets Topology

### Manifolds on which Analysis Meets Topology by Eiko Nakayama Tyler Book Summary:

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## Computational Approach to Riemann Surfaces

### Computational Approach to Riemann Surfaces by Alexander I. Bobenko,Christian Klein Book Summary:

This volume is a well structured overview of existing computational approaches to Riemann surfaces as well as those under development. It covers the software tools currently available and provides solutions to partial differential equations and surface theory.

## Real Algebraic Geometry and Ordered Structures

### Real Algebraic Geometry and Ordered Structures by AMS Special Session on Real Algebraic Geometry,Charles N. Delzell,James J. Madden Book Summary:

This volume contains 16 carefully refereed articles by participants in the Special Session on Real Algebraic Geometry and Ordered Algebraic Structures at the Sectional Meeting of the AMS in Baton Rouge, April 1996, and the associated Special Semester in the spring of 1996 at Louisiana State University and Southern University, Baton Rouge.The 23 contributors to this volume were among the 75 mathematicians from 15 countries who participated. The topics include the topology of real algebraic curves (Hilbert's 16th problem), moduli of real algebraic curves, effective sums of squares of real forms (Hilbert's 17th problem), efficient real quantifier elimination, subanalytic sets and stratifications, semialgebraic singularity theory, radial vector fields, exponential functions and valuations on nonarchimedean ordered fields, valued field extensions, partially ordered and lattice-ordered rings, rings of continuous functions, spectra of rings, and abstract spaces of (higher-level) orderings and real places.This volume provides a good overview of the state of the art in this area in the 1990s. It includes both expository and original research papers by top workers in this thriving field. The authors and editors strived to make the volume useful to a wide audience (including students and researchers) interested in real algebraic geometry and ordered structures - two subjects that are obviously related, but seldom brought together.

## Algebraic Curves and Riemann Surfaces

### Algebraic Curves and Riemann Surfaces by Rick Miranda Book Summary:

The book was easy to understand, with many examples. The exercises were well chosen, and served to give further examples and developments of the theory. --William Goldman, University of Maryland 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 center stage. But the main examples come from projective curves, and slowly but surely the text moves toward the algebraic category. Proofs of the Riemann-Roch and Serre Duality 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 and cohomology are introduced as a unifying device in the latter chapters, so that their utility and naturalness are immediately obvious. Requiring a background of one semester of complex variable theory and a year of abstract algebra, this is an excellent graduate textbook for a second-semester course in complex variables or a year-long course in algebraic geometry.

## International Books in Print

### International Books in Print by N.A Book Summary:

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## Algebraic Surfaces and Holomorphic Vector Bundles

### Algebraic Surfaces and Holomorphic Vector Bundles by Robert Friedman Book Summary:

A novel feature of the book is its integrated approach to algebraic surface theory and the study of vector bundle theory on both curves and surfaces. While the two subjects remain separate through the first few chapters, they become much more tightly interconnected as the book progresses. Thus vector bundles over curves are studied to understand ruled surfaces, and then reappear in the proof of Bogomolov's inequality for stable bundles, which is itself applied to study canonical embeddings of surfaces via Reider's method. Similarly, ruled and elliptic surfaces are discussed in detail, before the geometry of vector bundles over such surfaces is analysed. Many of the results on vector bundles appear for the first time in book form, backed by many examples, both of surfaces and vector bundles, and over 100 exercises forming an integral part of the text. Aimed at graduates with a thorough first-year course in algebraic geometry, as well as more advanced students and researchers in the areas of algebraic geometry, gauge theory, or 4-manifold topology, many of the results on vector bundles will also be of interest to physicists studying string theory.

## Introduction to Compact Riemann Surfaces and Dessins D'Enfants

### Introduction to Compact Riemann Surfaces and Dessins D'Enfants by Ernesto Girondo,Gabino González-Diez Book Summary:

An elementary account of the theory of compact Riemann surfaces and an introduction to the Belyi-Grothendieck theory of dessins d'enfants.

## Complex Analysis, Riemann Surfaces and Integrable Systems

### Complex Analysis, Riemann Surfaces and Integrable Systems by Sergey M. Natanzon Book Summary:

This book is devoted to classical and modern achievements in complex analysis. In order to benefit most from it, a first-year university background is sufficient; all other statements and proofs are provided. We begin with a brief but fairly complete course on the theory of holomorphic, meromorphic, and harmonic functions. We then present a uniformization theory, and discuss a representation of the moduli space of Riemann surfaces of a fixed topological type as a factor space of a contracted space by a discrete group. Next, we consider compact Riemann surfaces and prove the classical theorems of Riemann-Roch, Abel, Weierstrass, etc. We also construct theta functions that are very important for a range of applications. After that, we turn to modern applications of this theory. First, we build the (important for mathematics and mathematical physics) Kadomtsev-Petviashvili hierarchy and use validated results to arrive at important solutions to these differential equations. We subsequently use the theory of harmonic functions and the theory of differential hierarchies to explicitly construct a conformal mapping that translates an arbitrary contractible domain into a standard disk – a classical problem that has important applications in hydrodynamics, gas dynamics, etc. The book is based on numerous lecture courses given by the author at the Independent University of Moscow and at the Mathematics Department of the Higher School of Economics.

## Complex Algebraic Curves

### Complex Algebraic Curves by Frances Clare Kirwan,Frances Kirwan Book Summary:

This development of the theory of complex algebraic curves was one of the peaks of nineteenth century mathematics. They have many fascinating properties and arise in various areas of mathematics, from number theory to theoretical physics, and are the subject of much research. By using only the basic techniques acquired in most undergraduate courses in mathematics, Dr. Kirwan introduces the theory, observes the algebraic and topological properties of complex algebraic curves, and shows how they are related to complex analysis.

## Handbook of Teichmüller Theory

### Handbook of Teichmüller Theory by Athanase Papadopoulos Book Summary:

The subject of this handbook is Teichmuller theory in a wide sense, namely the theory of geometric structures on surfaces and their moduli spaces. This includes the study of vector bundles on these moduli spaces, the study of mapping class groups, the relation with $3$-manifolds, the relation with symmetric spaces and arithmetic groups, the representation theory of fundamental groups, and applications to physics. Thus the handbook is a place where several fields of mathematics interact: Riemann surfaces, hyperbolic geometry, partial differential equations, several complex variables, algebraic geometry, algebraic topology, combinatorial topology, low-dimensional topology, theoretical physics, and others. This confluence of ideas toward a unique subject is a manifestation of the unity and harmony of mathematics. This volume contains surveys on the fundamental theory as well as surveys on applications to and relations with the fields mentioned above. It is written by leading experts in these fields. Some of the surveys contain classical material, while others present the latest developments of the theory as well as open problems. This volume is divided into the following four sections: The metric and the analytic theory The group theory The algebraic topology of mapping class groups and moduli spaces Teichmuller theory and mathematical physics This handbook is addressed to graduate students and researchers in all the fields mentioned.

## Moduli Spaces of Stable Sheaves on Schemes

### Moduli Spaces of Stable Sheaves on Schemes by Masaki Maruyama Book Summary:

The notion of stability for algebraic vector bundles on curves was originally introduced by Mumford, and moduli spaces of semi-stable vector bundles were studied intensively by Indian mathematicians. The notion of stability for algebraic sheaves was generalized to higher dimensional varieties. The study of moduli spaces of algebraic sheaves not only on curves but also on higher dimensional algebraic varieties has attracted much interest for decades and its importance has been increasing not only in algebraic geometry but also in related fields as differential geometry, mathematical physics.Masaki Maruyama is one of the pioneers in the theory of algebraic vector bundles on higher dimensional algebraic varieties. This book is a posthumous publication of his manuscript. It starts with basic concepts such as stability of sheaves, Harder-Narasimhan filtration and generalities on boundedness of sheaves. It then presents fundamental theorems on semi-stable sheaves: restriction theorems of semi-stable sheaves, boundedness of semi-stable sheaves, tensor products of semi-stable sheaves. Finally, after constructing quote-schemes, it explains the construction of the moduli space of semi-stable sheaves. The theorems are stated in a general setting and the proofs are rigorous.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets

## Gazette - Australian Mathematical Society

### Gazette - Australian Mathematical Society by Australian Mathematical Society Book Summary:

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## Lectures on Introduction to Moduli Problems and Orbit Spaces

### Lectures on Introduction to Moduli Problems and Orbit Spaces by P. E. Newstead Book Summary:

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## Topics on Riemann Surfaces and Fuchsian Groups

### Topics on Riemann Surfaces and Fuchsian Groups by Paul Allan Mirecki,Emilio Bujalance García,E. Bujalance,Emilio Bujalance Garcia,A. F. Costa,Jason BeDuhn,E. Martínez,Ernesto Martínez,E. Martinez,J. W. S. Cassels,Savilian Professor of Geometry N J Hitchin Book Summary:

Introduction to Riemann surfaces for graduates and researchers, giving refreshingly new insights into the subject.

## Vector Bundles in Algebraic Geometry

### Vector Bundles in Algebraic Geometry by N. J. Hitchin,Savilian Professor of Geometry N J Hitchin,P. E. Newstead,W. M. Oxbury Book Summary:

This book is a collection of survey articles by the main speakers at the 1993 Durham symposium on vector bundles in algebraic geometry.

## Lectures on Riemann Surfaces

### Lectures on Riemann Surfaces by Otto Forster Book Summary:

This book grew out of lectures on Riemann surfaces given by Otto Forster at the universities of Munich, Regensburg, and Münster. It provides a concise modern introduction to this rewarding subject, as well as presenting methods used in the study of complex manifolds in the special case of complex dimension one. From the reviews: "This book deserves very serious consideration as a text for anyone contemplating giving a course on Riemann surfaces."—-MATHEMATICAL REVIEWS

## Lectures on Vector Bundles

### Lectures on Vector Bundles by J. Le Potier Book Summary:

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