## Hodge Theory and Complex Algebraic Geometry I:

**Author**: Claire Voisin

**Publisher:**Cambridge University Press

**ISBN:**9781139437691

**Category:**Mathematics

**Page:**N.A

**View:**3883

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The first of two volumes offering a modern introduction to Kaehlerian geometry and Hodge structure. The book starts with basic material on complex variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory, the latter being treated in a more theoretical way than is usual in geometry. The author then proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The book culminates with the Hodge decomposition theorem. The meanings of these results are investigated in several directions. Completely self-contained, the book is ideal for students, while its content gives an account of Hodge theory and complex algebraic geometry as has been developed by P. Griffiths and his school, by P. Deligne, and by S. Bloch. The text is complemented by exercises which provide useful results in complex algebraic geometry.

## Period Mappings and Period Domains

**Author**: N.A

**Publisher:**N.A

**ISBN:**1108422624

**Category:**

**Page:**N.A

**View:**2455

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## Introduction to Hodge Theory

**Author**: José Bertin

**Publisher:**American Mathematical Soc.

**ISBN:**9780821820407

**Category:**Mathematics

**Page:**232

**View:**2219

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Hodge theory is a powerful tool in analytic and algebraic geometry. This book consists of expositions of aspects of modern Hodge theory, with the purpose of providing the nonexpert reader with a clear idea of the current state of the subject. The three main topics are: $L^2$ Hodge theory and vanishing theorems; Hodge theory in characteristic $p$; and variations of Hodge structures and mirror symmetry. Each section has a detailed introduction and numerous references. Many open problems are also included. The reader should have some familiarity with differential and algebraic geometry, with other prerequisites varying by chapter. The book is suitable as an accompaniment to a second course in algebraic geometry.

## Algebraic Geometry over the Complex Numbers

**Author**: Donu Arapura

**Publisher:**Springer Science & Business Media

**ISBN:**1461418097

**Category:**Mathematics

**Page:**329

**View:**3750

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This is a relatively fast paced graduate level introduction to complex algebraic geometry, from the basics to the frontier of the subject. It covers sheaf theory, cohomology, some Hodge theory, as well as some of the more algebraic aspects of algebraic geometry. The author frequently refers the reader if the treatment of a certain topic is readily available elsewhere but goes into considerable detail on topics for which his treatment puts a twist or a more transparent viewpoint. His cases of exploration and are chosen very carefully and deliberately. The textbook achieves its purpose of taking new students of complex algebraic geometry through this a deep yet broad introduction to a vast subject, eventually bringing them to the forefront of the topic via a non-intimidating style.

## Positivity in Algebraic Geometry I

*Classical Setting: Line Bundles and Linear Series*

**Author**: R.K. Lazarsfeld

**Publisher:**Springer Science & Business Media

**ISBN:**9783540225331

**Category:**Mathematics

**Page:**387

**View:**4089

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This two volume work on Positivity in Algebraic Geometry contains a contemporary account of a body of work in complex algebraic geometry loosely centered around the theme of positivity. Topics in Volume I include ample line bundles and linear series on a projective variety, the classical theorems of Lefschetz and Bertini and their modern outgrowths, vanishing theorems, and local positivity. Volume II begins with a survey of positivity for vector bundles, and moves on to a systematic development of the theory of multiplier ideals and their applications. A good deal of this material has not previously appeared in book form, and substantial parts are worked out here in detail for the first time. At least a third of the book is devoted to concrete examples, applications, and pointers to further developments. Volume I is more elementary than Volume II, and, for the most part, it can be read without access to Volume II.

## 3264 and All That

*A Second Course in Algebraic Geometry*

**Author**: David Eisenbud,Joe Harris

**Publisher:**Cambridge University Press

**ISBN:**1316679381

**Category:**Mathematics

**Page:**N.A

**View:**3597

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This book can form the basis of a second course in algebraic geometry. As motivation, it takes concrete questions from enumerative geometry and intersection theory, and provides intuition and technique, so that the student develops the ability to solve geometric problems. The authors explain key ideas, including rational equivalence, Chow rings, Schubert calculus and Chern classes, and readers will appreciate the abundant examples, many provided as exercises with solutions available online. Intersection is concerned with the enumeration of solutions of systems of polynomial equations in several variables. It has been an active area of mathematics since the work of Leibniz. Chasles' nineteenth-century calculation that there are 3264 smooth conic plane curves tangent to five given general conics was an important landmark, and was the inspiration behind the title of this book. Such computations were motivation for Poincaré's development of topology, and for many subsequent theories, so that intersection theory is now a central topic of modern mathematics.

## Birational Geometry of Algebraic Varieties

**Author**: Janos Kollár,Shigefumi Mori

**Publisher:**Cambridge University Press

**ISBN:**9780521060226

**Category:**Mathematics

**Page:**254

**View:**4656

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This book provides the first comprehensive introduction to the circle of ideas developed around Mori's program.

## D-Modules, Perverse Sheaves, and Representation Theory

**Author**: Ryoshi Hotta,Toshiyuki Tanisaki

**Publisher:**Springer Science & Business Media

**ISBN:**081764363X

**Category:**Mathematics

**Page:**412

**View:**7133

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D-modules continues to be an active area of stimulating research in such mathematical areas as algebraic, analysis, differential equations, and representation theory. Key to D-modules, Perverse Sheaves, and Representation Theory is the authors' essential algebraic-analytic approach to the theory, which connects D-modules to representation theory and other areas of mathematics. To further aid the reader, and to make the work as self-contained as possible, appendices are provided as background for the theory of derived categories and algebraic varieties. The book is intended to serve graduate students in a classroom setting and as self-study for researchers in algebraic geometry, representation theory.

## Lectures on Logarithmic Algebraic Geometry

**Author**: Arthur Ogus

**Publisher:**Cambridge University Press

**ISBN:**1107187737

**Category:**Mathematics

**Page:**558

**View:**5854

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A self-contained introduction to logarithmic geometry, a key tool for analyzing compactification and degeneration in algebraic geometry.

## Complex Manifolds and Deformation of Complex Structures

**Author**: K. Kodaira

**Publisher:**Springer Science & Business Media

**ISBN:**1461385903

**Category:**Mathematics

**Page:**467

**View:**5043

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## Geometric Invariant Theory

**Author**: David Mumford,John Fogarty,Frances Kirwan

**Publisher:**Springer Science & Business Media

**ISBN:**9783540569633

**Category:**Mathematics

**Page:**292

**View:**8703

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"Geometric Invariant Theory" by Mumford/Fogarty (the firstedition was published in 1965, a second, enlarged editonappeared in 1982) is the standard reference on applicationsof invariant theory to the construction of moduli spaces.This third, revised edition has been long awaited for by themathematical community. It is now appearing in a completelyupdated and enlarged version with an additional chapter onthe moment map by Prof. Frances Kirwan (Oxford) and a fullyupdated bibliography of work in this area.The book deals firstly with actions of algebraic groups onalgebraic varieties, separating orbits by invariants andconstructionquotient spaces; and secondly with applicationsof this theory to the construction of moduli spaces.It is a systematic exposition of the geometric aspects ofthe classical theory of polynomial invariants.

## An Introduction to Invariants and Moduli

**Author**: Shigeru Mukai,W. M. Oxbury

**Publisher:**Cambridge University Press

**ISBN:**9780521809061

**Category:**Mathematics

**Page:**503

**View:**2313

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Incorporated in this volume are the first two books in Mukai's series on Moduli Theory. The notion of a moduli space is central to geometry. However, it's influence is not confined there; for example the theory of moduli spaces is a crucial ingredient in the proof of Fermat's last theorem. An accurate account of Mukai's influential Japanese texts, this tranlation will be a valuable resource for researchers and graduate students working in a range of areas.

## Hodge Theory (MN-49)

**Author**: Eduardo Cattani,Fouad El Zein,Phillip A. Griffiths,Lê Dũng Tráng

**Publisher:**Princeton University Press

**ISBN:**1400851475

**Category:**Mathematics

**Page:**608

**View:**5940

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This book provides a comprehensive and up-to-date introduction to Hodge theory—one of the central and most vibrant areas of contemporary mathematics—from leading specialists on the subject. The topics range from the basic topology of algebraic varieties to the study of variations of mixed Hodge structure and the Hodge theory of maps. Of particular interest is the study of algebraic cycles, including the Hodge and Bloch-Beilinson Conjectures. Based on lectures delivered at the 2010 Summer School on Hodge Theory at the ICTP in Trieste, Italy, the book is intended for a broad group of students and researchers. The exposition is as accessible as possible and doesn't require a deep background. At the same time, the book presents some topics at the forefront of current research. The book is divided between introductory and advanced lectures. The introductory lectures address Kähler manifolds, variations of Hodge structure, mixed Hodge structures, the Hodge theory of maps, period domains and period mappings, algebraic cycles (up to and including the Bloch-Beilinson conjecture) and Chow groups, sheaf cohomology, and a new treatment of Grothendieck’s algebraic de Rham theorem. The advanced lectures address a Hodge-theoretic perspective on Shimura varieties, the spread philosophy in the study of algebraic cycles, absolute Hodge classes (including a new, self-contained proof of Deligne’s theorem on absolute Hodge cycles), and variation of mixed Hodge structures. The contributors include Patrick Brosnan, James Carlson, Eduardo Cattani, François Charles, Mark Andrea de Cataldo, Fouad El Zein, Mark L. Green, Phillip A. Griffiths, Matt Kerr, Lê Dũng Tráng, Luca Migliorini, Jacob P. Murre, Christian Schnell, and Loring W. Tu.

## Algebraic Geometry: From algebraic varieties to schemes

**Author**: 健爾·上野

**Publisher:**American Mathematical Soc.

**ISBN:**0821808621

**Category:**Mathematics

**Page:**154

**View:**1229

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This is the first of three volumes on algebraic geometry. The second volume, Algebraic Geometry 2: Sheaves and Cohomology, is available from the AMS as Volume 197 in the Translations of Mathematical Monographs series. Early in the 20th century, algebraic geometry underwent a significant overhaul, as mathematicians, notably Zariski, introduced a much stronger emphasis on algebra and rigor into the subject. This was followed by another fundamental change in the 1960s with Grothendieck's introduction of schemes. Today, most algebraic geometers are well-versed in the language of schemes, but many newcomers are still initially hesitant about them. Ueno's book provides an inviting introduction to the theory, which should overcome any such impediment to learning this rich subject. The book begins with a description of the standard theory of algebraic varieties. Then, sheaves are introduced and studied, using as few prerequisites as possible. Once sheaf theory has been well understood, the next step is to see that an affine scheme can be defined in terms of a sheaf over the prime spectrum of a ring. By studying algebraic varieties over a field, Ueno demonstrates how the notion of schemes is necessary in algebraic geometry. This first volume gives a definition of schemes and describes some of their elementary properties. It is then possible, with only a little additional work, to discover their usefulness. Further properties of schemes will be discussed in the second volume. Ueno's book is a self-contained introduction to this important circle of ideas, assuming only a knowledge of basic notions from abstract algebra (such as prime ideals). It is suitable as a text for an introductory course on algebraic geometry.

## Recent Advances in Algebraic Geometry

**Author**: Christopher D. Hacon,Mircea Mustaţă,Mihnea Popa

**Publisher:**Cambridge University Press

**ISBN:**110764755X

**Category:**Mathematics

**Page:**447

**View:**7005

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A comprehensive collection of expository articles on cutting-edge topics at the forefront of research in algebraic geometry.

## Toric Varieties

**Author**: David A. Cox,John B. Little,Henry K. Schenck

**Publisher:**American Mathematical Soc.

**ISBN:**0821848194

**Category:**Mathematics

**Page:**841

**View:**5579

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Toric varieties form a beautiful and accessible part of modern algebraic geometry. This book covers the standard topics in toric geometry; a novel feature is that each of the first nine chapters contains an introductory section on the necessary background material in algebraic geometry. Other topics covered include quotient constructions, vanishing theorems, equivariant cohomology, GIT quotients, the secondary fan, and the minimal model program for toric varieties. The subject lends itself to rich examples reflected in the 134 illustrations included in the text. The book also explores connections with commutative algebra and polyhedral geometry, treating both polytopes and their unbounded cousins, polyhedra. There are appendices on the history of toric varieties and the computational tools available to investigate nontrivial examples in toric geometry. Readers of this book should be familiar with the material covered in basic graduate courses in algebra and topology, and to a somewhat lesser degree, complex analysis. In addition, the authors assume that the reader has had some previous experience with algebraic geometry at an advanced undergraduate level. The book will be a useful reference for graduate students and researchers who are interested in algebraic geometry, polyhedral geometry, and toric varieties.

## Lectures on K3 Surfaces

**Author**: Daniel Huybrechts

**Publisher:**Cambridge University Press

**ISBN:**1316797252

**Category:**Mathematics

**Page:**N.A

**View:**5744

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K3 surfaces are central objects in modern algebraic geometry. This book examines this important class of Calabi–Yau manifolds from various perspectives in eighteen self-contained chapters. It starts with the basics and guides the reader to recent breakthroughs, such as the proof of the Tate conjecture for K3 surfaces and structural results on Chow groups. Powerful general techniques are introduced to study the many facets of K3 surfaces, including arithmetic, homological, and differential geometric aspects. In this context, the book covers Hodge structures, moduli spaces, periods, derived categories, birational techniques, Chow rings, and deformation theory. Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin–Tate, are discussed in general and for K3 surfaces in particular, and each chapter ends with questions and open problems. Based on lectures at the advanced graduate level, this book is suitable for courses and as a reference for researchers.

## Algebraic Geometry

**Author**: Robin Hartshorne

**Publisher:**Springer Science & Business Media

**ISBN:**1475738498

**Category:**Mathematics

**Page:**496

**View:**7777

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An introduction to abstract algebraic geometry, with the only prerequisites being results from commutative algebra, which are stated as needed, and some elementary topology. More than 400 exercises distributed throughout the book offer specific examples as well as more specialised topics not treated in the main text, while three appendices present brief accounts of some areas of current research. This book can thus be used as textbook for an introductory course in algebraic geometry following a basic graduate course in algebra. Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris. He is the author of "Residues and Duality", "Foundations of Projective Geometry", "Ample Subvarieties of Algebraic Varieties", and numerous research titles.

## p-adic Differential Equations

**Author**: Kiran S. Kedlaya

**Publisher:**Cambridge University Press

**ISBN:**1139489208

**Category:**Mathematics

**Page:**N.A

**View:**4180

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Over the last 50 years the theory of p-adic differential equations has grown into an active area of research in its own right, and has important applications to number theory and to computer science. This book, the first comprehensive and unified introduction to the subject, improves and simplifies existing results as well as including original material. Based on a course given by the author at MIT, this modern treatment is accessible to graduate students and researchers. Exercises are included at the end of each chapter to help the reader review the material, and the author also provides detailed references to the literature to aid further study.

## An Introduction to Lie Groups and Lie Algebras

**Author**: Alexander Kirillov

**Publisher:**Cambridge University Press

**ISBN:**0521889693

**Category:**Mathematics

**Page:**222

**View:**9710

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This book is an introduction to semisimple Lie algebras; concise and informal, with numerous exercises and examples.