Overview

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.

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9781107153042

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Reviews

'K3 surfaces play something of a magical role in algebraic geometry and neighboring areas. They arise in astonishingly varied contexts, and the study of K3 surfaces has propelled the development of many of the most powerful tools in the field. The present lectures provide a comprehensive and wide-ranging survey of this fascinating subject. Suitable both for study and as a reference work, and written with Huybrechts's usual clarity of exposition, this book is destined to become the standard text on K3 surfaces.' Rob Lazarsfeld, State University of New York, Stony Brook 'This book will be extremely valuable to all mathematicians who are interested in K3 surfaces and related topics. It not only serves as an excellent introduction, but also covers a wide variety of advanced subjects, ranging from complex geometry to derived geometry and arithmetic.' Klaus Hulek, Leibniz Universität Hannover 'Since the nineteenth century, K3 surfaces have been a source of intriguing examples, problems and theorems. Huybrechts' book is a beautiful and reader-friendly presentation of the main results regarding this special class of varieties. The author fully succeeded in illustrating the richness of concepts and techniques which come into play in the theory of K3 surfaces.' Kieran G. O'Grady, Università degli Studi di Roma 'La Sapienza', Italy 'K3 surfaces play a ubiquitous role in algebraic geometry. At first glance they seem to be well understood and easy to describe, still they provide non-trivial examples of the most fundamental concepts: Hodge structures, moduli spaces, Chow ring, vector bundles, Picard and Brauer groups … Huybrechts' book, written with the usual talent of the author, is the first to cover systematically all these aspects. It will be an invaluable reference for algebraic geometers.' Arnaud Beauville, Université de Nice, Sophia Antipolis '… the book covers many subjects and recent developments, and contains an encyclopedic total of 655 references, which will be very useful for researchers and graduate students. A reader who opens any page of the book will enjoy the subject there. This book will become one's favorite book.' Shigeyuki Kondo, MathSciNet 'The book is a welcome addition to the literature, especially since its scope ranges from a very good introduction to K3 surfaces to the more recent advances on these surfaces and related topics.' Felipe Zaldivar, MAA Reviews

Advance praise: 'K3 surfaces play something of a magical role in algebraic geometry and neighboring areas. They arise in astonishingly varied contexts, and the study of K3 surfaces has propelled the development of many of the most powerful tools in the field. The present lectures provide a comprehensive and wide-ranging survey of this fascinating subject. Suitable both for study and as a reference work, and written with Huybrechts's usual clarity of exposition, this book is destined to become the standard text on K3 surfaces.' Rob Lazarsfeld, State University of New York, Stony Brook Advance praise: 'This book will be extremely valuable to all mathematicians who are interested in K3 surfaces and related topics. It not only serves as an excellent introduction, but also covers a wide variety of advanced subjects, ranging from complex geometry to derived geometry and arithmetic.' Klaus Hulek, Leibniz Universitat Hannover Advance praise: 'Since the nineteenth century, K3 surfaces have been a source of intriguing examples, problems and theorems. Huybrechts' book is a beautiful and reader-friendly presentation of the main results regarding this special class of varieties. The author fully succeeded in illustrating the richness of concepts and techniques which come into play in the theory of K3 surfaces.' Kieran G. O'Grady, Universit... degli Studi di Roma 'La Sapienza', Italy Advance praise: 'K3 surfaces play a ubiquitous role in algebraic geometry. At first glance they seem to be well understood and easy to describe, still they provide non-trivial examples of the most fundamental concepts: Hodge structures, moduli spaces, Chow ring, vector bundles, Picard and Brauer groups ... Huybrechts' book, written with the usual talent of the author, is the first to cover systematically all these aspects. It will be an invaluable reference for algebraic geometers.' Arnaud Beauville, Universite de Nice, Sophia Antipolis Advance praise: 'K3 surfaces, which were formally defined by Andre Weil in 1958, turned out to be among the most fascinating mathematical objects and had considerable impact outside algebraic geometry. For example, the differential manifold underlying a K3 surface has the kind of intersection pairing that is responsible for its remarkable role in the development of 4-dimensional topology in the 1980s. But such surfaces also admit a hyperkahler metric, and this largely accounts for their importance in complex differential geometry and string theory. Today the period map for K3 surfaces is well-understood, but deeper algebro-geometric properties, including those of a motivic and arithmetic nature, are not. Indeed, that these areas are currently the subject of intense research has increased the need for an up-to-date introduction to the recent literature. This systematic exposition by a renowned expert meets that need wonderfully.' Eduard Looijenga, Tsinghua University, Beijing and Universiteit Utrecht, The Netherlands

'K3 surfaces play something of a magical role in algebraic geometry and neighboring areas. They arise in astonishingly varied contexts, and the study of K3 surfaces has propelled the development of many of the most powerful tools in the field. The present lectures provide a comprehensive and wide-ranging survey of this fascinating subject. Suitable both for study and as a reference work, and written with Huybrechts's usual clarity of exposition, this book is destined to become the standard text on K3 surfaces.' Rob Lazarsfeld, State University of New York, Stony Brook 'This book will be extremely valuable to all mathematicians who are interested in K3 surfaces and related topics. It not only serves as an excellent introduction, but also covers a wide variety of advanced subjects, ranging from complex geometry to derived geometry and arithmetic.' Klaus Hulek, Leibniz Universitat Hannover 'Since the nineteenth century, K3 surfaces have been a source of intriguing examples, problems and theorems. Huybrechts' book is a beautiful and reader-friendly presentation of the main results regarding this special class of varieties. The author fully succeeded in illustrating the richness of concepts and techniques which come into play in the theory of K3 surfaces.' Kieran G. O'Grady, Universit... degli Studi di Roma 'La Sapienza', Italy 'K3 surfaces play a ubiquitous role in algebraic geometry. At first glance they seem to be well understood and easy to describe, still they provide non-trivial examples of the most fundamental concepts: Hodge structures, moduli spaces, Chow ring, vector bundles, Picard and Brauer groups ... Huybrechts' book, written with the usual talent of the author, is the first to cover systematically all these aspects. It will be an invaluable reference for algebraic geometers.' Arnaud Beauville, Universite de Nice, Sophia Antipolis '... the book covers many subjects and recent developments, and contains an encyclopedic total of 655 references, which will be very useful for researchers and graduate students. A reader who opens any page of the book will enjoy the subject there. This book will become one's favorite book.' Shigeyuki Kondo, MathSciNet

Author Information

Daniel Huybrechts is a professor at the Mathematical Institute of the University of Bonn. He previously held positions at the Université Denis Diderot Paris 7 and the University of Cologne. He is interested in algebraic geometry, particularly special geometries with rich algebraic, analytic, and arithmetic structures. His current work focuses on K3 surfaces and higher dimensional analogues. He has published four books.

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