K3 Surfaces and Their Moduli

K3 Surfaces and Their Moduli
Author: Carel Faber
Publisher: Birkhäuser
Total Pages: 403
Release: 2016-04-22
Genre: Mathematics
ISBN: 331929959X

This book provides an overview of the latest developments concerning the moduli of K3 surfaces. It is aimed at algebraic geometers, but is also of interest to number theorists and theoretical physicists, and continues the tradition of related volumes like “The Moduli Space of Curves” and “Moduli of Abelian Varieties,” which originated from conferences on the islands Texel and Schiermonnikoog and which have become classics. K3 surfaces and their moduli form a central topic in algebraic geometry and arithmetic geometry, and have recently attracted a lot of attention from both mathematicians and theoretical physicists. Advances in this field often result from mixing sophisticated techniques from algebraic geometry, lattice theory, number theory, and dynamical systems. The topic has received significant impetus due to recent breakthroughs on the Tate conjecture, the study of stability conditions and derived categories, and links with mirror symmetry and string theory. At the same time, the theory of irreducible holomorphic symplectic varieties, the higher dimensional analogues of K3 surfaces, has become a mainstream topic in algebraic geometry. Contributors: S. Boissière, A. Cattaneo, I. Dolgachev, V. Gritsenko, B. Hassett, G. Heckman, K. Hulek, S. Katz, A. Klemm, S. Kondo, C. Liedtke, D. Matsushita, M. Nieper-Wisskirchen, G. Oberdieck, K. Oguiso, R. Pandharipande, S. Rieken, A. Sarti, I. Shimada, R. P. Thomas, Y. Tschinkel, A. Verra, C. Voisin.

Lectures on K3 Surfaces

Lectures on K3 Surfaces
Author: Daniel Huybrechts
Publisher: Cambridge University Press
Total Pages: 499
Release: 2016-09-26
Genre: Mathematics
ISBN: 1316797252

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.

The Geometry of Moduli Spaces of Sheaves

The Geometry of Moduli Spaces of Sheaves
Author: Daniel Huybrechts
Publisher: Cambridge University Press
Total Pages: 345
Release: 2010-05-27
Genre: Mathematics
ISBN: 1139485822

This edition has been updated to reflect recent advances in the theory of semistable coherent sheaves and their moduli spaces. The authors review changes in the field and point the reader towards further literature. An ideal text for graduate students or mathematicians with a background in algebraic geometry.

Complex Analysis and Algebraic Geometry

Complex Analysis and Algebraic Geometry
Author: Kunihiko Kodaira
Publisher: CUP Archive
Total Pages: 424
Release: 1977
Genre: Mathematics
ISBN: 9780521217774

The articles in this volume cover some developments in complex analysis and algebraic geometry. The book is divided into three parts. Part I includes topics in the theory of algebraic surfaces and analytic surface. Part II covers topics in moduli and classification problems, as well as structure theory of certain complex manifolds. Part III is devoted to various topics in algebraic geometry analysis and arithmetic. A survey article by Ueno serves as an introduction to the general background of the subject matter of the volume. The volume was written for Kunihiko Kodaira on the occasion of his sixtieth birthday, by his friends and students. Professor Kodaira was one of the world's leading mathematicians in algebraic geometry and complex manifold theory: and the contributions reflect those concerns.

K3 Surfaces

K3 Surfaces
Author: Shigeyuki Kondō
Publisher:
Total Pages: 250
Release: 2020
Genre: Geometry, Algebraic
ISBN: 9783037197080

$K3$ surfaces are a key piece in the classification of complex analytic or algebraic surfaces. The term was coined by A. Weil in 1958 - a result of the initials Kummer, Kähler, Kodaira, and the mountain K2 found in Karakoram. The most famous example is the Kummer surface discovered in the 19th century.$K3$ surfaces can be considered as a 2-dimensional analogue of an elliptic curve, and the theory of periods - called the Torelli-type theorem for $K3$ surfaces - was established around 1970. Since then, several pieces of research on $K3$ surfaces have been undertaken and more recently $K3$ surfaces have even become of interest in theoretical physics.The main purpose of this book is an introduction to the Torelli-type theorem for complex analytic $K3$ surfaces, and its applications. The theory of lattices and their reflection groups is necessary to study $K3$ surfaces, and this book introduces these notions. The book contains, as well as lattices and reflection groups, the classification of complex analytic surfaces, the Torelli-type theorem, the subjectivity of the period map, Enriques surfaces, an application to the moduli space of plane quartics, finite automorphisms of $K3$ surfaces, Niemeier lattices and the Mathieu group, the automorphism group of Kummer surfaces and the Leech lattice.The author seeks to demonstrate the interplay between several sorts of mathematics and hopes the book will prove helpful to researchers in algebraic geometry and related areas, and to graduate students with a basic grounding in algebraic geometry.

K3 Projective Models in Scrolls

K3 Projective Models in Scrolls
Author: Trygve Johnsen
Publisher: Springer Science & Business Media
Total Pages: 180
Release: 2004
Genre: Projective modules (Algebra)
ISBN: 9783540215059

Mirror Symmetry II

Mirror Symmetry II
Author: Brian Greene
Publisher: American Mathematical Soc.
Total Pages: 862
Release: 1997
Genre: Mathematics
ISBN: 0821827448

Mirror Symmetry has undergone dramatic progress since the Mathematical Sciences Research Institute (MSRI) workshop in 1991, whose proceedings constitute voluem I of this continuing collection. Tremendous insight has been gained on a number of key issues. This volume surveys these results. Some of the contributions in this work have appeared elsewhere, while others were written specifically for this collection. The areas covered are organized into 4 sections, and each presents papers by both physicists and mathematicians. This volume collects the most important developments that have taken place in mathematical physics since 1991. It is an essential reference tool for both mathematics and physics libraries and for students of physics and mathematics. Titles in this series are co-published, between the American Mathematical Society and International Press, Cambridge, MA, USA.

Birational Geometry of Hypersurfaces

Birational Geometry of Hypersurfaces
Author: Andreas Hochenegger
Publisher: Springer Nature
Total Pages: 301
Release: 2019-10-08
Genre: Mathematics
ISBN: 3030186385

Originating from the School on Birational Geometry of Hypersurfaces, this volume focuses on the notion of (stable) rationality of projective varieties and, more specifically, hypersurfaces in projective spaces, and provides a large number of open questions, techniques and spectacular results. The aim of the school was to shed light on this vast area of research by concentrating on two main aspects: (1) Approaches focusing on (stable) rationality using deformation theory and Chow-theoretic tools like decomposition of the diagonal; (2) The connection between K3 surfaces, hyperkähler geometry and cubic fourfolds, which has both a Hodge-theoretic and a homological side. Featuring the beautiful lectures given at the school by Jean-Louis Colliot-Thélène, Daniel Huybrechts, Emanuele Macrì, and Claire Voisin, the volume also includes additional notes by János Kollár and an appendix by Andreas Hochenegger.