\(K3\) surfaces. Translated from the Japanese by the author
type: Item
instance of: scholarly article
title: \(K3\) surfaces. Translated from the Japanese by the author
zbMATH Open document ID: 1453.14095
DOI: 10.4171/208
author: Shigeyuki Kondō
published in: EMS Tracts in Mathematics
publication date: 2020-07-06T00:00:00Z
review text: \(K3\) surfaces are an ubiquitous object of study in several areas of modern research, from algebra and geometry to physics. Despite being widely studied, some of their properties still remain mysterious. This book focuses on complex \(K3\) surfaces, and its goal is to introduce and prove the Torelli-type theorem for \(K3\) surfaces. This is then used to review several important results about moduli spaces and automorphism groups of \(K3\) surfaces. The book is self-contained, and only requires a basic knowledge of algebraic geometry. All the necessary background is thoroughly covered in the first part. All the proofs and the many examples are explained in great detail, making them very easy to follow. Therefore the book will be particularly helpful for young researchers, approaching this topic for the first time. However, even experts will find several interesting concepts and ideas in the last part, that covers some of the authors' works. In the following we provide a short summary of each chapter. Chapters 1--5 provide an introduction to classical topics, relevant to the study of \(K3\) surfaces. Chapter 1 reviews the basics of lattice theory, as the classification of root lattices and the classification of unimodular lattices. It then moves to the study of the embeddings of even lattices into unimodular lattices, following [\textit{V. V. Nikulin}, Math. USSR, Izv. 14, 103--167 (1980; Zbl 0427.10014)]. Chapter 2 introduces the fundamental domain of a reflection group, and explains the decomposition in chambers, with special interest towards reflection groups associated with even lattices. In Chapter 3 the author reviews the classification of complex analytic surfaces, giving several examples. He then focuses on elliptic surfaces and on the Kodaira's list of their singular fibers. Chapter 4 introduces \(K3\) surfaces. It reviews the basic examples and the main properties of the Néron-Severi lattice and the Kähler cone. It defines Kummer surfaces and studies in det…
reviewed by: Giacomo Mezzedimi
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14C34
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14J15
Mathematics Subject Classification ID: 14J50
zbMATH DE Number: 7217918
zbMATH Keywords: \(K3\) surfaces
zbMATH Keywords: Enriques surfaces
zbMATH Keywords: Torelli theorem
zbMATH Keywords: elliptic surfaces
zbMATH Keywords: automorphism group
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/208
OpenAlex ID: W4256079863
description: scientific article; zbMATH DE number 7217918