\(K3\) surfaces. Translated from the Japanese by the author
\(K3\) 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 detail a few classical examples. Finally, the Torelli theorem for complex tori is proved. Chapter 5 gives first a short overview of bounded symmetric domains of type \(\mathrm{IV}\). It then applies some concepts of deformation theory for complex surfaces to the case of \(K3\) surfaces. Chapters 6--7 are the heart of the book. Chapter 6 is devoted to the proof of the Torelli-type theorem for (Kähler, projective, or polarized) \(K3\) surfaces. First, the local Torelli theorem is proved. Then the Torelli-type theorem is proved for Kummer surfaces, and combining it with the density of Kummer surfaces in the period domain of \(K3\) surfaces, the Torelli-type theorem is obtained. Chapter 7 proves the surjectivity of the period map of (marked Kähler, or projective) \(K3\) surfaces. Chapters 8--12 examine some applications of the Torelli-type theorem. Chapter 8 studies the automorphism group of a projective \(K3\) surface. More precisely, the actions of the automorphism group on the Néron-Severi and on the transcendental lattices are studied, and some explicit computations of automorphism groups of \(K3\) surfaces are provided. Finally, possible fixed loci of (symplectic and non-symplectic) involutions on \(K3\) surfaces are classified. Chapter 9 deals with Enriques surfaces. After reviewing their basic properties, the author proves the Torelli-type theorem for Enriques surfaces and the surjectivity of the period map. Then he studies smooth rational and elliptic curves on Enriques surfaces, and computes the automorphism group of some Enriques surfaces. He concludes the chapter with some classical and important examples. Chapter 10 reviews the main result of [\textit{S. Kondō}, J. Reine Angew. Math. 525, 219--232 (2000; Zbl 0990.14007)]. To any plane quartic \(C=V(f)\) we can associate the quartic \(K3\) surface \(X=V(w^4-f)\). This induces an isomorphism between \(\mathcal{M}_3\backslash \mathcal{H}_3\) and a certain moduli space of \(K3\) surfaces, that has the structure of a \(6\)-dimensional complex ball. Chapter 11 studies finite groups of symplectic automorphisms of \(K3\) surfaces, and shows that they can all be embedded in the Mathieu group \(M_{23}\). Chapter 12 investigates the automorphism group of the Kummer surface associated with a curve of genus \(2\). After recalling the classically known types of automorphisms of such \(K3\) surfaces, the author explicitly determines their automorphism group in the general case.
scientific article; zbMATH DE number 7217918