K3 Surfaces

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. Keywords: K3 surface, Enriques surface, Kummer surface, Torelli-type theorem, period, lattice, reflection group, automorphism group Introduction Lattice theory Basic properties Classification of indefinite unimodular lattices Embeddings of lattices Reflection groups and their fundamental domains Reflection groups and fundamental domains Reflection groups associated with lattices Complex analytic surfaces Basics of complex analytic surfaces Classification of complex analytic surfaces Elliptic surfaces and their singular fibers K3 surfaces and examples Definition and examples of K3 surfaces Reflection group associated with non-singular rational curves and the Kähler cone Kummer surfaces The Kummer surface associated with a curve of genus 2 Torelli theorem for 2-dimensional complex tori Bounded symmetric domains of type IV and deformations of complex structures Bounded symmetric domains of type IV Deformations of complex structures and the Kodaira–Spencer map The Torelli-type theorem for K3 surfaces Periods of K3 surfaces and the Torelli-type theorem Local isomorphism of the period map (local Torelli theorem) The Torelli-type theorem for Kummer surfaces Density of the periods of Kummer surfaces Behavior of the Kähler cones under a deformation Proof of the Torelli-type theorem for K3 surfaces Surjectivity of the period map of K3 surfaces The period map of marked Kähler K3 surfaces Surjectivity of the period map of K3 surfaces Outline of a proof of the surjectivity of the period map of projective K3 surfaces Application of the Torelli-type theorem to automorphisms Automorphism group of a projective K3 surface Action of the automorphism group on the transcendental lattice A finite group that can be realized as an automorphism group of a K3 surface Automorphisms of K3 surfaces of order 2 Enriques surfaces Periods of Enriques surfaces Non-singular rational curves and elliptic curves on Enriques surfaces Automorphism groups of Enriques surfaces Examples of Enriques surfaces Application to the moduli space of plane quartic curves Plane quartics and del Pezzo surfaces of degree 2 K3 surfaces associated with plane quartics The moduli space of plane quartics and a complex ball Finite groups of symplectic automorphisms of K3 surfaces and the Mathieu group Niemeier lattices and the Mathieu group Finite symplectic automorphisms and the Mathieu group Automorphism group of the Kummer surface associated with a curve of genus 2 The Leech lattice and the even unimodular lattice of signature (1,25) Néron–Severi lattice of the Kummer surface Classical automorphisms of the Kummer surface The Kummer surface and the Leech roots Automorphism group of a generic Kummer surface Bibliography Index