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
Author(s): Shigeyuki Kondo
Series: Tracts in Mathematics 32
Publisher: European Mathematical Society
Year: 2020
Language: English
Pages: 252
City: S.l.
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
