K3 Surfaces

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