This book provides an overview of the latest developments concerning the moduli of K3 surfaces. It is aimed at algebraic geometers, but is also of interest to number theorists and theoretical physicists, and continues the tradition of related volumes like "The Moduli Space of Curves" and "Moduli of Abelian Varieties," which originated from conferences on the islands Texel and Schiermonnikoog and which have become classics. K3 surfaces and their moduli form a central topic in algebraic geometry and arithmetic geometry, and have recently attracted a lot of attention from both mathematicians and theoretical physicists. Advances in this field often result from mixing sophisticated techniques from algebraic geometry, lattice theory, number theory, and dynamical systems. The topic has received significant impetus due to recent breakthroughs on the Tate conjecture, the study of stability conditions and derived categories, and links with mirror symmetry and string theory. At the same time, the theory of irreducible holomorphic symplectic varieties, the higher dimensional analogues of K3 surfaces, has become a mainstream topic in algebraic geometry. Contributors: S. Boissière, A. Cattaneo, I. Dolgachev, V. Gritsenko, B. Hassett, G. Heckman, K. Hulek, S. Katz, A. Klemm, S. Kondo, C. Liedtke, D. Matsushita, M. Nieper-Wisskirchen, G. Oberdieck, K. Oguiso, R. Pandharipande, S. Rieken, A. Sarti, I. Shimada, R. P. Thomas, Y. Tschinkel, A. Verra, C. Voisin.

Inhalt

Introduction.- Samuel Boissière, Andrea Cattaneo, Marc Nieper-Wisskirchen, and Alessandra Sarti: The automorphism group of the Hilbert scheme of two points on a generic projective K3 surface.- Igor Dolgachev: Orbital counting of curves on algebraic surfaces and sphere packings.- V. Gritsenko and K. Hulek: Moduli of polarized Enriques surfaces.- Brendan Hassett and Yuri Tschinkel: Extremal rays and automorphisms of holomorphic symplectic varieties.- Gert Heckman and Sander Rieken: An odd presentation for W(E6).- S. Katz, A. Klemm, and R. Pandharipande, with an appendix by R. P. Thomas: On the motivic stable pairs invariants of K3 surfaces.- Shigeyuki Kondö: The Igusa quartic and Borcherds products.- Christian Liedtke: Lectures on supersingular K3 surfaces and the crystalline Torelli theorem.- Daisuke Matsushita: On deformations of Lagrangian fibrations.- G. Oberdieck and R. Pandharipande: Curve counting on K3 x E,the Igusa cusp form X10, and descendent integration.- Keiji Oguiso: Simple abelian varieties and primitive automorphisms of null entropy of surfaces.- Ichiro Shimada: The automorphism groups of certain singular K3 surfaces and an Enriques surface.- Alessandro Verra: Geometry of genus 8 Nikulin surfaces and rationality of their moduli.- Claire Voisin: Remarks and questions on coisotropic subvarieties and 0-cycles of hyper-Kähler varieties.