K3 Surfaces and Their Moduli

This bookprovides an overview of the latest developments concerning the moduli of K3surfaces. It is aimed at algebraic geometers, but is also of interest to numbertheorists and theoretical physicists, and continues the tradition of relatedvolumes like "The Moduli Space of Curves" and "Moduli of Abelian Varieties,"which originated from conferences on the islands Texel and Schiermonnikoog andwhich have become classics.

K3 surfacesand their moduli form a central topic in algebraic geometry and arithmeticgeometry, and have recently attracted a lot of attention from bothmathematicians and theoretical physicists. Advances in this field often resultfrom mixing sophisticated techniques from algebraic geometry, lattice theory,number theory, and dynamical systems. The topic has received significantimpetus due to recent breakthroughs on the Tate conjecture, the study ofstability conditions and derived categories, and links with mirror symmetry andstring theory. At the sametime, the theory of irreducible holomorphicsymplectic varieties, the higher dimensional analogues of K3 surfaces, hasbecome 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.

Introduction.-Samuel Boissière, Andrea Cattaneo, MarcNieper-Wisskirchen, and Alessandra Sarti: The automorphism group of theHilbert scheme of two points on a generic projective K3 surface.- Igor Dolgachev: Orbital counting ofcurves on algebraic surfaces and sphere packings.- V. Gritsenko and K. Hulek: Moduli of polarized Enriques surfaces.- Brendan Hassett and Yuri Tschinkel: Extremalrays and automorphisms of holomorphic symplectic varieties.- Gert Heckman and Sander Rieken: An oddpresentation for W(E_6).- S. Katz, A.Klemm, and R. Pandharipande, with an appendix by R. P. Thomas: On themotivic stable pairs invariants of K3 surfaces.- Shigeyuki Kondö: The Igusa quartic and Borcherds products.- Christian Liedtke: Lectures onsupersingular K3 surfaces and the crystalline Torelli theorem.- Daisuke Matsushita: On deformations ofLagrangian fibrations.- G. Oberdieck andR. Pandharipande: Curve counting on K3 x E,the Igusa cusp form X_10, anddescendent integration.- Keiji Oguiso:Simple abelian varieties and primitive automorphisms of null entropy ofsurfaces.- Ichiro Shimada: Theautomorphism groups of certain singular K3 surfaces and an Enriques surface.- Alessandro Verra: Geometry of genus 8Nikulin surfaces and rationality of their moduli.- Claire Voisin: Remarks and questions on coisotropic subvarietiesand 0-cycles of hyper-Kähler varieties.