Geometry Over Nonclosed Fields

Description

Based on the Simons Symposia held in 2015, the proceedings in this volume focus on rational curves on higher-dimensional algebraic varieties and applications of the theory of curves to arithmetic problems. There has been significant progress in this field with major new results, which have given new impetus to the study of rational curves and spaces of rational curves on K3 surfaces and their higher-dimensional generalizations. One main recent insight the book covers is the idea that the geometry of rational curves is tightly coupled to properties of derived categories of sheaves on K3 surfaces. The implementation of this idea led to proofs of long-standing conjectures concerning birational properties of holomorphic symplectic varieties, which in turn should yield new theorems in arithmetic. This proceedings volume covers these new insights in detail.  Front Matter....Pages i-ix On the Kobayashi Pseudometric, Complex Automorphisms and Hyperkähler Manifolds....Pages 1-17 Lines on Cubic Hypersurfaces Over Finite Fields....Pages 19-51 Perverse Sheaves of Categories and Non-rationality....Pages 53-96 Divisor Classes and the Virtual Canonical Bundle for Genus 0 Maps....Pages 97-126 A Stronger Derived Torelli Theorem for K3 Surfaces....Pages 127-156 Morphisms to Brauer–Severi Varieties, with Applications to Del Pezzo Surfaces....Pages 157-196 Arithmetic of K3 Surfaces....Pages 197-248 Odd-Dimensional Cohomology with Finite Coefficients and Roots of Unity....Pages 249-261