Geometry Over Nonclosed Fields

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Fedor Bogomolov, Brendan Hassett, Yuri Tschinkel
Springer, Feb 9, 2017 - Mathematics - 261 pages
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.

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On the Kobayashi Pseudometric Complex Automorphisms and Hyperkähler Manifolds
Lines on Cubic Hypersurfaces Over Finite Fields
Perverse Sheaves of Categories and Nonrationality
Divisor Classes and the Virtual Canonical Bundle for Genus 0 Maps
A Stronger Derived Torelli Theorem for K3 Surfaces
Morphisms to BrauerSeveri Varieties with Applications to Del Pezzo Surfaces
Arithmetic of K3 Surfaces
OddDimensional Cohomology with Finite Coefficients and Roots of Unity

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About the author (2017)

Fedor Bogomolov, Courant Institute of Mathematical Sciences, New York, NYBrendan Hassett, Brown University, Providence, Rhode IslandYuri Tshinkel,Courant Institute of Mathematical Sciences, New York, NY

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