Geometry Over Nonclosed Fields

1. Front Matter

   Pages i-ix

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2. On the Kobayashi Pseudometric, Complex Automorphisms and Hyperkähler Manifolds

   • Fedor Bogomolov, Ljudmila Kamenova, Steven Lu, Misha Verbitsky

   Pages 1-17

3. Lines on Cubic Hypersurfaces Over Finite Fields

   • Olivier Debarre, Antonio Laface, Xavier Roulleau

   Pages 19-51

4. Perverse Sheaves of Categories and Non-rationality

   • Andrew Harder, Ludmil Katzarkov, Yijia Liu

   Pages 53-96

5. Divisor Classes and the Virtual Canonical Bundle for Genus 0 Maps

   • A. J. de Jong, Jason Starr

   Pages 97-126

6. A Stronger Derived Torelli Theorem for K3 Surfaces

   • Max Lieblich, Martin Olsson

   Pages 127-156

7. Morphisms to Brauer–Severi Varieties, with Applications to Del Pezzo Surfaces

   • Christian Liedtke

   Pages 157-196

8. Arithmetic of K3 Surfaces

   • Anthony Várilly-Alvarado

   Pages 197-248

9. Odd-Dimensional Cohomology with Finite Coefficients and Roots of Unity

   • Yuri G. Zarhin

   Pages 249-261

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. 

• algebraic geometry
• rational curves
• K3 surfaces
• cubic four-folds
• hyper-Kahler manifolds

• Courant Institute of Mathematical Sciences, New York University, New York, USA

  Fedor Bogomolov

• Department of Mathematics, Brown University, Providence, USA

  Brendan Hassett

• Courant Institute of Mathematical Sciences and Simons Foundation, New York University, New York, USA

  Yuri Tschinkel

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

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