Geometry Over Nonclosed Fields: Geometry and Arithmetic of Holomorphic Symplectic Varieties

March 22–28, 2015

 
Organizing committee:
Fedor Bogomolov, Courant Institute of Mathematical Sciences
Brendan Hassett, Rice University
Yuri Tschinkel, Simons Foundation

The second Simons Symposium on Geometry over Nonclosed Fields took place March 22-28. The first symposium in this series focused on rational curves on higher-dimensional algebraic varieties and outlined applications of the theory of curves to arithmetic problems. Since then, there has been significant progress in this field, with major new results obtained by participants:

  • Proof of the Tate conjecture for K3 surfaces by Maulik and Madapusi Pera,
  • Proof of the integral Tate conjecture for cubic fourfolds over finite fields by Charles and Pirutka,
  • Proof of the ample and effective cone conjecture for deformations of punctual Hilbert schemes of K3 surfaces by Bayer-Macrì and Bayer-Hassett-Tschinkel,
  • Proof of the Morrison-Kawamata cone conjecture for general hyperkähler manifolds, by Verbitsky,
  • Proof of vanishing of Kobayashi pseudo-metric on hyperkähler manifolds, by Kamenova, Lu, and Verbitsky.

Some of these developments were discussed at the Symposium by Pirutka, Kamenova, and Verbitsky. These results have given new impetus to the study of rational curves and spaces of rational curves on K3 surfaces and their higher-dimensional generalizations. One of the main recent insights is 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. The Symposium featured several talks concerning the derived categories approach, by Bayer, Katzarkov, Macrì, and Stellari. It is expected that this approach will lead to deeper understanding of arithmetic properties of K3 surfaces over local fields, number fields, and function fields. Some of these ideas were outlined in talks by Liedtke and Olsson. Another source of ideas comes from Galois theory: distribution properties of Frobenius classes in Galois representations attached to curves or surfaces have striking geometric applications. This was the topic of talks by Charles, Katz, and Zarhin. Finally, moduli spaces continue to play an important role in arithmetic. The talks by Farkas, Hulek, and Várilly-Alvarado introduced the participants to new theorems in this area.

 

Agenda & Lecture Notes

Monday K3 Surfaces
Christian Liedtke
Good Reduction of K3 Surfaces (PDF)
Daniel Huybrecht
Curves and Cycles on K3 Surfaces (PDF)
Martin Olsson
The Derived Torelli Theorem for K3 Surfaces and Applications (PDF)
Arend Bayer
Derived Automorphism Groups of K3 Surfaces of Picard Rank One (PDF)
Tuesday Moduli Spaces
Gavril Farkas
A Uniformization for the Moduli Space of Abelian Varieties of Dimension Six
Klaus Hulek
Moduli of Enriques Surfaces (PDF)
Anthony Várilly-Alvarado
Kodaira Dimension of Moduli of special cubic fourfolds (PDF)
Wednesday Geometry of Holomorphic Symplectic Varieties
Ljudmila Kamenova
Vanishing of the Kobayashi Pseudometric on K3 Surfaces and Hyperkähler Manifolds (PDF)
Misha Verbitsky
Proof of Morrison-Kawamata cone conjecture for holomorphically symplectic manifolds (PDF)
Thursday Arithmetic Problems
Francois Charles
Exceptional isogenies of Elliptic Curves and Frobenius Distribution
Alena Pirutka
Integral Tate Conjecture for Cubic Fourfolds (PDF)
Nick Katz
Simple Things We Don’t Know (PDF)
Yuri Zarhin
Families of Abelian Varieties with Big ℓ-adic Monodromy
Friday Categorical Approaches
Ludmil Katzarkov
Sheaf of Categories and 4-Dimensional Cubics (PDF)
Emanuele Macrì
Stability Conditions on Threefolds (PDF)
Paolo Stellari
Stability Conditions on Threefolds 2 (PDF)

Download full agenda PDF

Participants

Arend Bayer University of Edinburgh
Fedor Bogomolov New York University
François Charles Orsay
Ivan Cheltsov University of Edinburgh
Izzet Coskun University of Illinois, Chicago
Olivier Debarre Université Paris Diderot
Gavril Farkas University of Berlin
Brendan Hassett Rice University
Klaus Hulek Institute for Advanced Study
Daniel Huybrechts University of Bonn
Ljudmila Kamenova SUNY Stony Brook
Nick Katz Princeton University
Ludmil Katzarkov Miami & Vienna
Christian Liedtke University of Munich
Emanuele Macri Ohio State University
Martin Olsson UC Berkeley
Alena Pirutka École polytechnique
Jason Starr SUNY Stony Brook
Paolo Stellari Università degli Studi di Milano
Yuri Tschinkel Simons Foundation
Ravi Vakil Stanford University
Anthony Várilly-Alvarado Rice University
Misha Verbitsky Higher School of Economics
Yuri Zarhin Penn State University

Download Participants PDF

Photographs

(Photo credit: Prof. Debarre)
Simons Symposia 2015 - Geometry Over Nonclosed Fields - Photo Credit:  Prof. Olivier Debarre
Simons Symposia 2015 - Geometry Over Nonclosed Fields - Photo Credit:  Prof. Olivier Debarre
Simons Symposia 2015 - Geometry Over Nonclosed Fields - Photo Credit:  Prof. Olivier Debarre
Simons Symposia 2015 - Geometry Over Nonclosed Fields - Photo Credit:  Prof. Olivier Debarre
Simons Symposia 2015 - Geometry Over Nonclosed Fields - Photo Credit:  Prof. Olivier Debarre
Simons Symposia 2015 - Geometry Over Nonclosed Fields - Photo Credit:  Prof. Olivier Debarre
Simons Symposia 2015 - Geometry Over Nonclosed Fields - Photo Credit:  Prof. Olivier Debarre
Simons Symposia 2015 - Geometry Over Nonclosed Fields - Photo Credit:  Prof. Olivier Debarre
Simons Symposia 2015 - Geometry Over Nonclosed Fields - Photo Credit:  Prof. Olivier Debarre
Simons Symposia 2015 - Geometry Over Nonclosed Fields - Photo Credit:  Prof. Olivier Debarre
Simons Symposia 2015 - Geometry Over Nonclosed Fields - Photo Credit:  Prof. Olivier Debarre
Simons Symposia 2015 - Geometry Over Nonclosed Fields - Photo Credit:  Prof. Olivier Debarre
Simons Symposia 2015 - Geometry Over Nonclosed Fields - Photo Credit:  Prof. Olivier Debarre
Simons Symposia 2015 - Geometry Over Nonclosed Fields - Photo Credit:  Prof. Olivier Debarre
Simons Symposia 2015 - Geometry Over Nonclosed Fields - Photo Credit:  Prof. Olivier Debarre
Simons Symposia 2015 - Geometry Over Nonclosed Fields - Photo Credit:  Prof. Olivier Debarre
Simons Symposia 2015 - Geometry Over Nonclosed Fields - Photo Credit:  Prof. Olivier Debarre
Simons Symposia 2015 - Geometry Over Nonclosed Fields - Photo Credit:  Prof. Olivier Debarre
Simons Symposia 2015 - Geometry Over Nonclosed Fields - Photo Credit:  Prof. Olivier Debarre
Simons Symposia 2015 - Geometry Over Nonclosed Fields - Photo Credit:  Prof. Olivier Debarre
Simons Symposia 2015 - Geometry Over Nonclosed Fields - Photo Credit:  Prof. Olivier Debarre

 
« Back to Simons Symposia