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