Periodic Reporting for period 3 - ZETA-FM (Zeta functions and Fourier-Mukai transforms)
Reporting period: 2023-09-01 to 2025-02-28
Arithmetic geometry and the study of derived categories of coherent sheaves are two central areas of research in algebraic geometry. Despite their many points of contact, they have until recently remained largely disjoint.
The zeta function of an algebraic variety over a finite field is one of the most studied invariants in arithmetic geometry, and a conjecture of Orlov predicts that this invariant can be detected by the derived category of coherent sheaves on the variety. One of the principal aims of this project is to prove this for large classes of varieties. A secondary aim is to develop techniques that will allow for further interaction between arithmetic geometry and derived categories.
The zeta function of an algebraic variety over a finite field is one of the most studied invariants in arithmetic geometry, and a conjecture of Orlov predicts that this invariant can be detected by the derived category of coherent sheaves on the variety. One of the principal aims of this project is to prove this for large classes of varieties. A secondary aim is to develop techniques that will allow for further interaction between arithmetic geometry and derived categories.
One of the first main achievements of the project is a proof of Orlov's conjecture over the complex numbers, for hyperkähler varieties. This is an important class of algebraic varieties admitting interesting derived equivalences. This gives a promising starting point to prove Orlov's conjecture in more arithmetic settings through techniques from deformation theory and classical arithmetic geometry. As a first step in this direction, several results in deformation theory have been obtained. This includes new results on deformations of algebraic varieties with trivial canonical class, and new results on deformations of derived equivalences between algebraic varieties with trivial canonical class.
In the remainder of the project we will both focus on generalising our results to non-commutative deformations, and to prove Orlov's conjecture over the complex numbers for broader classes of varieties.