Zeta functions and Fourier-Mukai transforms

Informations projet

ZETA-FM

N° de convention de subvention: 864145

DOI

10.3030/864145

Date de signature de la CE 20 Decembre 2019

Date de début 1 Septembre 2020

Date de fin 31 Août 2026

Financé au titre de

• EXCELLENT SCIENCE - European Research Council (ERC)

Coût total

€ 2 000 000,00

Contribution de l’UE

€ 2 000 000,00

Coordonné par

UNIVERSITEIT VAN AMSTERDAM
Netherlands

Periodic Reporting for period 3 - ZETA-FM (Zeta functions and Fourier-Mukai transforms)

Période du rapport: 2023-09-01 au 2025-02-28

Résumé du contexte et des objectifs généraux du projet

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.

Travail effectué depuis le début du projet jusqu’à la fin de la période considérée dans le rapport et principaux résultats atteints jusqu’à présent

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.

Progrès au-delà de l’état des connaissances et impact potentiel prévu (y compris l’impact socio-économique et les conséquences sociétales plus larges du projet jusqu’à présent)

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.