Zeta functions and Fourier-Mukai transforms

Projektinformationen

ZETA-FM

ID Finanzhilfevereinbarung: 864145

DOI

10.3030/864145

EK-Unterschriftsdatum 20 Dezember 2019

Startdatum 1 September 2020

Enddatum 31 August 2026

Finanziert unter

• EXCELLENT SCIENCE - European Research Council (ERC)

Gesamtkosten

€ 2 000 000,00

EU-Beitrag

€ 2 000 000,00

Koordiniert durch

UNIVERSITEIT VAN AMSTERDAM
Netherlands

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

Berichtszeitraum: 2023-09-01 bis 2025-02-28

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

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.

Arbeit, die ab Beginn des Projekts bis zum Ende des durch den Bericht erfassten Berichtszeitraums geleistet wurde, und die wichtigsten bis dahin erzielten Ergebnisse

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.

Fortschritte, die über den aktuellen Stand der Technik hinausgehen und voraussichtliche potenzielle Auswirkungen (einschließlich der bis dato erzielten sozioökonomischen Auswirkungen und weiter gefassten gesellschaftlichen Auswirkungen des Projekts)

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.