RationAlgic
Financiado
Rationality of varieties and algebraic cycles
This project uses algebraic cycles and unramified cohomology to attack fundamental questions about the rationality, stable rationality and unirationality of rationally connected varieties, the integral Hodge conjecture for abelian...
This project uses algebraic cycles and unramified cohomology to attack fundamental questions about the rationality, stable rationality and unirationality of rationally connected varieties, the integral Hodge conjecture for abelian varieties, as well as the Griffiths-Harris conjecture about curves on three-dimensional hypersurfaces.
A breakthrough of Voisin, with improvements by Colliot-Thélène--Pirutka and myself, recently led to tremendous advances in our understanding of (stable) rationality of rationally connected varieties. For instance, this allowed me to solve the rationality problem for hypersurfaces under a logarithmic degree bound, improving previous linear bounds of Kollár and Totaro. This project pushes this circle of ideas further, aiming in particular at a solution of the rationality problem beyond my logarithmic bound.
One of the most powerful (stable) birational invariants of smooth projective varieties is unramified cohomology. In general, this invariant is notoriously hard to compute and we aim to develop new tools which allow to compute unramified cohomology more efficiently. We will use this to analyse the third unramified cohomology of abelian varieties and of hypersurfaces in projective 4-space. By a result of Colliot-Thélène and Voisin, this will allow us to attack the integral Hodge conjecture for abelian varieties, and hence, by work of Voisin, the longstanding open problem whether cubic threefolds are stably rational, as well as an old conjecture of Griffiths and Harris concerning curves on three-dimensional hypersurfaces.
We also introduce a cycle-theoretic approach, using the torsion order of symmetric products, to construct an obstruction for the unirationality of rationally connected varieties. We aim to use this to show that not every rationally connected variety is unirational, thereby solving a longstanding open problem in the field.
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Duración del proyecto: 65 meses
Fecha Inicio: 2020-10-08
Fecha Fin: 2026-03-31
Fecha Fin: 2026-03-31
Línea de financiación: concedida
El organismo H2020 notifico la concesión del proyecto el día 2020-10-08
Línea de financiación objetivo
El proyecto se financió a través de la siguiente ayuda:
ERC-2020-STG:
ERC STARTING GRANTS
Cerrada
hace 5 años
Presupuesto
El presupuesto total del proyecto asciende a
1M€
Líder del proyecto
GOTTFRIED WILHELM LEIBNIZ UNIVERSITAET HANNOV...
No se ha especificado una descripción o un objeto social para esta compañía.
Perfil tecnológico
TRL
4-5
Perfil tecnológico
TRL
4-5
798.84K€ |
Participante