Séminaire de Géométrie Arithmétique Paris-Pékin-Tokyo

${\Bbb P}^1$-Localisation and a Possible Definition of Arithmetic Kodaira-Spencer Classes

by Joseph Ayoub (University of Zurich)

Amphithéâtre Léon Motchane (IHES)

Amphithéâtre Léon Motchane


Le Bois Marie 35, route de Chartres 91440 Bures-sur-Yvette

${\mathbb A}^1$-localisation is a universal construction which produces "cohomology theories" for which the affine line ${\mathbb A}^1$ is contractible. It plays a central role in the theory of motives à la Morel-Voevodsky. In this talk, I'll discuss the analogous construction where the affine line is replaced by the projective line ${\mathbb P}^1$. This is the ${\mathbb P}^1$-localisation which is arguably an unnatural construction since it produces "cohomology theories" for which the projective line ${\mathbb P}^1$ is contractible. Nevertheless, I'll explain a few positive results and some computations around this construction which naturally lead to a definition of Kodaira-Spencer classes of arithmetic nature. (Unfortunately, it is yet unclear if these classes are really interesting and nontrivial.)

Organized by

Ahmed Abbes