Seminar on A1-topology, motives and K-theory

Motivic nearby cycles and quadratic conductor formulas

by Ran Azouri (Universität Duisburg-Essen)

Europe/Moscow
818-1526-4739 (Zoom)

818-1526-4739

Zoom

Description

We review some tools to investigate degenerations that can be adapted to a motivic setting: The nearby cycles functor of Ayoub in motivic homotopy theory; nearby cycles in the context of motivic integration; comparing the Euler characteristics of the singular and generic fibers. In that context we have a quadratic conductor formula for hypersurfaces in a local setting (a recent work by Levine, Pepin Lehalleur and Srinivas) with the motivic Euler characteristic, taking values at the Grothendieck-Witt ring of the base field, i.e. in quadratic forms. We discuss how reinterpreting the terms in the formula with motivic nearby cycles and computing them on a semi-stable reduction of the singular fiber, allows us to extend the formula to a more general degeneration with a few (quasi-)homogeneous singularities.