Séminaire d'arithmétique à Lyon

Motivic nearby cycles and quadratic conductor formulas

by Ran Azouri


Several tools for studying degenerations can be applied to a motivic setting: The nearby cycles functor of Ayoub in the motivic stable homotopy category; nearby cycles in the context of motivic integration; computing the Euler characteristics of the singular and generic fibers. In that context we report on a recent work by Levine, Pepin Lehalleur and Srinivas developing a quadratic conductor formula for hypersurfaces, using a motivic version for the Euler characteristic, which takes values at the Grothendieck-Witt ring of the base field, i.e. in quadratic forms. We discuss how reinterpreting  the formula in terms of motivic nearby cycles and computing it on a semi-stable reduction, allows us to extend the formula to a more general degeneration with a few (quasi-)homogeneous singularities.