Ran Azouri (Paris): Computing motivic nearby cycles and their mondromy

This talk was part of the workshop "Motivic and non-commutative aspects of enumerative geometry" held in July 2023 at Radboud University, Nijmegen, The Netherlands.

Abstract: A major result in Ayoub’s thesis is on the motivic nearby cycles functor of semi-stable schemes; the value of the functor at the unit, on a stratum of the special fibre, is determined by its smooth locus. In the talk I will explain how to make motivic nearby cycles computable by reducing to the case of the theorem. One result is a formula for the quadratic Euler characteristic of nearby cycles around a semi-homogeneous singularity. This allows to extend the quadratic conductor formula for projective hypersurfaces of Levine, Pepin Lehalleur and Srinivas, to more general schemes. In another work in progress, joint with Emil Jacobsen, we compute the motivic monodromy operator around a singularity. By adapting Illusie’s approach to the setting of Voevodsky motives, we obtain a motivic version for the Picard-Lefschetz formula of Deligne and Katz.