Motivic Euler Characteristic of Nearby Cycles and a Generalized Quadratic Conductor Formula

Abstract: We compute the motivic Euler characteristic of Ayoub's nearby cycles spectrum in terms of strata of a semi-stable reduction, for a degeneration to multiple semi-quasi-homogeneous singularities. This allows us to compare the local picture at the singularities with the global conductor formula for hypersurfaces developed by Levine, Pepin Lehalleur and Srinivas, revealing that the formula is local in nature, thus extending it to the more general setting considered in this paper. The result is a quadratic refinement to the Milnor number formula with multiple singularities.