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

TL;DR

This paper computes the motivic Euler characteristic of nearby cycles for complex degenerations, revealing a local quadratic refinement of the Milnor number formula applicable to multiple singularities.

Contribution

It introduces a method to compute the motivic Euler characteristic in terms of strata, extending the global conductor formula to a local setting with multiple singularities.

Findings

Motivic Euler characteristic expressed via strata of semi-stable reduction

Comparison of local singularity data with global conductor formula

Extension of Milnor number formula to multiple singularities with quadratic refinement

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.