Motivic nearby cycles and their monodromy at a singular point

TL;DR

This survey discusses methods to compute motivic nearby cycles and monodromy at quasi-homogeneous singularities, providing refined formulas that enhance classical results in algebraic geometry.

Contribution

It introduces quadratic and motivic refinements of classical formulas for singularities, extending the Deligne--Milnor and Picard--Lefschetz formulas within the motivic framework.

Findings

Quadratic Euler characteristic of nearby cycles computed

Motivic monodromy at singularities analyzed

Refined formulas extend classical results

Abstract

In this survey, we explain how to compute both the quadratic Euler characteristic of nearby cycles, and the motivic monodromy, at a quasi-homogeneous singularity. This gives, for such singularity, a quadratic refinement to the Deligne--Milnor formula in characteristic zero, and an enhancement of the Picard--Lefschetz formula to Voevodsky motives with rational coefficients.