Ran Azouri: Enumerating motivic nearby cycles

Abstract:

\(\mathbb{A}^1\) homotopy theory provides tools to refine geometric invariants from integers to quadratic forms. A key such invariant is the quadratic Euler characteristic. Ayoub's motivic nearby cycles are a tool to study singularities in the same world of \(\mathbb{A}^1\)-homotopy theory.

In the talk I will explain how to compute the quadratic Euler characteristic on the motivic sheaf of nearby cycles for certain singularities, using an explicit construction for semistable reduction . This, together with a recent work of Levine, Pepin Lehalleur and Srinivas, adds up to a quadratic conductor formula on schemes with semi-quasihomogeneous singularities, refining the integral formulas of Milnor and Deligne.

Time permitting, I will describe how, in a work in progress with Emil Jacobsen, we use a similar semistable reduction argument to compute the motivic monodromy on nearby cycles, generalising to motives the Picard-Lefschetz formula of Deligne.