Seminar on Arithmetic Geometry

Ran Azouri (Sorbonne Paris North University)

A^1 homotopy theory provides tools to refine geometric invariants on integers to quadratic forms. A key such invariant is the quadratic Euler characteristic; Ayoub’s motivic nearby cycles provide a tool to study singularities in the world of A^1 homotopy.
In the talk I will explain how to compute the quadratic Euler characteristic on the motivic nearby cycles spectrum around certain singularities, using an explicit semistable reduction construction. This, together with a work of Levine, Pepin Lehalleur and Srinivas, adds up to a quadratic conductor formula on schemes with semi-quasihomogeneous singularities, refining formulas of Milnor and Deligne.
Later 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 and Katz.

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