Some quadratic conductor formulas (joint work with Pepin Lehalleur and Srinivas)

A smooth projective variety over a field k defines a dualizable object in the motivic stable homotopy over k, and thereby an Euler characteristic in the endomorphism ring of the unit. A theorem of Morel identifies this endomorphism ring with the Grothendieck-Witt ring of quadratic forms over k and with A. Raksit, we have shown that this quadratic Euler characteristic is given by the intersection form on Hodge cohomology. We use the computation of Hodge cohomology of hypersurfaces via the Jacobian ring to give an explicit description of the categorical Euler characteristic of a smooth hypersurface and use this to give examples of a quadratic conductor formula for certain degenerations. There are mysterious "error terms" that makes the formula deviate from what one might expect at first glance. These error terms disappear over the complex numbers and also over the real numbers, but are in general non-zero.