Motivic measures from at-the-characteristic cohomology theories

I will discuss recent joint work with Satriano and Sundbo, in which we use non-A¹-invariant motivic homotopy theory to construct new realization homomorphisms from the Grothendieck ring of varieties in positive characteristic, conditional on resolution of singularities. These include realizations associated with Hodge polynomials, among others. These new realization homomorphisms reveal previously unknown properties of the Grothendieck ring of varieties. For example, we show that universally homeomorphic varieties need not have the same motivic class, and that the classes of curves of different genus are pairwise algebraically independent. Under the same assumption, we also show that K-equivalent varieties have the same Hodge numbers in positive characteristic.

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