Grothendieck ring of varieties and at-the-characteristic numerical invariants

Assuming a weak form of resolution of singularities in positive characteristic, we show several at-the-characteristic numerical invariants, including Hodge numbers, are given by homomorphisms out of the Grothendieck ring of varieties. Under the same assumption, we obtain two consequences: (i) $\widehat{K}$-equivalent varieties have the same Hodge numbers, answering a question of de Fernex–Mere; and (ii) universally homeomorphic varieties need not have the same class in the Grothendieck ring of varieties, answering a question of Nicaise–Sebag. In the appendix, we provide unconditional evidence for (i) by proving that birational Calabi–Yau varieties of dimension at most four have the same Hodge numbers.