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

Toni Annala; Matthew Satriano; Evan Sundbo

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arXiv · 2610.02690

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

Toni Annala·Matthew Satriano·Evan Sundbo

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Abstract

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.

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Toni Annala, Matthew Satriano, Evan Sundbo. 2026-10-02. Grothendieck ring of varieties and at-the-characteristic numerical invariants. https://arxiv.org/abs/2610.02690

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