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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) K̂-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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Cite

BibTeXRIS
BibTeX
@misc{annala2026grothendieck,
  title = {{Grothendieck ring of varieties and at-the-characteristic numerical invariants}},
  author = {Toni Annala and Matthew Satriano and Evan Sundbo},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02690},
  eprint = {2610.02690},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02690},
}
RIS
TY  - UNPB
TI  - Grothendieck ring of varieties and at-the-characteristic numerical invariants
AU  - Toni Annala
AU  - Matthew Satriano
AU  - Evan Sundbo
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.02690
UR  - https://arxiv.org/abs/2610.02690
AB  - 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) K̂-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.
ER  -