PreprintCC BY 4.0arXiv
Integral Hodge structures and atoms of fourfolds
Abstract
Following the recent work by OpenAI on the irrationality of very general cubic fourfolds in Hassett divisors, we prove some compatibility of Iritani's blow-up formula with the classical blow-up decomposition of integral Hodge structures. As a consequence, we express OpenAI's birational invariant in terms of the theory of atoms developped by Katzarkov--Kontsevich--Pantev--Yu. We also show that, for every smooth complex rational cubic fourfold, the primitive cohomology is isometric as an integral Hodge structure to the twisted middle cohomology of a K3 surface and we prove irrationality of the very general Verra fourfold.
Related
- The Tate conjecture for powers of abelian fourfolds and the Hodge conjecture for powers of CM fourfolds
- The Hodge conjecture for Fermat fourfolds of degree at most 100, and the semi-standard obstruction at degree 70
- Classification of can-Var diagrams of mixed Hodge structures and monodromic mixed Hodge modules on ℂ
- mixed Hodge structures on the strictly upper triangular Lie algebras
- Limiting Hodge Structures for Cubic Hypersurfaces of Secant Type
BibTeX
@misc{guere2026integral,
title = {{Integral Hodge structures and atoms of fourfolds}},
author = {Jérémy Guéré},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40811},
eprint = {2610.40811},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40811},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Integral Hodge structures and atoms of fourfolds
AU - Jérémy Guéré
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40811
UR - https://arxiv.org/abs/2610.40811
AB - Following the recent work by OpenAI on the irrationality of very general cubic fourfolds in Hassett divisors, we prove some compatibility of Iritani's blow-up formula with the classical blow-up decomposition of integral Hodge structures. As a consequence, we express OpenAI's birational invariant in terms of the theory of atoms developped by Katzarkov--Kontsevich--Pantev--Yu. We also show that, for every smooth complex rational cubic fourfold, the primitive cohomology is isometric as an integral Hodge structure to the twisted middle cohomology of a K3 surface and we prove irrationality of the very general Verra fourfold.
ER -