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arXiv · 2605.30450

Quantum cohomology and birational geometry of Verra fourfolds

Vladimiro Benedetti·Jérémy Guéré·Laurent Manivel·Nicolas Perrin
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Abstract

We compute the small quantum cohomology ring of a Verra fourfold. Using the theory of atoms recently developped by Katzarkov--Kontsevich--Pantev--Yu, and building on recent papers of the authors, we deduce that a Verra fourfold is never birational to a very general cubic fourfold, nor to a very general Gushel--Mukai fourfold, whereas it was previously known that a general Verra fourfold is birational to a general nodal Gushel--Mukai fourfold. More precisely, we show that for every smooth cubic fourfold or smooth Gushel--Mukai fourfold that is birational to some Verra fourfold, the primitive cohomology is isomorphic, as a rational Hodge structure, to the middle cohomology of some projective K3 surface.

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Vladimiro Benedetti, Jérémy Guéré, Laurent Manivel, Nicolas Perrin. 2026-05-28. Quantum cohomology and birational geometry of Verra fourfolds. https://arxiv.org/abs/2605.30450

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