The Very General Verra Fourfold is Irrational

Abstract

We show that the very general Verra fourfold is irrational, using the Hodge atom framework of Katzarkov--Kontsevich--Pantev--Yu. Two novel points are: a refined analysis of Hodge atoms, based on the involution on the cohomology of $X$, and a derivation of the quantum multiplication matrix from the quantum differential operator. This gives the first successful application of the method of Hodge atoms to a space with Picard rank greater than one.

The proof of Theorem 1 uses ordinary weak factorization as if compatible with the Verra factor-swap symmetry. Proposition 12 implicitly needs Z/2-equivariant surface blow-up centres, not supplied by weak factorization or Iritani's blowup formula. The technique naturally belongs to G-equivariant atoms, proving Z/2-irrationality of the very general symmetric Verra fourfold; submitted separately (arXiv:2605.30439)