New counterexamples to the birational Torelli theorem for Calabi--Yau manifolds

TL;DR

This paper constructs high-dimensional counterexamples to the birational Torelli theorem for Calabi-Yau manifolds, showing they can have isometric middle cohomologies despite not being birationally equivalent.

Contribution

It provides the first explicit high-dimensional counterexamples to the birational Torelli theorem for Calabi-Yau manifolds, including their cohomological and motivic properties.

Findings

Existence of non-birational Calabi-Yau pairs with isometric middle cohomologies

These pairs satisfy an L-equivalence relation in the Grothendieck ring

Counterexamples exist in arbitrarily high even dimensions

Abstract

We produce counterexamples to the birational Torelli theorem for Calabi-Yau manifolds in arbitrarily high dimension: this is done by exhibiting a series of non birational pairs of Calabi-Yau $(n^2-1)$-folds which, for $n \geq 2$ even, admit an isometry between their middle cohomologies. These varieties also satisfy an $\mathbb L$-equivalence relation in the Grothendieck ring of varieties, i.e. the difference of their classes annihilates a power of the class of the affine line. We state this last property for a broader class of Calabi-Yau pairs, namely all those which are realized as pushforwards of a general $(1,1)$-section on a homogeneous roof in the sense of Kanemitsu, along its two extremal contractions.