On an example of Aspinwall, Morrison, and Szendr\H{o}i

TL;DR

This paper investigates the cohomological properties of a family of Calabi-Yau 3-folds related to the mirror quintic, confirming that certain members have distinct integral Hodge structures despite sharing rational structures, thus providing insights into their birational and derived equivalence status.

Contribution

It proves that the integral Hodge structures of the family members differ, confirming Szendr"H{o}i's conjecture and enhancing understanding of their geometric relationships.

Findings

Integral Hodge structures differ among the family members.

Confirmed non-birational and non-derived equivalence.

Discussed implications for Torelli-type theorems.

Abstract

We study the cohomology of a 1-parameter family Y_t of Calabi-Yau 3-folds introduced by Aspinwall and Morrison, related to the mirror quintic family. Szendr\H{o}i proved that Y_t, Y_{xi t}, ..., Y_{xi^4 t}, where xi is a fifth root of unity, have the same rational Hodge structure but are not isomorphic, and conjectured that they are not birational or even derived equivalent. We confirm this by proving that their integral Hodge structures are different, and discuss how this fits with known Torelli-type theorems and counterexamples.