Non-birational Calabi-Yau threefolds that are derived equivalent

TL;DR

This paper explores the existence of non-birational Calabi-Yau threefolds that are derived equivalent, using genus one fibrations with high-degree multisections, and discusses implications for the Torelli problem.

Contribution

It demonstrates that certain Calabi-Yau threefolds with specific genus one fibrations can be non-birational yet derived equivalent, providing new insights into their classification.

Findings

Existence of non-birational derived equivalent Calabi-Yau threefolds

Implications for counterexamples to the Torelli problem

Connection between genus one fibrations and derived equivalences

Abstract

We argue that the existence of genus one fibrations with multisections of high degree on certain Calabi-Yau threefolds implies the existence of pairs of such varieties that are not birational, but are derived equivalent. It also (likely) implies the existence of non-birational counterexamples to the Torelli problem for Calabi-Yau threefolds.