Reduced Tate-Shafarevich group

TL;DR

This paper establishes a reconstruction theorem for generic elliptic Calabi-Yau 3-folds, linking their derived equivalences to those of their fibers, and confirms conjectures about their shared properties and derived equivalences.

Contribution

It proves a reconstruction theorem for generic elliptic Calabi-Yau 3-folds and confirms conjectures relating their derived equivalences and shared Jacobians.

Findings

Two generic elliptic Calabi-Yau 3-folds are derived-equivalent over the base iff their fibers are derived-equivalent.

Pairs of elliptic Calabi-Yau 3-folds share the relative Jacobian.

Such pairs are proven to be P^2-linear derived-equivalent.

Abstract

We prove a sort of reconstruction theorem for generic elliptic Calabi-Yau $3$-folds in the sense of C\u{a}ld\u{a}raru. From our argument it follows that two generic elliptic Calabi-Yau $3$-folds are derived-equivalent linear over the base if and only if their generic fibers are derived-equivalent. As an application, we give affirmative answers to the conjectures raised by Knapp-Scheidegger-Schimannek. Namely, for each pair of elliptic Calabi-Yau $3$-folds in their list we prove that they share the relative Jacobian and are $P^2$-linear derived-equivalent.