A counterexample of the birational Torelli problem via Fourier--Mukai transforms

TL;DR

This paper constructs a specific example of two minimal 3-folds that are derived and deformation equivalent but not birationally equivalent, providing a counterexample to the birational Torelli problem using Fourier--Mukai transforms.

Contribution

It presents the first known counterexample to the birational Torelli problem by leveraging Fourier--Mukai transforms on rational elliptic surfaces.

Findings

Two minimal 3-folds are derived equivalent but not birationally equivalent.

The example demonstrates the limitations of the Torelli problem in birational geometry.

Fourier--Mukai numbers are studied for rational elliptic surfaces.

Abstract

We study the Fourier--Mukai numbers of rational elliptic surfaces. As its application, we give an example of a pair of minimal 3-folds with Kodaira dimensions 1, $h^1(\mc O)=h^2(\mc O)=0$ such that they are mutually derived equivalent, deformation equivalent, but not birationally equivalent. It also supplies a counterexample of the birational Torelli problem.