The stability manifold of $E{\times} E{\times} E$

TL;DR

This paper identifies a specific 14-dimensional component of the stability conditions space for the derived category of a triple product of elliptic curves without complex multiplication, confirming a conjecture related to homological mirror symmetry.

Contribution

It explicitly describes a full component of the stability manifold for $D^b(E^3)$, providing a concrete description in terms of alternating trilinear forms and confirming a conjecture of Kontsevich.

Findings

Determined a 14-dimensional component of the stability manifold.

Provided a concrete description using alternating trilinear forms.

Confirmed Kontsevich's conjecture for dimension 3.

Abstract

We determine a full component of the space of stability conditions on $D^b(E^3)$ where $E$ is an elliptic curve without complex multiplication. The component has complex dimension 14 and a very concrete description in terms of alternating trilinear forms. This confirms a conjecture of Kontsevich, motivated by homological mirror symmetry, in the case of dimension $3$.