On the PSL(2,19)-invariant cubic sevenfold

TL;DR

This paper investigates the unique PSL(2,19)-invariant cubic sevenfold, analyzing its intermediate Jacobian and demonstrating that its associated 85-dimensional torus is an Abelian variety, with detailed study of the G-invariant abelian 9-fold.

Contribution

It establishes that the intermediate Jacobian of the invariant cubic sevenfold is an Abelian variety and characterizes the G-invariant abelian 9-fold in detail.

Findings

The 85-dimensional torus is an Abelian variety.

The G-invariant abelian 9-fold A(X) is uniquely defined and its period lattice is described.

The cubic sevenfold's symmetry group determines a special Abelian variety.

Abstract

It has been proved by Adler that there exists a unique cubic hypersurface X in P^8 which is invariant under the action of the simple group PSL(2,19). In the present note we study the intermediate Jacobian of X and in particular we prove that the subjacent 85-dimensional torus is an Abelian variety. The symmetry group G=PSL(2,19) defines uniquely a G-invariant abelian 9-fold A(X), which we study in detail and describe its period lattice.