Cubic fourfolds with symplectic automorphisms

TL;DR

This paper classifies smooth complex cubic fourfolds with symplectic automorphisms by analyzing group representations, determining their moduli space components, and discussing their fields of definition.

Contribution

It provides explicit projective equations for these fourfolds and classifies their automorphism groups, advancing understanding of their symmetries and moduli.

Findings

Number of moduli space components identified

Explicit equations for fourfolds with symplectic automorphisms derived

Fields of definition analyzed for six maximal cases

Abstract

We determine projective equations of smooth complex cubic fourfolds with symplectic automorphisms by classifying 6-dimensional projective representations of Laza and Zheng's 34 groups. In particular, we determine the number of irreducible components for moduli spaces of cubic fourfolds with symplectic actions by these groups. We also discuss the fields of definition of cubic fourfolds in six maximal cases.