On the deformation chirality of real cubic fourfolds

TL;DR

This paper classifies real cubic fourfolds based on their deformation types and introduces an arithmetical criterion for chirality, revealing the number of chiral classes among certain types of these fourfolds.

Contribution

It provides a method to determine chirality of real cubic fourfolds using eigen-sublattices of complex conjugation in homology, refining deformation classification.

Findings

Identifies one chiral class among M-cubics.

Finds three chiral classes among (M-1)-cubics.

Contrasts chirality distribution with achiral classes.

Abstract

According to our previous results, the conjugacy class of the involution induced by the complex conjugation in the homology of a real non-singular cubic fourfold determines the fourfold up to projective equivalence and deformation. Here, we show how to eliminate the projective equivalence and to obtain a pure deformation classification, that is how to respond to the chirality question: which cubics are not deformation equivalent to their image under a mirror reflection. We provide an arithmetical criterion of chirality, in terms of the eigen-sublattices of the complex conjugation involution in homology, and show how this criterion can be effectively applied taking as examples $M$-cubics (that is those for which the real locus has the richest topology) and $(M-1)$-cubics (the next case with respect to complexity of the real locus). It happens that there is one chiral class of $M$-cubics and three chiral classes of $(M-1)$-cubics, contrary to two achiral classes of $M$-cubics and three achiral classes of $(M-1)$-cubics.