Co--calibrated $G_2$ structure from cuspidal cubics

TL;DR

This paper constructs a twistor correspondence linking cuspidal cubic curves in complex projective planes to co-calibrated $G_2$ structures on seven-dimensional spaces, revealing new geometric insights and explicit structures.

Contribution

It establishes a novel twistor correspondence between cuspidal cubics and co-calibrated $G_2$ structures, including explicit examples on $SU(2,1)/U(1)$ and connections to Aloff--Wallach manifolds.

Findings

Explicit co-calibrated $G_2$ structure on $SU(2,1)/U(1)$

Characterization of cuspidal cubics via 7th order ODEs

Connection between projective orbits and Aloff--Wallach manifolds

Abstract

We establish a twistor correspondence between a cuspidal cubic curve in a complex projective plane, and a co-calibrated homogeneous $G_2$ structure on the seven--dimensional parameter space of such cubics. Imposing the Riemannian reality conditions leads to an explicit co-calibrated $G_2$ structure on $SU(2, 1)/U(1)$. This is an example of an SO(3) structure in seven dimensions. Cuspidal cubics and their higher degree analogues with constant projective curvature are characterised as integral curves of 7th order ODEs. Projective orbits of such curves are shown to be analytic continuations of Aloff--Wallach manifolds, and it is shown that only cubics lift to a complete family of contact rational curves in a projectivised cotangent bundle to a projective plane.