Co-calibrated \(G_2\) structure from cuspidal cubics (Q1759654)

The authors establish a correspondence between a cuspidal curve in a complex projective plane and a co-calibrated homogeneous \(G_2\) structure on the seven-dimensional parameter space of such cubics, using twistor theory. They actually use this twistor correspondence to construct a class of explicit examples of seven-dimensional manifolds \(M\) with a \(G_2\) structure. The main result of the paper is that the seven-dimensional space \(M=\mathrm{SL}(3,\mathbb{C})/\mathbb{C^*}\) of plane cuspidal cubics admits a canonical complexified co-calibrated \(G_2\) structure, whose three-form and metric are given explicitly. They show that this structure admits three homogeneous real forms: two with signature \((4,3)\), where \(M=\mathrm{SL}(3,\mathbb{R})/\mathbb{R}^*\) or \(M=\mathrm{SU}(3)/\mathrm{U}(1)\), and a Riemannian one with \(M=\mathrm{SU}(2,1)/\mathrm{U}(1)\). They finally discuss a differential equation approach showing that \(M\) arises as the solution space of an ODE. It is actually shown that cuspidal cubics and their higher-degree analogues with constant projective curvature are characterised as integral curves of certain seventh-order ODEs. Projective orbits of such curves are shown to be analytic continuations of Aloff-Wallach manifolds, and it is shown that only cubic lifts to a complete family of contact rational curves in a projectivised cotangent bundle to a projective plane. The whole setting is part of a more general theory, presented in [the second author and \textit{M. Godliński}, Q. J. Math. 63, No. 1, 101--132 (2012; Zbl 1258.53024)].