Classification of complete projective special real surfaces (Q2921107)

A \textit{projective special real manifold} is a hypersurface \({\mathcal H} \subset {\mathbb R}^{n+1}\) defined by an equation \(h(x)=1\) where \(h\) is a cubic polynomial on \(\mathbb R^{n+1}\) such that \(-\partial^2h\) induces a positive symmetric tensor field on \(\mathcal{H}\). Using supergravity constructions, an \(n\)-dimensional complete projective special real manifold defines a \((2n+2)\)-dimensional complete projective special Kähler manifold, and a \((4n+8)\)-dimensional complete quaternionic Kähler manifold [the first author et al., Commun. Math. Phys. 311, No. 1, 191--213 (2012; Zbl 1247.83224)]. The authors consider the case when \(n=2\) and they classify complete projective special real surfaces using homogeneous real cubic polynomials. These surfaces give rise to 6-dimensional complete projective special Kähler manifolds and to 16-dimensional complete quaternionic Kähler manifolds.