Integrable Systems In Four Dimensions Associated With Six-Folds In Gr(4,6)

Let Gr(d, n) be the Grassmannian of d-dimensional linear subspaces of an n-dimensional vector space V. A submanifold X subset of Gr(d, n) gives rise to a differential system Sigma(X) that governs d-dimensional submanifolds of V whose Gaussian image is contained in X. We investigate a special case of this construction where X is a six-fold in Gr(4, 6). The corresponding system Sigma(X) reduces to a pair of first-order PDEs for 2 functions of 4 independent variables. Equations of this type arise in self-dual Ricci-flat geometry. Our main result is a complete description of integrable systems Sigma(X). These naturally fall into two subclasses.Systems of Monge-Ampere type. The corresponding six-folds are codimension 2 linear sections of the Plucker embedding Gr(4, 6) hooked right arrow P-14.General linearly degenerate systems. The corresponding six-folds are the images of quadratic maps P-6 -> Gr(4, 6) given by a version of the classical construction of Chasles.We prove that integrability is equivalent to the requirement that the characteristic variety of system Sigma(X) gives rise to a conformal structure which is self-dual on every solution. In fact, all solutions carry hyper-Hermitian geometry.