Integrable Systems in Four Dimensions Associated with Six-Folds in Gr(4, 6)

Abstract

Let Gr(d, n) be the Grassmannian of d-dimensional linear subspaces of an n-dimensional vector space V. A submanifold X Gr(d, n) gives rise to a differential system ς(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 ς(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 ς(X). These naturally fall into two subclasses. Systems of Monge-Ampère type. The corresponding six-folds X are codimension 2 linear sections of the Plücker embedding Gr(4, 6)→P14. General linearly degenerate systems. The corresponding six-folds X are the images of quadratic mapsP6 → 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 ς(X) gives rise to a conformal structure which is self-dual on every solution. In fact, all solutions carry hyper-Hermitian geometry.