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

Abstract

Let Gr(d, n) be the Grassmannian of <i>d</i>-dimensional linear subspaces of an <i>n</i>-dimensional vector space <i>V</i>. A submanifold <i>X</i> ⊂ Gr(<i>d, n</i>) gives rise to a differential system Σ(X) that governs <i>d</i>-dimensional submanifolds of <i>V</i> whose Gaussian image is contained in <i>X</i>. We investigate a special case of this construction where <i>X</i> is a six-fold in Gr(4, 6). The corresponding system Σ(<i>X</i>) 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 <i>integrable</i> systems Σ(<i>X</i>). These naturally fall into two subclasses.<p>

<ul id = «mylist»><li>Systems of Monge–Ampère type. The corresponding six-folds <i>X</i> are codimension 2 linear sections of the Plücker embedding Gr(4, 6)<span>&#8618;</span>P¹⁴⁠.</li>

<li>General linearly degenerate systems. The corresponding six-folds <i>X</i> are the images of quadratic maps P⁶⇢ Gr(4, 6) given by a version of the classical construction of Chasles.</li></ul><p>

We prove that integrability is equivalent to the requirement that the characteristic variety of system Σ(<i>X</i>) gives rise to a conformal structure which is self-dual on every solution. In fact, all solutions carry hyper-Hermitian geometry.