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Download fileIntegrable systems in four dimensions associated with six-folds in Gr(4, 6)
journal contribution
posted on 2017-12-13, 14:34 authored by B. Doubrov, Evgeny FerapontovEvgeny Ferapontov, B. Kruglikov, Vladimir NovikovVladimir NovikovLet 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 di↵erential system ⌃(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 sixfold in Gr(4, 6). The corresponding system ⌃(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 ⌃(X). These naturally fall into two subclasses.
• Systems of Monge-Ampere type. The corresponding sixfolds X are codimension 2 linear sections of the Pl¨ucker embedding Gr(4, 6) ,! P14.
• General linearly degenerate systems. The corresponding sixfolds X are the images
of quadratic maps P6 99K 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 ⌃(X) gives rise to a conformal structure which is self-dual on every solution. In
fact, all solutions carry hyper-Hermitian geometry.
Funding
The research of EVF was partially supported by the EPSRC grant EP/N031369/1.
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