Integrable systems in four dimensions associated with six-folds in Gr(4, 6)
journal contributionposted on 2017-12-13, 14:34 authored by B. Doubrov, Evgeny FerapontovEvgeny Ferapontov, B. Kruglikov, Vladimir NovikovVladimir Novikov
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 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.
The research of EVF was partially supported by the EPSRC grant EP/N031369/1.
- Mathematical Sciences
Published inInternational Mathematics Research Notices
CitationDOUBROV, B. ...et al., 2018. Integrable systems in four dimensions associated with six-folds in Gr(4, 6). International Mathematics Research Notices, 2019(21), pp. 6585–6613.
PublisherOxford University Press © The authors
- AM (Accepted Manuscript)
Publisher statementThis work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/
NotesThis is a pre-copyedited, author-produced version of an article accepted for publication in International Mathematics Research Notices following peer review. The version of record DOUBROV, B. ...et al., 2018. Integrable systems in four dimensions associated with six-folds in Gr(4, 6). International Mathematics Research Notices, 2019(21), pp. 6585–6613, is available online at: https://doi.org/10.1093/imrn/rnx308.