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Integrable systems in four dimensions associated with six-folds in Gr(4, 6)

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posted on 13.12.2017 by B. Doubrov, Evgeny Ferapontov, B. Kruglikov, Vladimir 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.

Funding

The research of EVF was partially supported by the EPSRC grant EP/N031369/1.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

International Mathematics Research Notices

Volume

2019

Issue

21

Pages

6585–6613

Citation

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.

Publisher

Oxford University Press © The authors

Version

AM (Accepted Manuscript)

Publisher statement

This 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/

Acceptance date

14/11/2017

Publication date

2018-01-29

Copyright date

2019

Notes

This 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.

ISSN

1073-7928

eISSN

1687-0247

Language

en

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