Singular improper affine spheres from a given Lagrangian submanifold

Given a Lagrangian submanifold L of the affine symplectic 2n-space, one can canonically and uniquely define a center-chord and a special improper affine sphere of dimension 2n, both of whose sets of singularities contain L. Although these improper affine spheres (IAS) always present other singularit...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2020-11, Vol.374, p.107326, Article 107326
Main Authors: Craizer, Marcos, Domitrz, Wojciech, Rios, Pedro de M.
Format: Article
Language:English
Subjects:
Affine differential geometry
Geometric analysis
Lagrangian submanifolds
Lagrangian/Legendrian singularities
parabolic affine spheres
Singularity theory
solutions of Monge-Ampère equation
special Kähler manifolds
symmetric singularities
Symplectic geometry
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Summary:Given a Lagrangian submanifold L of the affine symplectic 2n-space, one can canonically and uniquely define a center-chord and a special improper affine sphere of dimension 2n, both of whose sets of singularities contain L. Although these improper affine spheres (IAS) always present other singularities away from L (the off-shell singularities studied in [8]), they may also present singularities other than L which are arbitrarily close to L, the so called singularities “on shell”. These on-shell singularities possess a hidden Z2 symmetry that is absent from the off-shell singularities. In this paper, we study these canonical IAS obtained from L and their on-shell singularities, in arbitrary even dimensions, and classify all stable Lagrangian/Legendrian singularities on shell that may occur for these IAS when L is a curve or a Lagrangian surface.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2020.107326