Elsevier

Advances in Mathematics

Volume 374, 18 November 2020, 107326
Advances in Mathematics

Singular improper affine spheres from a given Lagrangian submanifold

https://doi.org/10.1016/j.aim.2020.107326Get rights and content
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Abstract

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.

MSC

53A15
53D12
58K40
58K70

Keywords

Affine differential geometry
parabolic affine spheres
Geometric analysis
solutions of Monge-Ampère equation
Symplectic geometry
special Kähler manifolds
Lagrangian submanifolds
Singularity theory
Lagrangian/Legendrian singularities
symmetric singularities

Cited by (0)

The first author thanks CNPq (grant 304314/2017-0) and the third author thanks FAPESP (grants 2015/02029-1, 2016/09249-0) for partial support during the preparation of this manuscript. Both authors also thank CAPES (finance code 001) for partial support. The research of the second author was partially supported by NCN (grant DEC-2013/11/B/ST1/03080).