Singular Improper Affine Spheres from a given Lagrangian Submanifold

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 our previous paper), 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 ${\mathbb{Z}}_2$ 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.

30 pages, 3 figures

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Cited by in corpus (2)

• Singularities of 3-parameter line congruences in ${\mathbb{R}}^4$
• Asymptotic localization of symbol correspondences for spin systems and sequential quantizations of $S^2$