# Singular Improper Affine Spheres from a given Lagrangian Submanifold

## TL;DR

This paper explores the singularities of improper affine spheres derived from Lagrangian submanifolds, revealing a hidden symmetry and classifying stable singularities in various dimensions.

## Contribution

It introduces a canonical construction of improper affine spheres from Lagrangian submanifolds and classifies their stable on-shell singularities across dimensions.

## Key findings

- Identification of a hidden Z2 symmetry in on-shell singularities
- Classification of stable Lagrangian/Legendrian singularities for these spheres
- Analysis of singularities near Lagrangian curves and surfaces

## 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.