Even Dimensional Improper Affine Spheres

TL;DR

This paper generalizes the classification of two-dimensional improper affine spheres to higher even dimensions, linking them to Lagrangian submanifolds and special Kähler manifolds, and explores their differential systems and singularities.

Contribution

It extends the known models of improper affine spheres to arbitrary even dimensions, connecting geometric constructions with advanced differential geometry concepts.

Findings

Both types of 2D improper affine spheres are generalized to higher dimensions.

The higher-dimensional spheres relate to Lagrangian submanifolds and special Kähler manifolds.

Solutions to certain exterior differential systems are identified as improper affine spheres.

Abstract

There are exactly two different types of bi-dimensional improper affine spheres: the non-convex ones can be modeled by the center-chord transform of a pair of planar curves while the convex ones can be modeled by a holomorphic map. In this paper, we show that both constructions can be generalized to arbitrary even dimensions: the former class corresponds to the center-chord transform of a pair of Lagrangian submanifolds while the latter is related to special K\"ahler manifolds. Furthermore, we show that the improper affine spheres obtained in this way are solutions of certain exterior differential systems. Finally, we also discuss the problem of realization of simple stable Legendrian singularities as singularities of these improper affine spheres.