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 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.

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来自给定拉格朗日子流形的奇异仿射球

摘要 给定仿射辛 2n 空间的拉格朗日子流形 L，可以规范且唯一地定义中心弦和维数为 2n 的特殊不适当仿射球体，它们的奇点集都包含 L。尽管这些不适当仿射球体（IAS ) 总是呈现远离 L 的其他奇点（[8] 中研究的壳外奇点），它们也可能呈现与 L 任意接近的 L 以外的奇点，即所谓的“壳上”奇点。这些壳上奇点具有隐藏的 Z 2 对称性，这是壳外奇点所不具备的。在本文中，我们研究了这些从 L 获得的规范 IAS 及其壳上奇点，在任意偶数维度上，