In this paper we give an exposition of several recent results on local symmetries of real submanifolds in complex space, featuring new examples and important corollaries. Departing from Levi non-degenerate hypersurfaces, treated in the classical Chern–Moser theory, we explore three important classes of manifolds, which naturally extend the classical case. We start with quadratic models for real submanifolds of higher codimension and review some recent striking results, which demonstrate that such higher codimension models may possess symmetries of arbitrarily high jet degree. This disproves the long held belief that the fundamental 2-jet determination results from Chern–Moser theory extend to this case. As a second case, we consider hypersurfaces with singular Levi form at a point, which are of finite multitype. This leads to the study of holomorphically nondegenerate polynomial models. We outline several results on their symmetry algebras including a characterization of models admitting nonlinear symmetries. In the third part we consider the class of structures with everywhere singular Levi forms that has received the most attention recently, namely everywhere 2-nondegenerate structures. We present a computation of their Catlin multitype and results on symmetry algebras of their weighted homogeneous (w.r.t. multitype) models.

在本文中，我们阐述了关于复杂空间中实子流形局部对称性的几个最新结果，其中包括新的例子和重要的推论。与经典 Chern-Moser 理论中处理的 Levi 非简并超曲面不同，我们探索了三类重要的流形，它们自然地扩展了经典情况。我们从更高余维的真实子流形的二次模型开始，并回顾了最近的一些引人注目的结果，这些结果表明这种更高的余维模型可能具有任意高射流度的对称性。这反驳了长期以来的信念，即陈-莫泽理论的基本 2 射流测定结果适用于这种情况。作为第二种情况，我们考虑在一点处具有奇异 Levi 形式的超曲面，它们是有限多类型的。这引发了全纯非简并多项式模型的研究。我们概述了它们的对称代数的几个结果，包括承认非线性对称性的模型的表征。在第三部分中，我们考虑最近最受关注的处处奇异李维形式的结构类，即处处2-非简并结构。我们提出了他们的卡特林多类型的计算以及他们的加权齐次（w.r.t.多类型）模型的对称代数结果。