Geometry of affine immersions is the study of hypersurfaces that are invariant under affine transformations. As with the hypersurface theory on the Euclidean space, an affine immersion can induce a torsion-free affine connection and a (pseudo)-Riemannian metric on the hypersurface. Moreover, an affine immersion can induce a statistical manifold, which plays a central role in information geometry. Recently, a statistical manifold with a complex structure is actively studied since it connects information geometry and Kähler geometry. However, a holomorphic complex affine immersion cannot induce such a statistical manifold with a Kähler structure. In this paper, we introduce complex affine distributions, which are non-integrable generalizations of complex affine immersions. We then present the fundamental theorem for a complex affine distribution, and show that a complex affine distribution can induce a statistical manifold with a Kähler structure.

仿射浸没的几何形状是仿射变换下不变的超曲面的研究。与欧几里德空间上的超曲面理论一样，仿射浸入可以在超曲面上引发无扭转仿射连接和（伪）-黎曼度量。此外，仿射沉浸可以引发统计流形，这在信息几何中起着核心作用。最近，由于它连接信息几何和Kähler几何，因此正在积极研究具有复杂结构的统计流形。但是，全纯复仿射浸没不能产生具有Kähler结构的统计流形。在本文中，我们介绍了复杂仿射分布，这是复杂仿射浸没的不可积分概括。然后，我们提出了复杂仿射分布的基本定理，