We study geometric structures on the complement of a toric mirror arrangement associated with a root system. Inspired by those root system hypergeometric functions found by Heckman–Opdam, and in view of the work of Couwenberg–Heckman–Looijenga on the geometric structures on projective arrangement complements, we consider a family of connections on a total space, namely, a \({\mathbb {C}}^{\times }\)-bundle on the complement of a toric mirror arrangement (=finite union of hypertori, determined by a root system). We prove that these connections are torsion free and flat, and hence define a family of affine structures on the total space, which is equivalent to a family of projective structures on the toric arrangement complement. We then determine a parameter region for which the projective structure admits a locally complex hyperbolic metric. In the end, we show that the space in question can be biholomorphically mapped onto a divisor complement of a ball quotient if the Schwarz conditions are invoked.

我们研究了与根系相关的复曲面镜排列的补充上的几何结构。受 Heckman-Opdam 发现的那些根系超几何函数的启发，并鉴于 Couwenberg-Heckman-Looijenga 在射影排列补集的几何结构上的工作，我们考虑了一个总空间上的连接族，即\( {\mathbb {C}}^{\times }\)-捆绑在复曲面镜排列的补充上（=hypertori 的有限联合，由根系统决定）。我们证明这些连接是无扭转和平面的，因此在整个空间上定义了一个仿射结构族，这相当于复曲面排列补集上的一个射影结构族。然后，我们确定一个参数区域，其中射影结构允许局部复双曲度量。最后，我们表明，如果调用 Schwarz 条件，则所讨论的空间可以双全纯映射到球商的除数补上。