Exponential actions defined by vector configurations, Gale duality, and moment-angle manifolds

Exponential actions defined by vector configurations provide a universal framework for several constructions of holomorphic dynamics, non-Kähler complex geometry, toric geometry and topology. These include leaf spaces of holomorphic foliations, intersections of real and Hermitian quadrics, the quotient construction of simplicial toric varieties, LVM and LVMB manifolds, complex-analytic structures on moment-angle manifolds and their partial quotients, reviewed in this survey. In all cases, the geometry and topology of the appropriate quotient object can be described by combinatorial data including a pair of Gale dual vector configurations.