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

TL;DR

This paper introduces a unifying framework using exponential actions defined by vector configurations, linking various complex geometric and topological constructions through Gale duality and moment-angle manifolds.

Contribution

It establishes a universal approach to describe diverse geometric structures via Gale duality and exponential actions, connecting multiple areas in complex geometry and topology.

Findings

Unified description of holomorphic foliations and toric varieties

Characterization of moment-angle manifolds using Gale duality

Framework applicable to non-Kaehler complex geometry

Abstract

Exponential actions defined by vector configurations provide a universal framework for several constructions of holomorphic dynamics, non-Kaehler 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. In all of these cases, the geometry and topology of the appropriate quotient object can be described by combinatorial data including a pair of Gale dual vector configurations.