2013/02/04 by Hiroaki Ishida, Ishida, Hiroaki
Mathematics · #57S25 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Primary 14M25 #Secondary 32M05 #math.AG #math.CV #math.DG #msc:14M25 #msc:32M05 #msc:57S25
paper · pdf · doi:10.48550/arxiv.1302.0633
62 pages. The definition of $\mathcal{C}_1$ has been modified. Typos have been fixed. Some notations have been modified
openalex publication_date 2013/02/04 · arxiv created 2015/04/30 · arxiv updated 2015/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we introduce the notion of maximal actions of compact tori on smooth manifolds and study compact connected complex manifolds equipped with maximal actions of compact tori. We give a complete classification of such manifolds, in terms of combinatorial objects, which are triples (Δ, \mathfrakh, G) of nonsingular complete fan Δ in \mathfrakg, complex vector subspace \mathfrakh of \mathfrakgℂ and compact torus G satisfying certain conditions. We also give an equivalence of categories with suitable definitions of morphisms in these families, like toric geometry. We obtain several results as applications of our equivalence of categories; complex structures on moment-angle manifolds, classification of holomorphic nondegenerate ℂn-actions on compact connected complex manifolds of complex dimension n, and construction of concrete examples of non-Kähler manifolds.