Complex manifolds with maximal torus actions

Abstract

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 $(\Delta ,\mathfrak{h},G)$ of nonsingular complete fan $\Delta$ in $\mathfrak{g}$, complex vector subspace $\mathfrak{h}$ of ${\mathfrak{g}}^{\mathbb{C}}$ 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 ${\mathbb{C}}^n$-actions on compact connected complex manifolds of complex dimension $n$, and construction of concrete examples of non-Kähler manifolds.

62 pages. The definition of ${\mathcal{C}}_1$ has been modified. Typos have been fixed. Some notations have been modified

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• Geometric structures on moment-angle manifolds
• Torus orbifolds, slice-maximal torus actions and rational ellipticity
• Complex geometry of moment-angle manifolds
• On the cohomology of quotients of moment-angle complexes