Geometry of compact complex manifolds with maximal torus action

Abstract

In this present paper we study geometry of compact complex manifolds equipped with a \emph{maximal} torus $T=(S^1{)}^k$ action. We give two equivalent constructions providing examples of such manifolds given a simplicial fan $\Sigma$ and a compelx subgroup $H\subset T_{\mathbb{C}}=({\mathbb{C}}^{\ast }{)}^k$. On every manifold $M$ we define the canonical holomorphic foliation $\mathcal{F}$ and under additional restrictions construct transverse-Kähler form ${\omega }_{\mathcal{F}}$. As an application of these constructions, we prove some results on geometry of manifolds~$M$ regarding its analytic subsets.