Geometry of compact complex manifolds with maximal torus action

TL;DR

This paper explores the geometry of compact complex manifolds with maximal torus actions, providing new constructions and analyzing their foliations and transverse-Kähler structures to understand their analytic subsets.

Contribution

It introduces two equivalent constructions of such manifolds from fans and complex subgroups, and studies their holomorphic foliations and transverse-Kähler forms.

Findings

Constructed examples from fans and subgroups

Defined canonical holomorphic foliations

Proved results on analytic subsets

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 $H\subset T_\mathbb C=(\mathbb C^*)^k$. On every manifold $M$ we define the canonical holomorphic foliation $\mathcal F$ and under additional restrictions construct transverse-K\"{a}hler form $\omega_\mathcal F$. As an application of these constructions, we prove some results on geometry of manifolds~$M$ regarding its analytic subsets.