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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

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1435-5345
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Volume 2019, Issue 751

# Complex manifolds with maximal torus actions

Hiroaki Ishida
Published Online: 2016-07-22 | DOI: https://doi.org/10.1515/crelle-2016-0023

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

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Revised: 2016-04-04

Published Online: 2016-07-22

Published in Print: 2019-06-01

The author was supported by JSPS Research Fellowships for Young Scientists.

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2019, Issue 751, Pages 121–184, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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