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Complex manifolds with maximal torus actions

  • Hiroaki Ishida ORCID logo EMAIL logo


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,𝔥,G) of nonsingular complete fan Δ in 𝔤, complex vector subspace 𝔥 of 𝔤 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 n-actions on compact connected complex manifolds of complex dimension n, and construction of concrete examples of non-Kähler manifolds.

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


The author would like to thank Yael Karshon for fruitful and stimulating discussions. He also would like to thank Laurent Battisti, Mikiya Masuda and Taras Panov for valuable comments.


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Received: 2015-04-30
Revised: 2016-04-04
Published Online: 2016-07-22
Published in Print: 2019-06-01

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