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 of nonsingular complete fan Δ in , complex vector subspace of 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 -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.
 F. Bosio, Variétés complexes compactes: une généralisation de la construction de Meersseman et López de Medrano–Verjovsky, Ann. Inst. Fourier (Grenoble) 51 (2001), no. 5, 1259–1297. 10.5802/aif.1855Search in Google Scholar
 V. M. Buchstaber and T. E. Panov, Torus actions and their applications in topology and combinatorics, Univ. Lecture Ser. 24, American Mathematical Society, Providence 2002. 10.1090/ulect/024Search in Google Scholar
 G. Hochschild, The structure of Lie groups, Holden-Day, San Francisco 1965. Search in Google Scholar
 H. Ishida and Y. Karshon, Completely integrable torus actions on complex manifolds with fixed points, Math. Res. Lett. 19 (2012), no. 6, 1283–1295. 10.4310/MRL.2012.v19.n6.a9Search in Google Scholar
 S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts Pure Appl. Math. 15, John Wiley & Sons, New York 1969. Search in Google Scholar
 S. López de Medrano and A. Verjovsky, A new family of complex, compact, non-symplectic manifolds, Bol. Soc. Brasil. Mat. (N.S.) 28 (1997), no. 2, 253–269. 10.1007/BF01233394Search in Google Scholar
 N. T. Zung and N. V. Minh, Geometry of nondegenerate -actions on n-manifolds, J. Math. Soc. Japan 66 (2014), no. 3, 839–894. Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston