Osaka Journal of Mathematics

Uniqueness of the direct decomposition of toric manifolds

Miho Hatanaka

Full-text: Open access


In this paper, we study the uniqueness of the direct decomposition of a toric manifold. We first observe that the direct decomposition of a toric manifold as algebraic varieties is unique up to order of the factors. An algebraically indecomposable toric manifold happens to decompose as smooth manifold and no criterion is known for two toric manifolds to be diffeomorphic, so the unique decomposition problem for toric manifolds as smooth manifolds is highly nontrivial and nothing seems known for the problem so far. We prove that this problem is affirmative if the complex dimension of each factor in the decomposition is less than or equal to two. A similar argument shows that the direct decomposition of a smooth manifold into copies of $\mathbb{C}P^{1}$ and simply connected closed smooth 4-manifolds with smooth actions of $(S^{1})^{2}$ is unique up to order of the factors.

Article information

Osaka J. Math., Volume 52, Number 2 (2015), 439-453.

First available in Project Euclid: 24 March 2015

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55N40: Axioms for homology theory and uniqueness theorems
Secondary: 14M25: Toric varieties, Newton polyhedra [See also 52B20]


Hatanaka, Miho. Uniqueness of the direct decomposition of toric manifolds. Osaka J. Math. 52 (2015), no. 2, 439--453.

Export citation


  • L.S. Charlap: Compact flat riemannian manifolds, I, Ann. of Math. (2) 81 (1965), 15–30.
  • S. Choi, M. Masuda and S. Oum: Classification of real Bott manifolds and acyclic digraphs, arXiv:1006.4658.
  • S. Choi, M. Masuda and D.Y. Suh: Rigidity problems in toric topology: a survey, Proc. Steklov Inst. Math. 275 (2011), 177–190.
  • W. Fulton: Introduction to Toric Varieties, Annals of Mathematics Studies 131, Princeton Univ. Press, Princeton, NJ, 1993.
  • H. Ishida, Y. Fukukawa and M. Masuda: Topological toric manifolds, Mosc. Math. J. 13 (2013), 57–98, arXiv:1012.1786.
  • M. Masuda: Toric topology, Sūgaku 62 (2010), 386–411 (Japanese), English translation will appear in Sugaku Expositions, arxiv:1203.4399.
  • M. Masuda and D.Y. Suh: Classification problems of toric manifolds via topology; in Toric Topology, Contemp. Math. 460, Amer. Math. Soc., Providence, RI, 2008, 273–286.
  • T. Oda: Convex Bodies and Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 15, Springer, Berlin, 1988.
  • P. Orlik and F. Raymond: Actions of the torus on $4$-manifolds, I, Trans. Amer. Math. Soc. 152 (1970), 531–559. \endthebibliography*