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March 2010 Characterization of homogeneous torus manifolds
Shintarô Kuroki
Osaka J. Math. 47(1): 285-299 (March 2010).

Abstract

This is the first of a series of papers which will be devoted to the study of the extended $G$-actions on torus manifolds $(M^{2n}, T^{n})$, where $G$ is a compact, connected Lie group whose maximal torus is $T^{n}$. The goal of this paper is to characterize codimension $0$ extended $G$-actions up to essential isomorphism. For technical reasons, we do not assume that torus manifolds are omnioriented. The main result of this paper is as follows: a homogeneous torus manifold $M^{2n}$ is (weak equivariantly) diffeomorphic to a product of complex projective spaces $\prod\mathbb{C}P(l)$ and quotient spaces of a product of spheres $\bigl(\prod S^{2m}\bigr)/\mathcal{A}$ with standard torus actions, where $\mathcal{A}$ is a subgroup of $\prod \mathbb{Z}_{2}$ generated by the antipodal involutions on $S^{2m}$. In particular, if the homogeneous torus manifold $M^{2n}$ is a compact (non-singular) toric variety or a quasitoric manifold, then $M^{2n}$ is just a product of complex projective spaces $\prod \mathbb{C}P(l)$.

Citation

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Shintarô Kuroki. "Characterization of homogeneous torus manifolds." Osaka J. Math. 47 (1) 285 - 299, March 2010.

Information

Published: March 2010
First available in Project Euclid: 19 February 2010

zbMATH: 1238.57033
MathSciNet: MR2666135

Subjects:
Primary: 57S25
Secondary: 22F30

Rights: Copyright © 2010 Osaka University and Osaka City University, Departments of Mathematics

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Vol.47 • No. 1 • March 2010
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