Characterization of homogeneous torus manifolds

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 $\prod \mathbb{C}P(l)$.