Osaka Journal of Mathematics

Quasitoric manifolds over a product of simplices

Suyoung Choi, Mikiya Masuda, and Dong Youp Suh

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A quasitoric manifold (resp. a small cover) is a $2n$-dimensional (resp. an $n$-dimensional) smooth closed manifold with an effective locally standard action of $(S^{1})^{n}$ (resp. $(\mathbb{Z}_{2})^{n}$) whose orbit space is combinatorially an $n$-dimensional simple convex polytope $P$. In this paper we study them when $P$ is a product of simplices. A generalized Bott tower over $\mathbb{F}$, where $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$, is a sequence of projective bundles of the Whitney sum of $\mathbb{F}$-line bundles starting with a point. Each stage of the tower over $\mathbb{F}$, which we call a generalized Bott manifold, provides an example of quasitoric manifolds (when $\mathbb{F}=\mathbb{C}$) and small covers (when $\mathbb{F}=\mathbb{R}$) over a product of simplices. It turns out that every small cover over a product of simplices is equivalent (in the sense of Davis and Januszkiewicz [5]) to a generalized Bott manifold. But this is not the case for quasitoric manifolds and we show that a quasitoric manifold over a product of simplices is equivalent to a generalized Bott manifold if and only if it admits an almost complex structure left invariant under the action. Finally, we show that a quasitoric manifold $M$ over a product of simplices is homeomorphic to a generalized Bott manifold if $M$ has the same cohomology ring as a product of complex projective spaces with $\mathbb{Q}$ coefficients.

Article information

Osaka J. Math., Volume 47, Number 1 (2010), 109-129.

First available in Project Euclid: 19 February 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57S15: Compact Lie groups of differentiable transformations 14M25: Toric varieties, Newton polyhedra [See also 52B20]
Secondary: 57S25: Groups acting on specific manifolds


Choi, Suyoung; Masuda, Mikiya; Suh, Dong Youp. Quasitoric manifolds over a product of simplices. Osaka J. Math. 47 (2010), no. 1, 109--129.

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