Translator Disclaimer
March 2010 Quasitoric manifolds over a product of simplices
Suyoung Choi, Mikiya Masuda, Dong Youp Suh
Osaka J. Math. 47(1): 109-129 (March 2010).

Abstract

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.

Citation

Download Citation

Suyoung Choi. Mikiya Masuda. Dong Youp Suh. "Quasitoric manifolds over a product of simplices." Osaka J. Math. 47 (1) 109 - 129, March 2010.

Information

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

zbMATH: 1237.57036
MathSciNet: MR2666127

Subjects:
Primary: 14M25, 57S15
Secondary: 57S25

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

JOURNAL ARTICLE
21 PAGES


SHARE
Vol.47 • No. 1 • March 2010
Back to Top