Quasitoric manifolds over a product of simplices
Quasitoric manifolds over a product of simplices
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DOI：
10.18910/7012
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发表时间：
2008-03
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影响因子：
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通讯作者：
Suyoung Choi;M. Masuda;D. Suh
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文献类型：
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作者：
Suyoung Choi;M. Masuda;D. Suh
一个拟素流形（quasitoric manifold）一个小的覆盖）是一个2n维的（相应地，一个$n$维光滑闭流形，其有效局部标准作用为$（S^1）^n$（分别$（\mathbb Z_2）^n$），其轨道空间组合为$n$维单凸多胞形P$。在本文中，我们研究当$P$是一个产品的单形。环$\F$上的广义Bott塔，其中$\F=\C$或$\R$，是从一点开始的$\F$-线丛的Whitney和的投射丛序列。在$\F$上的塔的每一级，我们称之为广义Bott流形，提供了一个在单形乘积上的拟流形（当$\F=\C$）和小覆盖（当$\F=\R$）的例子。事实证明，单形乘积上的每个小覆盖都等价于（在Davis和Januszkiewicz \cite{DJ}的意义上）广义Bott流形。但这是不是情况下的拟流形，我们表明，一个拟流形上的产品的单形是等价的广义Bott流形的当且仅当它允许一个几乎复杂的结构下的行动保持不变。最后，我们证明了单形乘积上的拟流形M同胚于广义Bott流形，如果M与系数为\Q的复射影空间的乘积具有相同的上同调环.
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 $\F$, where $\F=\C$ or $\R$, is a sequence of projective bundles of the Whitney sum of $\F$-line bundles starting with a point. Each stage of the tower over $\F$, which we call a generalized Bott manifold, provides an example of quasitoric manifolds (when $\F=\C$) and small covers (when $\F=\R$) 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 \cite{DJ}) 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 $\Q$ coefficients.