环面流形直分解的唯一性
Uniqueness of the direct decomposition of toric manifolds
摘要：
本文研究复曲面流形的直分解的唯一性。我们首先观察到，直接分解的环面流形\emph{algebraic varieties}是唯一的顺序的因素。一个代数不可分解的环面流形恰好可以分解为光滑流形，而两个环面流形是同构的又没有一个判据，所以环面流形的唯一分解问题如\emph{smooth manifolds}是高度非平凡的，而且迄今为止对这个问题还一无所知.我们证明了当分解中每个因子的复维数小于或等于2时，这个问题是肯定的。一个类似的论证表明，一个光滑流形直接分解为$\mathbb{C}P^1$的副本和具有$(S^1)^2$光滑作用的单连通闭光滑4-流形是唯一的，直到因子的阶。
英文摘要：
In this paper, we study the uniqueness of the direct decomposition of a toric manifold. We first observe that the direct decomposition of a toric manifold as \emph{algebraic varieties} is unique up to order of the factors. An algebraically indecomposable toric manifold happens to decompose as smooth manifold and no criterion is known for two toric manifolds to be diffeomorphic, so the unique decomposition problem for toric manifolds as \emph{smooth manifolds} is highly nontrivial and nothing seems known for the problem so far. We prove that this problem is affirmative if the complex dimension of each factor in the decomposition is less than or equal to two. A similar argument shows that the direct decomposition of a smooth manifold into copies of $\mathbb{C}P^1$ and simply connected closed smooth 4-manifolds with smooth actions of $(S^1)^2$ is unique up to order of the factors.
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