复环面簇的链接束与相交空间
Link Bundles and Intersection Spaces of Complex Toric Varieties
AI总结：
本文计算了复三维环面簇相交空间的有理同调，发现其与相交同调不同，不具有组合不变性，从而保留了关于定义扇形的更精细信息。
摘要：
存在几种满足广义 Poincaré 对偶的奇异空间同调理论，包括 Goresky-MacPherson 的相交同调、Cheeger 的 $L^2$ 上同调以及相交空间的同调。环面簇的相交同调和 $L^2$ 上同调是已知的。在此，我们计算复三维环面簇相交空间的有理同调，并将其与相交同调进行比较。为此，我们分析了这些簇的胞腔结构和拓扑分层，并确定了其奇点链接上的相容结构。特别地，我们计算了三维环面簇中链接束的同调。我们发现使用有理同调分层的概念十分方便。结果表明，环面簇的相交空间同调与其相交同调相反，不具有组合不变性，从而保留了关于定义扇形的更精细信息。
英文摘要：
There exist several homology theories for singular spaces that satisfy generalized Poincaré duality, including Goresky-MacPherson's intersection homology, Cheeger's $L^2$ cohomology and the homology of intersection spaces. The intersection homology and $L^2$ cohomology of toric varieties is known. Here, we compute the rational homology of intersection spaces of complex 3-dimensional toric varieties and compare it to intersection homology. To achieve this, we analyze cell structures and topological stratifications of these varieties and determine compatible structures on their singularity links. In particular, we compute the homology of links in 3-dimensional toric varieties. We find it convenient to use the concept of a rational homology stratification. It turns out that the intersection space homology of a toric variety, contrary to its intersection homology, is not combinatorially invariant and thus retains more refined information on the defining fan.
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