On the topology of complex algebraic maps
On the topology of complex algebraic maps
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论复代数映射的拓扑
DOI：
10.1007/bfb0071279
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发表时间：
1982
期刊：
影响因子：
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通讯作者：
R. Macpherson
中科院分区：
文献类型：
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作者：
M. Goresky;R. Macpherson
在这主要是暂时的说明，我们给一些同调性质的代数映射的复代数簇，而令人惊讶的拓扑观点。其中包括将曲线映射的不变循环定理推广到更高维度。这些性质都是德利涅、加伯、贝林森和伯恩斯坦最近提出的一个深定理的推论，该定理在§2中陈述。这个定理涉及到交同调和导出范畴。我们在这里的目的之一是通过给出只涉及普通同调的推论来推广它。出于这个原因，有些读者可能希望从第3节开始开始。§]。交叉同源性。对于任何复代数簇V，设Db（v）是V上Q-模的范畴层的代数上c可构造的有界导出范畴。（Db（v）的对象是V上q-模的层的有界复，c在V的某些复代数子流形的分层上是上同调局部常数;参见（[GM 2]，§l.]）l）。如果u ′ CD ~（V）和U c V，Hk（u，~“）（分别为H~（U，U '）表示超上同调（分别为H ~（U，U'））。紧支集的超上同调）的限制。如果p E V，设T °是距离p小于P E的点的“开圆盘”，其中距离是使用P在~N中的邻域的某些局部解析嵌入的通常的欧几里得距离。对于= S 'C Db（V）c，Hk（~，S“）=和H~（~，S”）与c的选择无关。空间U上的局部系是U上Q-模的局部常数层。
In this largely expository note we give some homological properties of algebraic maps of complex algebraic varieties which are rather surprising from the topological point of view. These include a generalisation to higher dimension of the invariant cycle theorem for maps to curves. These properties are all corollaries of a recent deep theorem of Deligne, Gabber, Beilinson, and Bernstein which is stated in §2. This theorem involves intersection homology and the derived category. One of our objects here is to popularize it by giving corollaries involving only ordinary homology. For this reason some readers may wish to begin with §3. §]. Intersection homology. For any complex algebraic variety V , let Db(v) be the algebraically c constructible bounded derived category of the category sheaves of Q-module on V. (Objects of Db(v) are bounded complexes of sheaves of q-modules on V that c are cohomologically locally constant on the strata for some stratification of V by complex algebraic submanifolds; see ([GM2], §l.]l). If ~'C D~(V) and U c V , Hk(u,~ ")(resp. H~(U,~')) denotes the hyper-cohomology (resp. hypercohomology with compact supports) of the restriction of S" = to U. If p E V , let T ° be the "open disk" of points at distance less than P E from p , where distance is the usual Euclidean distance using some local analytic embedding of a neighborhood of P in ~N. For =S'C Db(V)c , Hk(~,S ")= and H~(~,~') are independent of the choices for small enough c • A local system on a space U is a locally constant sheaf of Q-module on U .