Hypercohomology of Milnor fibres
Hypercohomology of Milnor fibres
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Milnor 纤维的超上同调
DOI：
10.1016/0040-9383(95)00054-2
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
D. Massey
中科院分区：
文献类型：
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作者：
D. Massey
在本文中，我们证明了一些有助于描述有界的、可构造的束状复合体中带系数的一般Milnor纤维的超上同调的结果。所有这些结果的主要目的是提供一种代数方法来计算一些典型的与复解析奇点相关的数据：函数的米尔诺纤维的上同调群，空间在一点上的复链路的上同调，以及复束的特征循环。其中一些结果以较弱的形式出现在我们之前的论文b[35]中。本文的结果分为三大类:如果X是一个分析空间,F有界成就,可构成的复杂的捆在X,和F: X + @与分层是一个复杂的解析函数孤立的临界点在X,然后我们将会显示在第三节相对hypercohomology系数F ',一个小球的X模米尔诺尔纤维在断裂的直和权力的hypercohomology正常数据的每一个阶层。当系数为常时，这与[23]中的L&、[40]中的Tibgr以及当X本身在原点也有孤立奇点时，[38]中的Siersma的结果密切相关；然而，这三篇论文没有提供确定每个地层对直接总和贡献多少次的方法。我们在这个问题上的主要结果是在这个直接和中出现的指数的显式计算（定理3.2）。这种计算的一个含义是，它允许人们指定代数数据，这些数据暗示了广义孤立奇点族中消失环束的茎上同调的常数（见第6节关于这个和相关问题的讨论）。其次，在第4节中，我们研究了函数f: X+@具有任意维的分层临界轨迹的一般情况；在这里，描述米尔诺纤维的超上同性变得更加困难。我们通过一个充分泛型的超曲面片的模化，研究了这样一个f的Milnor纤维。我们证明，如果g: X+@是满足超曲面V (f)的第二个函数，那么Milnor纤维的相对超上同调离模flvcB的Milnor纤维，分解为正规数据对X的每个层的超上同调的幂的直接和（定理4.2），这让人想起孤立临界点的情况。而且，我们可以再一次明确地描述在这个直接和中出现的指数。作为我们计算的一个推论，在例5.3中，我们恢复并推广了LE和Greuel的公式（参见[13,17,211]和[28,5.11]）。a])为孤立完全交点的Milnor数。
IN THIS paper, we prove a number of results which help describe the hypercohomology of general Milnor fibres with coefficients in bounded, constructible complexes of sheaves. The principal goal of all of these results is to provide a means of algebraically calculating some pieces of data typically associated with a complex analytic singularity: the cohomology groups of the Milnor fibre of a function, the cohomology of the complex link of a space at a point, and the characteristic cycle of a complex of sheaves. Some of these results have appeared in substantially weaker forms in our earlier paper [35]. The results of this paper fall into three main categories: If X is an analytic space, F’is a bounded, constructible complex of sheaves on X, and f: X+@ is a complex analytic function with a stratified isolated critical point at x EX, then we will show in Section 3 that the relative hypercohomology, with coefficients in F’, of a small ball around x modulo the Milnor fibre off at x breaks up as a direct sum of powers of the hypercohomology of the normal data to each of the strata. With constant coefficients, this is closely related to a result of L& in [23], Tibgr in [40], and, when X itself also has an isolated singular point at the origin, of Siersma in [38]; however, these three papers provide no method for determining how many times each stratum contributes to the direct sum. Our main result on this problem is the explicit calculation of the exponents that occur in this direct sum (Theorem 3.2). One implication of this calculation is that it allows one to specify algebraic data which implies the constancy of the stalk cohomology of the sheaf of vanishing cycles in a family of generalized isolated singularities (see Section 6 for a discussion of this and related problems).Secondly, in Section 4, we examine the general case where the function f: X+@ has a stratified critical locus of arbitrary dimension; here, the description of the hypercohomology of the Milnor fibre becomes more difficult. We study the Milnor fibre of such an f by modding-out by a sufficiently generic hypersurface slice. We prove that, if g: X+@ is a second function which meets the hypersurface V (f)“nicely”, then the relative hypercohomology of the Milnor fibre off modulo the Milnor fibre of flvcB, breaks up as a direct sum of powers of the hypercohomology of the normal data to each of the strata of X (Theorem 4.2 tthis is reminiscent of the isolated critical point case. Moreover, we can once again explicitly describe the exponents that occur in this direct sum. As a corollary of our calculations, in Example 5.3, we recover and generalize the formula of LE and Greuel (see [13, 17, 211 and [28, 5.11. a]) for the Milnor number of isolated complete intersections.