A characterization of the intersection form of a Milnor's fiber for a function with an isolated critical point
A characterization of the intersection form of a Milnor's fiber for a function with an isolated critical point
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具有孤立临界点的函数的 Milnor 纤维相交形式的表征
DOI：
10.3792/pjaa.58.79
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发表时间：
1982
期刊：
影响因子：
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通讯作者：
Kyoji Saito
中科院分区：
文献类型：
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作者：
Kyoji Saito
1o引言和主要成果的陈述。设f：C+x，0C，0是在0eC+处具有孤立临界点的全纯函数芽。由于Milnor[2]，对于足够小的具有0E<；<；r<；<；1的r和e，f的限制f：{xe C“+‘ix]<；r}n{ifl=}>；{t e C：itl-}定义了一个纤维，它的一般纤维F是n球的花束，使得中间同调群H，(F，Z)不为零。利用Poincar6对偶H，(F，Z)_H“(F，F，Z)，得到一个交形<；，>；H，(F，Z)H，(F，Z)-Z，它是对称的或偏对称的，根据n是偶数或奇数。对于交集形式的计算，我们在[3]中使用了以下事实。定理1.H上的复值双线性形式B，(F，Z)(R)C是交形式的常数倍，如果B在H，(F，Z)上的全单向群作用下不变，但当f在0处为非退化(即普通双点)且n为奇数时除外。这里的全单调群按定义是f的泛展开的判别轨迹的补的基本群的像。由于这一行为似乎仍然不是普遍熟知的，我们在这里公布它和另一个证明[3]。在文献2中，我们给出了刻画不变双线性范数的一个比较抽象的引理。
1o Introduction and the statements of the main results. Let f: C+x, 0C, 0 be a germ of holomorphie unction at 0 e C+ with an isolated critical point. Due to Milnor [2], for r and e sufficiently small with 0e<<r << 1, the restriction f: {x e C"+’ ix]<r} n {Ifl=} >{t e C: Itl-} of f defines a fibration whose general fiber F is a bouquet of n-spheres so that the middle homology group H,(F, Z) is nonvanishing. Using Poincar6 duality H,(F, Z)_H"(F, F, Z), one gets an intersection form <, > H,(F, Z) H,(F, Z)-.Z, which is symmetric or skewsymmetric according as n is even or odd. For a computation of the intersection form, we used in [3] the following fact. Theorem 1. A complex valued bilinear form B on H,(F, Z)(R)C is a constant multiple of the intersection form if B is invariant under the total monodromy group action on H,(F,Z), except for the case when f at 0 is nondegenerate (i.e. ordinary double point) and n is odd. Here the total monodromy group is by definition the image of the fundamental group of the complement of the discriminant loci of a universal unfolding of f. Since this act seems still not generally well-known, we publish it here with a proof separately rom [3]. In 2 we give a somewhat abstract lemma characterizing invariant bilinear orms.