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Critical values and the determinant of periods. (English. Russian original) Zbl 0716.32024

Russ. Math. Surv. 44, No. 4, 209-210 (1989); translation from Usp. Mat. Nauk 44, No. 4(268), 235-236 (1989).
Consider the Pham polynomial \(x_ 1^{k_ 1}+...+x_ n^{k_ n}\) and its deformation \(f(x)=x_ 1^{k_ 1}+...x_ n^{k_ n}+\sum_{| m| <1}\lambda_ mx_ 1^{m_ 1}...x_ n^{m_ n},\) where \(| (m_ 1,...,m_ n)| =m_ 1/k_ 1+...+m_ n/k_ n\) and \(\lambda_ m\in {\mathbb{C}}\). \(X=\{x\in {\mathbb{C}}^ n| f(x)=0\}\) has the homotopy type of a bouquet of \(\mu\) (n-1)-dimensional spheres, where \(\mu =(k_ 1-1)...(k_ n-1).\) Consider the canonical base \(\delta_ 1,...,\delta_{\mu}\in H_{n-1}(X,{\mathbb{Z}})\) and the base for the cohomology given by the forms \(\omega_ I/df\), where \(\omega_ I=x_ 1^{i_ 1}...x_ n^{i_ n}dx_ 1\wedge...\wedge dx_ n\), \(0\leq i_ j\leq k_ j-2.\)
The main result is a formula expressing det(\(\int_{\delta_ j}\omega_ I/df) \) as (n/2-1)-power of the product of the critical values of f, modulo the multiplication with a well determined constant. One asks for a similar formula when starting with an arbitrary quasi- homogeneous polynomial.
Other related results are also given, as part of a general principle; express the determinant of the matrix of periods in terms of the critical values of the equations defining the variety.
Reviewer: C.Bănică

MSC:

32S20 Global theory of complex singularities; cohomological properties
32G20 Period matrices, variation of Hodge structure; degenerations
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
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