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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