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On Steenbrink’s conjecture. (English) Zbl 0712.14002
Let f be a holomorphic function on a complex manifold X. By Steenbrink, we have the spectrum Sp(f,x) of f at $$x\in f^{-1}(0)$$ using the monodromy and the mixed Hodge structure on the cohomology of Milnor fiber. It is well studied in the isolated singularity case. Assume $$\dim (Sing(f))=1$$. Let g be a function defined on a neighborhood of x such that $$g(x)=0$$, dg$$\neq 0$$ and the restriction of f to $$g^{-1}(0)$$ has isolated singularity at x. Let $$f_ s$$ be the restriction of f to $$g^{-1}(s)$$. Then there is a $$\mu$$-constant deformation of isolated singularities for $$s\neq 0$$ sufficiently small. For r large enough, $$f+g^ r$$ has isolated singularity at x, and we can express $$Sp(f+g^ r,x)-Sp(f,x)$$ in terms of the spectrum of $$f_ s$$ and the monodromy along Sing(f). So we can calculate the spectrum of f if we know the spectrum of isolated singularity and the monodromy along Sing(f). The formula was conjectured, and proved in some cases, by Steenbrink.
Reviewer: M.Saito

##### MSC:
 14B07 Deformations of singularities 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 32G20 Period matrices, variation of Hodge structure; degenerations
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