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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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