On Steenbrink's conjecture
説明
Let f : X ~ C be a holomorphic function on a complex manifold. Using the monodromy and mixed Hodge structure on the cohomology of Milnor fiber, we can associate the spectrum Sp(f, x) of f at x e Xo = f 1(0). It is a fractional Laurent polynomial ~ n,t ~ with n~ ~ Z. This notion is first introduced by Steenbrink [15] cteQ in the isolated singularity case (cf. also [19]) and [16] in general. If f has isolated singularity, n~ are nonnegative and {0~: n ~ 0 } (counted with multiplicity) are called the exponents (or spectra [19]) of f. Assume f has one-dimensional singular locus, and consider a deformation of the form f + g', where g is a linear form [i.e. a function such that g(x) = 0 and dg =~ 0 at x]. Then f + g" has isolated singularity if g is generic. Let Zk be a local irreducible component of Singf at x, and mk its multiplicity. Along Z* = Zk\{X}, f is viewed as #-constant deformation of isolated singularity. Let ~k.j be its exponents (which are constant on Z~' [20]). We can also define ilk. j e (0, 1] in a compatible way with ~k, j SO that exp(--2ni[3k, j) are the eigenvalues of the monodromy along Z*. Then Steenbrink conjectured
収録刊行物
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- Mathematische Annalen
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Mathematische Annalen 289 703-716, 1991-03-01
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