Semistable reduction of a normal crossing $$\mathbb {Q}$$ Q -divisor

Abstract

In a previous work we have introduced the notion of embedded Q-resolution, which essentially consists in allowing the final ambient space to contain abelian quotient singularities, and A'Campo's formula was calculated in this setting. Here we study the semistable reduction associated with an embedded Q-resolution so as to compute the mixed Hodge structure on the cohomology of the Milnor fiber in the isolated case using a generalization of Steenbrink's spectral sequence. Examples of Yomdin-Lê surface singularities are presented as an application.Let us sketch some definitions and properties about V -manifolds, weighted projective spaces, and weighted blow-ups, see [4,15] for a more detailed exposition.Definition 1.1. Let H = {f = 0} ⊂ C n+1 . An embedded Q-resolution of (H, 0) ⊂ (C n+1 , 0) is a proper analytic map π : X → (C n+1 , 0) such that:1. X is a V -manifold with abelian quotient singularities.2. π is an isomorphism over X \ π −1 (Sing(H)).3. π * (H) is a hypersurface with Q-normal crossings on X.To deal with these resolutions, some notation needs to be introduced. Let G := µ d 0 × · · · × µ dr be an arbitrary finite abelian group written as a product of finite cyclic groups, that is, µ d i is the cyclic group of d i -th roots of unity. Consider a matrix of weight vectors A := (a ij ) i,j = [a 0 | · · · | a n ] ∈ M at((r + 1) × (n + 1), Z)

2-Jordan blocks for the eigenvalue λ=1 of Yomdin–Lê surface singularities

“…+ , définies dans (3) et calculées dans (4). La suite spectrale généralisée de Steenbrink dans [8] implique que le nombre de 2-blocs de Jordan pour λ = 1 est 1 − b 0 (D…”

“…The Jordan blocks of size 2 for λ = 1 appears in the 2nd and 4th W -graded parts of H. This is a consequence of the fact that i = 3 is the central index of the monodromy filtration for λ = 1, see [8] for a more careful explanation.…”

“…Proof. On the one hand, the generalized Steenbrink's spectral sequence of [8] gives rise the following exact sequences: 0 −→ Gr W 4 H −→ H 0 (D [2] ) −→ H 2 (D [1] ) −→ H 4 (D…”

“…The complete description, see (2), for the other eigenvalues is announced as a future work too, but no proof is provided here. Our method consists in applying the generalized Steenbrink's spectral sequence in [8] to the embedded Q-resolution of YLS obtained in [7].…”