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Weight filtration and integral monodromy. (Filtration par le poids et monodromie entière.) (French) Zbl 0771.14005

This paper presents a rather detailed investigation of the topology of the Milnor fibre \(F\) and the monodromy for a plane curve singularity.
Starting from the map-germ \(f:(\mathbb{C}^ 2,0)\to(\mathbb{C},0)\) take a local embedded resolution \(\pi:X\to Y\) of \(f^{-1}0\), with exceptional fibre \(E_ +=\bigcup E_ i\). An \(e\)-fold covering of \((\mathbb{C},0)\) gives a semistable reduction, and \(X'\) is the normalisation of \(X\times_ \mathbb{C}\mathbb{C}'\); the exceptional fibre here is \(D_ +=\bigcup D_ i\). A model for \(X'\) can be constructed by plumbing neighbourhoods of the \(D_ i\) along intersections \(B_{ij}\). There is a map \(F\to D_ +\) which is a homeomorphism outside the \(D_ i\cap D_ j\); this induces a decomposition \(F=\bigcup F_ i\). Then \(H_ 1F\) is filtered by \(M_{- 3}=0\), \(M_{-2}=\) image of \(\oplus H_ 1(F_ i\cap F_ j)\), \(M_{- 1}=\) image of \(\oplus H_ 1F_ i\), \(M_ 0=H_ 1F\).
There is a corresponding dual filtration of cohomology; a major result of the paper is that when tensored with \(\mathbb{Q}\) this coincides with Steenbrink’s mixed Hodge structure [J. H. M. Steenbrink in Real and complex Singul., Proc. Nordic Summer Sch., Symp. Math., Oslo 1976, 525- 563 (1977; Zbl 0373.14007)]. However these filtrations are defined over \(\mathbb{Z}\); the graded group is torsion free and is described explicitly by generators and relations. After a full development of this theory, it is illustrated with explicit calculations on certain examples, including one where the graded modules are not isomorphic but become so after tensoring with \(\mathbb{Q}\).

MSC:

14H20 Singularities of curves, local rings
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
14D07 Variation of Hodge structures (algebro-geometric aspects)
14B05 Singularities in algebraic geometry
57R45 Singularities of differentiable mappings in differential topology
32B10 Germs of analytic sets, local parametrization
32S55 Milnor fibration; relations with knot theory
32S35 Mixed Hodge theory of singular varieties (complex-analytic aspects)

Citations:

Zbl 0373.14007
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References:

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