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 \(\mathbb{Q}\).


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)


Zbl 0373.14007
Full Text: DOI Numdam EuDML


[1] BRIESKORN (E.) et KNORRER (H.) . - Plane Algebraic Curves . - Birkhaüser.
[2] DELIGNE (P.) . - Théorie de Hodge III , Inst. Hautes Études Sci. Publ. Math., t. 44, 1972 , p. 5-77. Numdam | Zbl 0237.14003 · Zbl 0237.14003 · doi:10.1007/BF02685881
[3] DURFEE (A.) . - Fibered Knots and Algebraic Singularities , Topology, t. 13, 1974 , p. 47-59. MR 49 #1523 | Zbl 0275.57007 · Zbl 0275.57007 · doi:10.1016/0040-9383(74)90037-8
[4] JACO (W.) et SHALEN (P.) . - Seifert Fibered Spaces in 3-Manifolds , A.M.S. Memoirs n^\circ 220. Zbl 0415.57005 · Zbl 0415.57005
[5] LE DUNG TRANG , MICHEL (F.) et WEBER (C.) . - Courbes Polaires et Topologie des Courbes Planes , Ann. Sci. École Norm. Sup. (4), t. 24, 1991 . Numdam | MR 92b:32043 | Zbl 0748.32018 · Zbl 0748.32018
[6] LE VAN THANH et STEENBRINK (J.H.M.) . - Le Spectre d’une Singularité d’un germe de Courbe Plane . - Preprint Math. Inst. Catholic University of Nijmegen, 1988 .
[7] MILNOR (J.) . - Singular Points of Complex Hypersurfaces , Ann. of Math. Stud., t. 61, 1968 . MR 39 #969 | Zbl 0184.48405 · Zbl 0184.48405
[8] MICHEL (F.) et WEBER (C.) . - Topologie des Germes de Courbes Planes à plusieurs Branches . - Université de Genève, 1985 .
[9] MICHEL (F.) et WEBER (C.) . - Sur le Rôle de la Monodromie Entière dans la Topologie des Singularités , Ann. Inst. Fourier (Grenoble), t. 36, 1, 1986 , p. 183-218. Numdam | MR 87k:32019 | Zbl 0557.57017 · Zbl 0557.57017 · doi:10.5802/aif.1043
[10] STEENBRINK (J.H.M.) . - Limits of Hodge Structures , Invent. Math., t. 31, 1976 , p. 229-257. MR 55 #2894 | Zbl 0303.14002 · Zbl 0303.14002 · doi:10.1007/BF01403146
[11] STEENBRINK (J.H.M.) . - Mixed Hodge Structure on the Vanishing Cohomology , Nordic Summer School/NAVF, Symposium in Math., Oslo, 1976 . Zbl 0373.14007 · Zbl 0373.14007
[12] STEENBRINK (J.H.M.) et ZUCKER (S.) . - Variation of Mixed Hodge Structure, I . - Preprint Math. Inst. University of Leiden, 1984 .
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.