Monodromy and weight filtration for smoothings of isolated singularities. (English) Zbl 0844.14001

Given a germ of a flat smoothing deformation of an isolated singularity of a complex variety, the author describes the weighted filtration of the limit mixed Hodge structure in the mean vanishing cohomology of a smooth generic fiber, generated by a logarithm of the unipotent part of the monodromy transformation around the critical value. In particular, it is given a formula for the signature of the intersection form via the Hodge numbers of the vanishing cohomology. This completes the results of the author’s earlier work [Proc. Symp. Pure Math. 40, Part 2, 513-536 (1983; Zbl 0515.14003)].


14B05 Singularities in algebraic geometry
32S35 Mixed Hodge theory of singular varieties (complex-analytic aspects)
14D07 Variation of Hodge structures (algebro-geometric aspects)


Zbl 0515.14003
Full Text: Numdam EuDML


[1] . M. Artin , Algebraic approximation of structures over complete local rings , Publ. Math. I.H.E.S., 36 (1969) 23-58. · Zbl 0181.48802
[2] . H. Hironaka , On the equivalence of singularities: I, Arithmetical algebraic geometry . Proceedings, New York: Harper and Row 1965. · Zbl 0147.20502
[3] . E.J.N. Looijenga , Isolated singular points on complete intersections , London Math. Soc. Lecture Note Series 77, Cambridge University Press 1984. · Zbl 0552.14002
[4] . E.J.N. Looijenga , Riemann-Roch and smoothings of singularities , Topology 25 (1986) 293-302. · Zbl 0654.32007
[5] . A. Némethi , The mixed Hodge structure of an ICIS,I . Preliminary version November 1994.
[6] . M. Saito , Hodge structure via filtered D-modules, in: Systèmes différentiels et singularités , A. Galligo, M. Granger, Ph. Maisonobe ed., Astérisque 130 (1985) 342-351. · Zbl 0621.14008
[7] . W. Schmid , Variation of Hodge structure: the singularities of the period mapping , Invent. Math. 22 (1973), 211-320. · Zbl 0278.14003
[8] . J.H.M.Steenbrink, Mixed Hodge structure on the vanishing cohomology . In: Real and complex singularities , Oslo 1976. P. Holm Ed. pp. 525-563, Sijthoff- Noordhoff, Alphen a/d Rijn 1977. · Zbl 0373.14007
[9] . J.H.M. Steenbrink , Mixed Hodge structures associated with isolated singularities . Proc. Symp. Pure Math. 40 Part 2 (1983), 513-536. · Zbl 0515.14003
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.