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Sur le théorème local des cycles invariants. (On the local invariant cycle theorem). (French) Zbl 0722.14002

This paper presents a streamlined overview of the arguments involved in the proof of the local monodromy-invariant cycle theorem. Let \(\{X_ t\}\) be an analytic family of complex projective varieties, t varying in the unit disk, with \(X_ t\) smooth for \(t\neq 0\) and \(Y:=X_ 0\) a divisor with normal crossings on the total space of the family. After certain insights of P. Deligne [cf. P. A. Griffiths, Bull. Am. Math. Soc. 76, 228-296 (1970; Zbl 0214.198); conjecture 9.17; this is the best reference for background as well], W. Schmid [Invent. Math. 22, 211-319 (1973; Zbl 0278.14003)] and J. Steenbrink [ibid. 31, 229-257 (1976; Zbl 0303.14002; see also Report 73-04 and 74-04. Dep. Math., Univ. Amsterdam (1973; Zbl 0329.14006 and 1974; Zbl 0329.14007)] clarified, the latter by purely algebraic means, the notion of “cohomology of the limit fiber” as a mixed Hodge structure, lim \(H^ n\), obtained as a “limit”, when \(t\to 0\), of the canonical Hodge structures on the n-th cohomology group of \(X_ t (t\neq 0)\). As noted by F. El Zein, however [Ann. Sci. Éc. Norm. Supér., IV. Sér. 19, 107-184 (1986; Zbl 0538.14003)], Steenbrink’s proof that the weight filtration of lim \(H^ n\) coincides with the filtration defined by the monodromy was not conclusive. Recently M. Saito [Publ. Res. Inst. Math. Sci. 24, No.6, 849-995 (1988; Zbl 0691.14007)] gave a proof and now, in the paper under review, the authors put forward a context that allows a simpler proof. It is crafted using ideas of Steenbrink (loc. cit.), P. Deligne (“Positivité: signes”, 16-Feb-84 and 6-Nov- 85, unpublished manuscripts) and Saito (loc. cit), the key new ingredient being that the \(E_ 1\) term of Steenbrink’s spectral sequence is a polarized Hodge-Lefschetz module. At this point an argument of Deligne (see J. Steenbrink, loc. cit. 5.2) shows that the theorem of Schmid and Steenbrink implies the local invariant cycle theorem, namely, that the image of \(H^ n(Y)\) in lim \(H^ n\) under the specialization map coincides with the invariant space of the map induced by the monodromy.

MSC:

14C25 Algebraic cycles
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14D07 Variation of Hodge structures (algebro-geometric aspects)
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