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Logarithmic embeddings of varieties with normal crossings and mixed Hodge structures. (English) Zbl 0814.14010
Recently, the theory of logarithmic structures on algebraic varieties has been developed by I. Illusie [“Logarithmic spaces (according to K. Kato)” in Barsotti Sympos. Algebraic Geometry, Abano Terme 1991, Perspect. Math. 15, 183-203 (1994)] and K. Kato [in Algebraic Analysis, Geometry and Number Theory, Proc. JAMI Inaugur. Conf., Baltimore 1988, 191-224 (1989; Zbl 0776.14004)]. Each embedding of a variety \(D\) as a divisor in a smooth ambient space \(X\) gives rise to logarithmic structures on \(D\) and \(X\). In this paper, the author considers logarithmic embeddings of varieties over \(\mathbb{C}\) with normal crossings; these are logarithmic structures which are locally isomorphic to the standard one obtained from an embedding of the variety as a divisor in a smooth space. To each logarithmic embedding of \(D\) is associated a filtered logarithmic de Rham complex and a counterpart of this with integral coefficients, which is a constructible complex. If \(D\) is compact, the hypercohomology groups of these form mixed Hodge structures, which in the case of genuine embeddings reduces to the mixed Hodge structure of a deleted neighborhood of \(D\) [cf. A. Durfee, Duke Math. J. 50, 1017-1040 (1983; Zbl 0545.14005) and F. Elzein, Trans. Am. Math. Soc. 275, 71-106 (1983; Zbl 0511.14003)].
A logarithmic embedding provided with a suitable global section of its logarithmic sheaf is called a logarithmic deformation. Such a structure exists exactly for \(d\)-semistable varieties in the sense of R. Friedman [Ann. Math., II. Ser. 118, 75-114 (1983; Zbl 0569.14002)]. The previous result is used for the construction of a “limit mixed Hodge structure” associated to a logarithmic deformation, which reduces to the construction by the reviewer [Invent. Math. 31, 229-257 (1976; Zbl 0305.14002)] in the case of a one-parameter smoothing with smooth total space.

MSC:
14D07 Variation of Hodge structures (algebro-geometric aspects)
14E25 Embeddings in algebraic geometry
14F99 (Co)homology theory in algebraic geometry
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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References:
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