zbMATH — the first resource for mathematics

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.

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)
Full Text: DOI EuDML
[1] Deligne, P.: Théorie de Hodge II. Publ. Math. IHES40 (1971), 5-57. · Zbl 0219.14007
[2] Durfee, A.: Mixed Hodge structures on punctured neighborhoods. Duke Math. J.50 (1983), 1017-1040. · Zbl 0545.14005 · doi:10.1215/S0012-7094-83-05043-3
[3] Elzein, F.: Mixed Hodge structures, Trans. Amer. Math. Soc.275 (1983), 71-106. · Zbl 0511.14003 · doi:10.2307/1999006
[4] Friedman, R.: Global smoothings of varieties with normal crossings. Ann. of Math.118 (1983) 75-114. · Zbl 0569.14002 · doi:10.2307/2006955
[5] Fujisawa, T.: private communication.
[6] Illusie, L.: Complexe cotangent et déformations I. Lecture Notes in Math. 239, Berlin, Heidelberg, New York: Springer Verlag 1971. · Zbl 0224.13014
[7] Illusie, L.: Logarithmic spaces (according to K. Kato). Séminaire Bourbaki 1992.
[8] Kato, K.: Logarithmic structures of Fontaine-Illusie, in Algebraic Analysis, Geometry and Number Theory, J.-I. Igusa ed., 1988, Johns Hopkins Univ., 191-224. · Zbl 0776.14004
[9] Kawamata, Y. and Y. Namikawa: Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties. Preprint 1993, 24 p. · Zbl 0848.14004
[10] Saito, M.: Modules de Hodge polarisables. Publ. RIMS Kyoto Univ.24 (1988), 849-995. · Zbl 0691.14007 · doi:10.2977/prims/1195173930
[11] Steenbrink, J.H.M.: Limits of Hodge structures. Invent. Math.31, (1976), 229-257. · Zbl 0312.14007 · doi:10.1007/BF01403146
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.