zbMATH — the first resource for mathematics

On log Hodge structures of higher direct images. (English) Zbl 1017.32017
Let $$f:X\to Y$$ be a proper smooth morphism of complex manifolds and let $$\omega^\bullet_{X/Y}$$ be the relative de Rham complex. Then Poincaré’s lemma asserts that the complex of sheaves $$\omega^\bullet_{X/Y}$$ is a resolution of $$f^{-1}{\mathcal O}_Y$$. From this it is easy to construct an isomorphism of $${\mathcal O}_Y$$-modules: $$\mathbb{R}^m f_*\mathbb{Q} \otimes{\mathcal O}_Y \simeq \mathbb{R}^m f_* \omega^c_{X/Y}$$.
In a paper by F. Kato [Duke Math. J. 93, No. 1, 179-206 (1998; Zbl 0947.32003)] a generalization of this result was proved for log analytic spaces as defined by L. Illusie [Perspect. Math. 15, 183-203 (1994; Zbl 0832.14015)]. The aim of this paper is to prove a similar result for log Hodge structures.
More precisely, let $$(Y,{\mathcal M}_Y)$$ be a log analytic space and let $$(Y^{\log}, {\mathcal O}^{\log}_Y$$ be the corresponding ringed space endowed with a continuous surjective map $$\tau: Y^{\log} \to Y$$. Let $$f:(X, {\mathcal M}_X)\to (Y,{\mathcal M}_X) \to(Y,{\mathcal M}_Y)$$ be a morphism of log analytic space (satisfying some extra conditions). Then the author shows, using a log version of the relative Poincaré lemma, that there is an isomorphism of $${\mathcal O}^{\log}_Y$$-modules: $\lambda: \mathbb{R}^m f^{\log}_* \mathbb{Q} \otimes {\mathcal O}^{\log}_Y \simeq \tau^*\mathbb{R}^m f_*\omega^\bullet _{X/Y}.$ Using this, he proves the following result for log Hodge structures: Let $${\mathcal H}_\mathbb{Q} =\mathbb{R}^mf^{\log}_* \mathbb{Q}$$ and $${\mathcal H}_{\mathcal O}= \mathbb{R}^m f_* \omega^\bullet_{X/Y}$$ endowed with a filtration $$\{\mathbb{R}^m_*\omega^\bullet\geq i_{X/Y}\}$$ and let $$\lambda$$ be the above isomorphism. Then the triplet $$({\mathcal H}_\mathbb{Q}, {\mathcal H}_{\mathcal O}, \lambda)$$ is a log Hodge structure on $$Y$$.

MSC:
 32J25 Transcendental methods of algebraic geometry (complex-analytic aspects) 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 14F40 de Rham cohomology and algebraic geometry
Full Text:
References:
 [1] DELIGNE, P., Equations Differentiells a Points Singuliers Reguliers, Lecture Notes in Math., 163, Sponger, 1970. · Zbl 0244.14004 · doi:10.1007/BFb0061194 [2] DELIGNE, P., Theooe de Hodge II, Inst. Hautes Etudes Sci. Publ. Math., 40 (1971), 5-57 · Zbl 0219.14007 · doi:10.1007/BF02684692 · numdam:PMIHES_1971__40__5_0 · eudml:103914 [3] DELIGNE, P., Cohomologie Etale, Seminaire de Geometric Algeboque du Bois-Marie SG · Zbl 0495.14024 [4] DELIGNE, P., Avec la collaboration de J. F. Boutot, A. Grothendieck, L. IIIusie et J. L. Verdier, Lecture Notes m Math., 569, Sponger, 1977 [5] GRIFFITHS, P., Topics in Transcendental Algebraic Geometry, Ann. of Math. Studies, Ponceton Umv. Press, 1984 · Zbl 0528.00004 [6] IVERSEN, B., Cohomology of Sheaves, Sponger-Verlag, 1986 · Zbl 1272.55001 [7] KATO, F., The relative log Pomcare lemma and relative log de Rham theory, Duke Math J., 93 (1998), 179-206. · Zbl 0947.32003 · doi:10.1215/S0012-7094-98-09307-3 [8] KATO, K., Logarithmic structures of Fontame-IIIusie, Algebraic Analysis, Geometry, an Number Theory, Igusa, J. -I. ed., Johns Hopkins Univ. Press, 1989, 191-224. · Zbl 0776.14004 [9] KATO, K., Problems concerning log Hodge structures, prepont [10] KATO, K. AND NAKAYAMA, C., Log Betti cohomology, log etale cohomology, and log d Rham cohomology of log schemes over C, · Zbl 0957.14015 · doi:10.2996/kmj/1138044041 [11] SCHMID, W., Vaoation of Hodge structure: The singularities of the peood mapping, Invent Math., 22 (1973), 211-319. · Zbl 0278.14003 · doi:10.1007/BF01389674 · eudml:142246 [12] STEENBRINK, J., Limits of Hodge structures, Invent. Math., 31 (1976), 229-257 · Zbl 0303.14002 · doi:10.1007/BF01403146 · eudml:142362 [13] STEENBRINK, J., Logaothmic embeddings of varieties with normal crossings and mixe Hodge structures, Math. Ann., 301 (1995), 105-118. · Zbl 0814.14010 · doi:10.1007/BF01446621 · eudml:165283 [14] Usui, S., Recovery of vanishing cycles by log geometry, prepont · Zbl 1015.14005 · doi:10.2748/tmj/1178207529
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.