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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 \(Y\).

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: DOI
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