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Direct image of logarithmic complexes and infinitesimal invariants of cycles

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Morihiko Saito

Algebraic Geometry · Submitted 1 Jun 2005

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We show that the direct image of the filtered logarithmic de Rham complex is a direct sum of filtered logarithmic complexes with coefficients in variations of Hodge structures, using a generalization of the decomposition theorem of Beilinson, Bernstein and Deligne to the case of filtered $D$-modules.

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arXiv IDmath/0506020 (version 4)
Pith IDKYKHQ7GT, Pith's permanent record of this paper and its review
CitationMorihiko Saito. "Direct image of logarithmic complexes and infinitesimal invariants of cycles." arXiv:math/0506020 (2005). https://pith.science/paper/KYKHQ7GT
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,132 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.68 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; .

We show that the direct image of the filtered logarithmic de Rham complex is a direct sum of filtered logarithmic complexes with coefficients in variations of Hodge structures, using a generalization of the decomposition theorem of Beilinson, Bernstein and Deligne to the case of filtered $D$-modules. The advantage of using the logarithmic complexes is that we have the strictness of the Hodge filtration by Deligne after taking the cohomology group in the projective case. As a corollary, we get the total infinitesimal invariant of a (higher) cycle in a direct sum of the cohomology of filtered logarithmic complexes with coefficients, and this is essentially equivalent to the cohomology class of the cycle.

background contribution method result interpretation main claim

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 We show that the direct image of the filtered logarithmic de Rham complex is a direct sum of filtered logarithmic complexes with coefficients in variations of Hodge structures, using a generalization of the decomposition theorem of Beilinson, Bernstein and Deligne to the case of filtered $D$-modules. result 0.81 0.95 0.85 asserted 1.00
s1 The advantage of using the logarithmic complexes is that we have the strictness of the Hodge filtration by Deligne after taking the cohomology group in the projective case. interpretation 0.76 0.23 0.05 asserted 0.99
s2 As a corollary, we get the total infinitesimal invariant of a (higher) cycle in a direct sum of the cohomology of filtered logarithmic complexes with coefficients, and this is essentially equivalent to the cohomology class of the cycle. result 0.93 0.91 0.27 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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The paper
Original abstract

We show that the direct image of the filtered logarithmic de Rham complex is a direct sum of filtered logarithmic complexes with coefficients in variations of Hodge structures, using a generalization of the decomposition theorem of Beilinson, Bernstein and Deligne to the case of filtered $D$-modules. The advantage of using the logarithmic complexes is that we have the strictness of the Hodge filtration by Deligne after taking the cohomology group in the projective case. As a corollary, we get the total infinitesimal invariant of a (higher) cycle in a direct sum of the cohomology of filtered logarithmic complexes with coefficients, and this is essentially equivalent to the cohomology class of the cycle.

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