Hostname: page-component-6bf8c574d5-b4m5d Total loading time: 0 Render date: 2025-02-20T12:13:19.991Z Has data issue: false hasContentIssue false
  • English
  • Français

MIXED HODGE STRUCTURES WITH MODULUS

Part of: Cycles and subschemes (Co)homology theory Representation theory of rings and algebras General theory of differentiable manifolds

Published online by Cambridge University Press:  02 March 2020

Florian Ivorra Open the ORCID record for Florian Ivorra [Opens in a new window]  and
Takao Yamazaki
Florian Ivorra
Affiliation:
Institut de recherche mathématique de Rennes, UMR 6625 du CNRS, Université de Rennes 1, Campus de Beaulieu, 35042Rennes Cedex, France ([email protected])
Takao Yamazaki
Affiliation:
Institute of Mathematics, Tohoku University, Aoba, Sendaï, 980-8578, Japan ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We define a notion of mixed Hodge structure with modulus that generalizes the classical notion of mixed Hodge structure introduced by Deligne and the level one Hodge structures with additive parts introduced by Kato and Russell in their description of Albanese varieties with modulus. With modulus triples of any dimension, we attach mixed Hodge structures with modulus. We combine this construction with an equivalence between the category of level one mixed Hodge structures with modulus and the category of Laumon 1-motives to generalize Kato–Russell’s Albanese varieties with modulus to 1-motives.

Keywords

enriched Hodge structureformal Hodge structureLaumon 1-motives

MSC classification

Primary: 16G20: Representations of quivers and partially ordered sets 14C30: Transcendental methods, Hodge theory , Hodge conjecture 58A14: Hodge theory
Secondary: 14F42: Motivic cohomology; motivic homotopy theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 1 , January 2022 , pp. 161 - 195
Copyright
© Cambridge University Press 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barbieri-Viale, L., Formal Hodge theory, Math. Res. Lett. 14(3) (2007), 385–394. MR 2318642.CrossRefGoogle Scholar
Barbieri-Viale, L. and Bertapelle, A., Sharp de Rham realization, Adv. Math. 222(4) (2009), 1308–1338. MR 2554937.CrossRefGoogle Scholar
Barbieri-Viale, L. and Srinivas, V., Albanese and Picard 1-motives, Mém. Soc. Math. Fr. (N.S.) 87 (2001), vi+104. MR 1891270.Google Scholar
Binda, F. and Saito, S., Relative cycles with moduli and regulator maps, J. Inst. Math. Jussieu 18(6) (2019), 1233–1293.CrossRefGoogle Scholar
Bloch, S. and Esnault, H., An additive version of higher Chow groups, Ann. Sci. Éc. Norm. Supér. (4) 36(3) (2003), 463–477.CrossRefGoogle Scholar
Bloch, S. and Srinivas, V., Enriched Hodge structures, in Algebra, Arithmetic and Geometry, Part I, II (Mumbai, 2000), Tata Institute of Fundamental Research Studies in Mathematics, Volume 16, pp. 171–184 (Tata Institute of Fundamental Research, Bombay, 2002). MR 1940668.Google Scholar
Deligne, P., Théorie de Hodge. II, Publ. Math. Inst. Hautes Études Sci. 40 (1971), 5–57. MR 0498551.CrossRefGoogle Scholar
Deligne, P., Théorie de Hodge. III, Publ. Math. Inst. Hautes Études Sci. 44 (1974), 5–77. MR 0498552.CrossRefGoogle Scholar
Deligne, P. and Illusie, L., Relèvements modulo p 2 et décomposition du complexe de de Rham, Invent. Math. 89(2) (1987), 247–270. MR 894379.CrossRefGoogle Scholar
El Zein, F. and Lê Dũng Tráng, Mixed Hodge Structures, Hodge theory, Math. Notes, Volume 49, pp. 123–216 (Princeton University Press, Princeton, NJ, 2014). MR 3290125.Google Scholar
Esnault, H., Srinivas, V. and Viehweg, E., The universal regular quotient of the Chow group of points on projective varieties, Invent. Math. 135(3) (1999), 595–664. MR 1669284.CrossRefGoogle Scholar
Fujiki, A., Duality of mixed Hodge structures of algebraic varieties, Publ. Res. Inst. Math. Sci. 16(3) (1980), 635–667. MR 602463.CrossRefGoogle Scholar
Hartshorne, R., Algebraic Geometry, Graduate Texts in Mathematics, Volume 52 (Springer, New York-Heidelberg, 1977). MR 0463157.CrossRefGoogle Scholar
Ivorra, F. and Yamazaki, T., Nori motives of curves with modulus and Laumon 1-motives, Canad. J. Math. 70(4) (2018), 868–897. MR 3813515.CrossRefGoogle Scholar
Kato, K. and Russell, H., Albanese varieties with modulus and Hodge theory, Ann. Inst. Fourier (Grenoble) 62(2) (2012), 783–806. MR 2985516.CrossRefGoogle Scholar
Laumon, G., Transformation de Fourier généralisée, Preprint, 1996, arXiv:math/9603004v1.Google Scholar
Lekaus, S., On Albanese and Picard 1-motives with 𝔾a-factors, Manuscripta Math. 130(4) (2009), 495–522. MR 2563147.CrossRefGoogle Scholar
Mazzari, N., Extensions of formal Hodge structures, Comm. Algebra 39(4) (2011), 1372–1393. MR 2782612.CrossRefGoogle Scholar
Peters, C. A. M. and Steenbrink, J. H. M., Mixed Hodge Structures, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Volume 52 (Springer, Berlin, 2008). MR 2393625.Google Scholar
Rülling, K. and Yamazaki, T., Suslin homology of relative curves with modulus, J. Lond. Math. Soc. (2) 93(3) (2016), 567–589. MR 3509954.CrossRefGoogle Scholar
Russell, H., Albanese varieties with modulus over a perfect field, Algebra Number Theory 7(4) (2013), 853–892. MR 3095229.CrossRefGoogle Scholar
Saito, M., Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26(2) (1990), 221–333. MR 1047415.CrossRefGoogle Scholar
Saito, M., Mixed Hodge complexes on algebraic varieties, Math. Ann. 316(2) (2000), 283–331. MR 1741272.CrossRefGoogle Scholar
Serre, J.-P., Faisceaux algébriques cohérents, Ann. of Math. (2) 61 (1955), 197–278. MR 0068874.CrossRefGoogle Scholar
Serre, J.-P., Algebraic Groups and Class Fields, Graduate Texts in Mathematics, Volume 117 (Springer, New York, 1988). Translated from the French. MR 918564.CrossRefGoogle Scholar
Vistoli, A., Grothendieck topologies, fibered categories and descent theory, in Fundamental Algebraic Geometry, Mathematical Surveys and Monographs, Volume 123, pp. 1–104 (American Mathematical Society, Providence, RI, 2005). MR 2223406.Google Scholar