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Lowest weights in cohomology of variations of Hodge structure

Published online by Cambridge University Press:  11 January 2016

Chris Peters  and
Morihiko Saito
Chris Peters
Affiliation:
Institut Fourier – UMR CNRS 5582 Université Grenoble 1, 38402-Saint-Martin d’Hères, France, [email protected]
Morihiko Saito
Affiliation:
Research Institute for Mathematical Sciences Kyoto University, Kyoto 606-8502, Japan, [email protected]
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Abstract

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Let X be an irreducible complex analytic space with j: U ↪ X an immersion of a smooth Zariski-open subset, and let 𝕍 be a variation of Hodge structure of weight n over U. Assume that X is compact Kähler. Then, provided that the local monodromy operators at infinity are quasi-unipotent, IHk (X, 𝕍) is known to carry a pure Hodge structure of weight k+n, while Hk (U, 𝕍) carries a mixed Hodge structure of weight at least k +n. In this note it is shown that the image of the natural map IHk (X, 𝕍) → Hk (U, 𝕍) is the lowest-weight part of this mixed Hodge structure. In the algebraic case this easily follows from the formalism of mixed sheaves, but the analytic case is rather complicated, in particular when the complement X — U is not a hypersurface.

Keywords

14C3032S35
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 206 , June 2012 , pp. 1 - 24
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2012

References

[BBD] Beilinson, A., Bernstein, J., and Deligne, P., “Faisceaux pervers” in Analyse et topologie sur les espaces singuliers, I (Luminy, 1981), Astérisque 100, Soc. Math. France, Paris, 1982.Google Scholar
[Bo] Borel, A. and Spaltenstein, N., “Sheaf theoretic intersection cohomology” in Intersection Cohomology (Bern, 1983), Progr. Math. 50, Birkhäuser, Boston, 1984.CrossRefGoogle Scholar
[CSP] Carlson, J., Müller-Stach, S., and Peters, C., Period Mappings and Period Domains, Cambridge Stud. Adv. Math. 85, Cambridge University Press, Cambridge, 2003.Google Scholar
[CKS] Cattani, E., Kaplan, A., and Schmid, W., L2 and intersection cohomologies for a polarizable variation of Hodge structures, Invent. Math. 87 (1987), 217–252.CrossRefGoogle Scholar
[D1] Deligne, P., Théorie de Hodge, II, Publ. Math. Inst. Hautes Etudes Sci. 40 (1971), 5–57.CrossRefGoogle Scholar
[D2] Deligne, P., Théorie de Hodge, III, Publ. Math. Inst. Hautes Etudes Sci. 44 (1974), 5–77.CrossRefGoogle Scholar
[D3] Deligne, P., “Décompositions dans la catégorie dérivée” in Motives (Seattle, 1991), Proc. Sympos. Pure Math. 55 (1994), 115–128.Google Scholar
[HS] Hanamura, M. and Saito, M., Weight filtration on the cohomology of algebraic varieties, arXiv:math/0605603 [math.AG] Google Scholar
[K] Kashiwara, M., Quasi-unipotent constructible sheaves, J. Math. Sci. Univ. Tokyo 28 (1982), 757–773.Google Scholar
[KK1] Kashiwara, M. and Kawai, T., Hodge structure and holonomic systems, Proc. Japan Acad. Ser. A Math. Sci. 62 (1986), 1–4.CrossRefGoogle Scholar
[KK2] Kashiwara, M. and Kawai, T., The Poincaré lemma for variations of polarized Hodge structure, Publ. Res. Inst. Math. Sci. 23 (1987), 345–407.CrossRefGoogle Scholar
[KK3] Kashiwara, M. and Kawai, T., “A particular partition of unity: An auxiliary tool in Hodge theory” in Theta Functions (Bowdoin, Maine, 1987), Proc. Sympos. Pure Math. 49, Pt. 1, Amer. Math. Soc., Providence, 1989, 19–26.Google Scholar
[M] Morel, S., Complexes pondérés sur les compactifications de Baily-Borel: le cas des variétés modulaires de Siegel, J. Amer. Math. Soc. 21 (2008), 23–61.CrossRefGoogle Scholar
[P] Peters, C., Lowest weights in cohomology of variations of Hodge structure, preprint, arXiv:0708.0130v2 [math.AG] Google Scholar
[PS] Peters, C. and Steenbrink, J., Mixed Hodge Structures, Ergeb. Math. Grenzgeb. (3) 52, Springer, Berlin, 2008.Google Scholar
[S1] Saito, M., Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), 849–995.CrossRefGoogle Scholar
[S2] Saito, M., Decomposition theorem for proper Kähler morphisms, Tohoku Math. J. (2) 42 (1990), 127–147.CrossRefGoogle Scholar
[S3] Saito, M., Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), 221–333.CrossRefGoogle Scholar
[S4] Saito, M., Mixed Hodge complexes on algebraic varieties, Math. Ann. 316 (2000), 283–331.CrossRefGoogle Scholar
[S5] Saito, M., On the formalism of mixed sheaves, preprint, arXiv:math/0611597 [math.ST] Google Scholar
[Sch] Schmid, W., Variation of Hodge structure: The singularities of the period mapping, Invent. Math. 22 (1973), 211–319.CrossRefGoogle Scholar
[V] Verdier, J.-L., “Catégories Dérivées (Etat 0)” in Cohomologie étale, Séminaire de Géométrie Algébrique du Bois-Marie (SGA) 4½, Lecture Notes in Math. 569, Springer, Berlin, 1977, 262–311.CrossRefGoogle Scholar
[Z] Zucker, S., Hodge theory with degenerating coefficients: L2-cohomology in the Poincaré-metric, Ann. of Math. (2) 109 (1979), 415–476.CrossRefGoogle Scholar