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P = W for Lagrangian fibrations and degenerations of hyper-Kähler manifolds

Part of: Families, fibrations Cycles and subschemes

Published online by Cambridge University Press:  25 June 2021

Andrew Harder ,
Zhiyuan Li ,
Junliang Shen  and
Qizheng Yin
Andrew Harder
Affiliation:
Lehigh University, Department of Mathematics; E-mail: anh318@lehigh.edu
Zhiyuan Li
Affiliation:
Fudan University, Shanghai Center for Mathematical Sciences; E-mail: zhiyuan_li@fudan.edu.cn
Junliang Shen
Affiliation:
Massachusetts Institute of Technology, Department of Mathematics; E-mail: jlshen@mit.edu
Qizheng Yin
Affiliation:
Peking University, Beijing International Center for Mathematical Research; E-mail: qizheng@math.pku.edu.cn

Abstract

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We identify the perverse filtration of a Lagrangian fibration with the monodromy weight filtration of a maximally unipotent degeneration of compact hyper-Kähler manifolds.

MSC classification

Secondary: 14C30: Transcendental methods, Hodge theory , Hodge conjecture 14D06: Fibrations, degenerations
Type
Algebraic and Complex Geometry
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e50
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

de Cataldo, M. A., Hausel, T. and Migliorini, L., ‘Topology of Hitchin systems and Hodge theory of character varieties: the case ${A}_1$ ’, Ann. Math. (2) 175 (3) (2012), 1329–1407.CrossRefGoogle Scholar
Fujisawa, T., ‘Polarizations on limiting mixed Hodge structures’, J. Singul. 8 (2014), 146–193.Google Scholar
Green, M., Kim, Y.-J., Laza, R. and Robles, C., ‘The LLV decomposition of hyper-Kähler cohomology’, Preprint, 2019, arXiv:1906.03432.Google Scholar
Harder, A., ‘Torus fibers and the weight filtration’, Preprint, 2019, arXiv:1908.05110.Google Scholar
Kollár, J., Laza, R., Saccà, G. and Voisin, C., ‘Remarks on degenerations of hyper-Kähler manifolds’, Ann. Inst. Fourier (Grenoble) 68 (7) (2018), 2837–2882.CrossRefGoogle Scholar
Kurnosov, N., Soldatenkov, A. and Verbitsky, M., ‘Kuga-Satake construction and cohomology of hyperkähler manifolds’, Adv. Math. 351 (2019), 275–295.CrossRefGoogle Scholar
Looijenga, E. and Lunts, V. A., ‘A Lie algebra attached to a projective variety’, Invent. Math. 129 (2) (1997), 361–412.CrossRefGoogle Scholar
Nagai, Y., ‘On monodromies of a degeneration of irreducible symplectic Kähler manifolds’, Math. Z. 258 (2) (2008), 407–426.CrossRefGoogle Scholar
Shen, J. and Yin, Q., ‘Topology of Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds’, Duke. Math. J., to appear. With an appendix by Claire Voisin.Google Scholar
Shende, V., ‘The weights of the tautological classes of character varieties’, Int. Math. Res. Not. IMRN (22) (2017), 6832–6840.Google Scholar
Simpson, C. T., ‘Higgs bundles and local systems’, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5–95.CrossRefGoogle Scholar
Soldatenkov, A., ‘Limit mixed Hodge structures of hyperkähler manifolds’, Mosc. Math. J. 20 (2) (2020), 423–436.CrossRefGoogle Scholar
Steenbrink, J., ‘Limits of Hodge structures’, Invent. Math. 31 (3) (1975–1976), 229–257.CrossRefGoogle Scholar
Verbitskiĭ, M. S., ‘Action of the Lie algebra of $\mathrm{SO}(5)$ on the cohomology of a hyper-Kähler manifold’, Funct. Anal. Appl. 24 (3) (1990), 229–230.CrossRefGoogle Scholar
Verbitsky, M., ‘Cohomology of compact hyper-Kähler manifolds and its applications’, Geom. Funct. Anal. 6 (4) (1996), 601–611.CrossRefGoogle Scholar