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Perverse Leray filtration and specialisation with applications to the Hitchin morphism

Part of: Families, fibrations Curves Algebraic geometry: Foundations

Published online by Cambridge University Press:  20 April 2021

MARK ANDREA A. DE CATALDO
MARK ANDREA A. DE CATALDO*
Affiliation:
Department of Mathematics, Stony Brook University, Stony Brook, NY, 11794-3651, U.S.A e-mail: mark.decataldo@stonybrook.edu
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Abstract

We initiate and develop a framework to handle the specialisation morphism as a filtered morphism for the perverse, and for the perverse Leray filtration, on the cohomology with constructible coefficients of varieties and morphisms parameterised by a curve. As an application, we use this framework to carry out a detailed study of filtered specialisation for the Hitchin morphisms associated with the compactification of Dolbeault moduli spaces in [de 2018].

MSC classification

Primary: 14A10: Varieties and morphisms 14D06: Fibrations, degenerations 14D20: Algebraic moduli problems, moduli of vector bundles 14H60: Vector bundles on curves and their moduli
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 2 , March 2022 , pp. 443 - 487
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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References

Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981), 5-171, Astérisque, 100, (Soc. Math. France, Paris, 1982).Google Scholar
de Cataldo, M.A., “The perverse filtration and the Lefschetz Hyperplane Theorem, II.,” J. Algebraic Geometry 21 (2012), 305–345.CrossRefGoogle Scholar
de Cataldo, M.A., “Decomposition theorem for semi-simples.,” J. Singul. 14 (2016), 194–197.Google Scholar
de Cataldo, M.A., “Compactification of Dolbeault moduli spaces,” to appear in Int. Mat. Res. Nat. arXiv:1812.07086.Google Scholar
de Cataldo, M.A., Maulik, D., “The perverse filtration for the Hitchin fibration is locally constant,” to appear in Pure and Applied Mathematics Quarterly, arXiv:1808.02235.Google Scholar
de Cataldo, M.A., Maulik, D., Shen, J. “Hitchin fibrations, abelian surfaces, and the P=W conjecture.,” arXiv:1909.11885.Google Scholar
de Cataldo, M.A., Migliorini, L., “The Hodge theory of algebraic maps.,” Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 5, 693–750.CrossRefGoogle Scholar
de Cataldo, M.A., Migliorini, L., “The decomposition theorem, perverse sheaves and the topology of algebraic maps.,” Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 4, 535–633.CrossRefGoogle Scholar
Grothendieck, A. et. al, SGA 7.I., Groupes de Monodromie en Géomtrie Algébrique., Lecture Notes in Math. 288, (Springer-Verlag, Heidelberg 1972).Google Scholar
Edidin, D., Graham, W., “Algebraic cuts,” Proc. Amer. Math. Soc. 126 (1998), no. 3, 677–685.CrossRefGoogle Scholar
El Zein, F., Lê, D.T., Migliorini, L., “A topological construction of the weight filtration,” Manuscripta Math. 133 (2010), no. 1-2, 173–182.CrossRefGoogle Scholar
Grothendieck, A., “Sur quelques points d’algèbre homologique.” Tôohoku Math. J. (2) 9 (1957), 119–221.Google Scholar
Illusie, L., Complexe cotangent et déformations, I., (French) Lecture Notes in Mathematics, Vol. 239. (Springer-Verlag, Berlin-New York, 1971). xv+355 pp.CrossRefGoogle Scholar
Illusie, L., “Autour du théorème de monodromie locale.,” in Périodes p-adiques (Bures-sur-Yvette, 1988)., Astérisque 223 (1994), 9–57.Google Scholar
Kashiwara, M., Schapira, P., Sheaves on manifolds. With a chapter in French by Christian Houzel., Grundlehren Math. Wiss., 292. (Springer-Verlag, Berlin, 1990). x+512 pp.CrossRefGoogle Scholar
Migliorini, L., Shende, V., “Higher discriminants and the topology of algebraic maps,” Algebr. Geom. 5 (2018), no. 1, 114–130.CrossRefGoogle Scholar
Mochizuki, T., Asymptotic behaviour of tame harmonic bundles and an application to pure twistor D-modules, I, Mem. Amer. Math. Soc. 185 (2007), no. 869, xii+324 pp.Google Scholar
Sabbah, C., Polarizable twistor D-modules, Asterisque No. 300 (2005), vi+208 pp.Google Scholar
Saito, M., “Modules de Hodge polarizables,” Publ. Res. Inst. Math. Sci. 24 (1988), no. 6, 849–995 (1989).CrossRefGoogle Scholar
Schürmann, J., Topology of singular spaces and constructible sheaves, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series), 63, (Birkhäuser Verlag, Basel, 2003). x+452 pp.CrossRefGoogle Scholar
Wu, B., “A topological characterization of the middle perversity intersection complex for arbitrary complex algebraic varieties.,” Journal of Singularities, 20 (2020), 54–78.CrossRefGoogle Scholar