The perverse filtration and the Lefschetz hyperplane theorem, II
Author:
Mark Andrea A. de Cataldo
Journal:
J. Algebraic Geom. 21 (2012), 305-345
DOI:
https://doi.org/10.1090/S1056-3911-2011-00566-3
Published electronically:
May 16, 2011
MathSciNet review:
2877437
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Abstract |
References |
Additional Information
Abstract: The perverse filtration in cohomology and in cohomology with compact supports is interpreted, in terms of kernels of restriction maps to suitable subvarieties by using the Lefschetz hyperplane theorem and spectral objects. Various mixed-Hodge-theoretic consequences for intersection cohomology and for the decomposition theorem are derived.
References
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References
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- G. Barthel, J-P. Brasselet, K-H. Fieseler, O. Gabber, L. Kaup, “Relèvement de cycles algébriques et homomorphismes associés en homologie d’intersection,” Ann. of Math. 141 (1995), 147–179. MR 1314034 (96a:14027)
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- A.A. Beilinson, J.N. Bernstein, P. Deligne, Faisceaux pervers, Astérisque 100, Paris, Soc. Math. Fr. 1982. MR 751966 (86g:32015)
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- M. de Cataldo, L. Migliorini, “The Hard Lefschetz Theorem and the topology of semismall maps,” Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 5, 759–772. MR 1951443 (2003m:32026)
- M. de Cataldo, L. Migliorini, “The Hodge Theory of Algebraic maps,” Ann. Sci. École Norm. Sup. (4) 38 (2005) no. 5, 693–750. MR 2195257 (2007a:14016)
- M. de Cataldo, L. Migliorini, “The Hodge Theory of Algebraic maps,” arXiv:math/ 0306030v1.
- M. de Cataldo, L. Migliorini, “Hodge-theoretic aspects of the decomposition theorem,” Algebraic geometry—Seattle 2005. Part 2, 489–504, Proc. Sympos. Pure Math., 80, Part 2, Amer. Math. Soc., Providence, RI, 2009. MR 2483945 (2009m:14008)
- M.A. de Cataldo, L. Migliorini, “The perverse filtration and the Lefschetz hyperplane theorem,” arXiv:0805.4634, Annals of Math., Vol. 171, No. 3, (2010), 2089–2113. MR 2680404
- M. de Cataldo, “The standard filtration on cohomology with compact supports with an appendix on the base change map and the Lefschetz hyperplane theorem,” in Interactions of Classical and Numerical Algebraic Geometry, 199–220, Contemporary Mathematics, 496 (2009), Amer. Math. Soc., Providence, RI, 2009. MR 2555955
- M. de Cataldo, L. Migliorini, “The decomposition theorem, perverse sheaves and the topology of algebraic maps”, Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 4, 535–633. MR 2525735
- P. Deligne, “Théorème de finitude en cohomologie $l$-adique," LNM 569, 233–261 (1977).
- P. Deligne, “Théorie de Hodge, III,” Publ. Math. IHES 44 (1974), 5–78. MR 0498552 (58:16653b)
- P. Deligne, “Décompositions dans la catégorie Dérivée,” Motives (Seattle, WA, 1991), 115–128, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994. MR 1265526 (95h:18013)
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- M. Kashiwara, P. Schapira, Sheaves on manifolds, Grundlehren der mathematischen Wissenschaften. Vol. 292, Springer-Verlag, Berlin Heidelberg, 1990. MR 1074006 (92a:58132)
- M.V. Nori, Constructible sheaves, in Algebra, arithmetic and geometry, Part I, II (Mumbai, 2000), pp. 471–491, Tata Inst. Fund. Res. Stud. Math., 16, Bombay, 2002. MR 1940678 (2003m:14027)
- M. Saito, “Mixed Hodge modules,” Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, 221–333. MR 1047415 (91m:14014)
- M. Saito, “Mixed Hodge complexes on algebraic varieties,” Math. Ann. 316 (2000), 283–331. MR 1741272 (2002h:14012)
- J. Schürmann, Topology of Singular Spaces and Constructible Sheaves, Monografie Matematyczne, Vol. 63, Birkhäuser, 2003. MR 2031639 (2005f:32053)
Additional Information
Mark Andrea A. de Cataldo
Affiliation:
Department of Mathematics, Stony Brook University, Stony Brook, New York 11794
Email:
mde@math.sunysb.edu
Received by editor(s):
June 16, 2009
Published electronically:
May 16, 2011
Additional Notes:
The author was supported in part by NSA and NSF grants.
Dedicated:
This paper is dedicated to the memory of Prakob Monkolchayut.
