Vanishing cycle sheaves of one-parameter smoothings and quasi-semistable degenerations

Dimca, Alexandru; Saito, Morihiko

doi:10.1090/S1056-3911-2011-00572-9

Authors: Alexandru Dimca and Morihiko Saito
Journal: J. Algebraic Geom. 21 (2012), 247-271
DOI: https://doi.org/10.1090/S1056-3911-2011-00572-9
Published electronically: April 18, 2011
MathSciNet review: 2877434
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Abstract | References | Additional Information

Abstract: We study the vanishing cycles of a one-parameter smoothing of a complex analytic space and show that the weight filtration on its perverse cohomology sheaf of the highest degree is quite close to the monodromy filtration so that its graded pieces have a modified Lefschetz decomposition. We describe its primitive part using the weight filtration on the perverse cohomology sheaves of the constant sheaves. As a corollary we show in the local complete intersection case that 1 is not an eigenvalue of the monodromy on the reduced Milnor cohomology at any points if and only if the total space and the singular fiber are both rational homology manifolds. Also, we introduce quasi-semistable degenerations and calculate the limit mixed Hodge structure by constructing the weight spectral sequence. As a corollary we show non-triviality of the space of vanishing cycles of the Lefschetz pencil associated with a tensor product of any two very ample line bundles except for the case of even-dimensional projective space where two has to be replaced by three.

References

D. Barlet and H.-M. Maire, Poles of the current $|f|^{2\lambda }$ over an isolated singularity, Internat. J. Math. 11 (2000), no. 5, 609–635. MR 1780733, DOI https://doi.org/10.1142/S0129167X00000313
D. Barlet and H.-M. Maire, Non-trivial simple poles at negative integers and mass concentration at singularity, Math. Ann. 323 (2002), no. 3, 547–587. MR 1923697, DOI https://doi.org/10.1007/s002080100315
A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
Patrick Brosnan, Hao Fang, Zhaohu Nie, and Gregory Pearlstein, Singularities of admissible normal functions, Invent. Math. 177 (2009), no. 3, 599–629. With an appendix by Najmuddin Fakhruddin. MR 2534102, DOI https://doi.org/10.1007/s00222-009-0191-9
Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57 (French). MR 498551

Deligne, P., Le formalisme des cycles évanescents, in SGA7 XIII and XIV, Lect. Notes in Math. 340, Springer, Berlin, 1973, pp. 82–115 and 116–164.

Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252 (French). MR 601520

Pierre Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245–274. MR 382702, DOI https://doi.org/10.1007/BF01389853

Alan H. Durfee and Morihiko Saito, Mixed Hodge structures on the intersection cohomology of links, Compositio Math. 76 (1990), no. 1-2, 49–67. Algebraic geometry (Berlin, 1988). MR 1078857

Mark Goresky and Robert MacPherson, Intersection homology theory, Topology 19 (1980), no. 2, 135–162. MR 572580, DOI https://doi.org/10.1016/0040-9383%2880%2990003-8

Mark Green, Phillip Griffiths, and Matt Kerr, Néron models and limits of Abel-Jacobi mappings, Compos. Math. 146 (2010), no. 2, 288–366. MR 2601630, DOI https://doi.org/10.1112/S0010437X09004400

Phillip A. Griffiths, On the periods of certain rational integrals. I, II, Ann. of Math. (2) 90 (1969), 460-495; ibid. (2) 90 (1969), 496–541. MR 0260733, DOI https://doi.org/10.2307/1970746

Phillip Griffiths and Joe Harris, On the Noether-Lefschetz theorem and some remarks on codimension-two cycles, Math. Ann. 271 (1985), no. 1, 31–51. MR 779603, DOI https://doi.org/10.1007/BF01455794

Helmut Hamm, Lokale topologische Eigenschaften komplexer Räume, Math. Ann. 191 (1971), 235–252 (German). MR 286143, DOI https://doi.org/10.1007/BF01578709

Katz, N., Etude cohomologique des pinceaux de Lefschetz, in Lect. Notes in Math., vol. 340, Springer Berlin, 1973, pp. 254–327.

Klaus Lamotke, The topology of complex projective varieties after S. Lefschetz, Topology 20 (1981), no. 1, 15–51. MR 592569, DOI https://doi.org/10.1016/0040-9383%2881%2990013-6

Lefschetz, S., L’analysis situs et la géométrie algébrique, Gauthier-Villars, Paris, 1924.

David B. Massey, Intersection cohomology, monodromy and the Milnor fiber, Internat. J. Math. 20 (2009), no. 4, 491–507. MR 2515051, DOI https://doi.org/10.1142/S0129167X0900539X

John W. Morgan, The algebraic topology of smooth algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 48 (1978), 137–204. MR 516917

Ania Otwinowska and Morihiko Saito, Monodromy of a family of hypersurfaces containing a given subvariety, Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 3, 365–386 (English, with English and French summaries). MR 2166338, DOI https://doi.org/10.1016/j.ansens.2005.03.003

Morihiko Saito, Hodge structure via filtered ${\scr D}$-modules, Astérisque 130 (1985), 342–351. Differential systems and singularities (Luminy, 1983). MR 804062

Morihiko Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), no. 6, 849–995 (1989) (French). MR 1000123, DOI https://doi.org/10.2977/prims/1195173930

Morihiko Saito, Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, 221–333. MR 1047415, DOI https://doi.org/10.2977/prims/1195171082

Morihiko Saito, On microlocal $b$-function, Bull. Soc. Math. France 122 (1994), no. 2, 163–184 (English, with English and French summaries). MR 1273899

Wilfried Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211–319. MR 382272, DOI https://doi.org/10.1007/BF01389674

Joseph Steenbrink, Limits of Hodge structures, Invent. Math. 31 (1975/76), no. 3, 229–257. MR 429885, DOI https://doi.org/10.1007/BF01403146

J. H. M. Steenbrink, Monodromy and weight filtration for smoothings of isolated singularities, Compositio Math. 97 (1995), no. 1-2, 285–293. Special issue in honour of Frans Oort. MR 1355129

Tristan Torrelli, Intersection homology $\scr D$-module and Bernstein polynomials associated with a complete intersection, Publ. Res. Inst. Math. Sci. 45 (2009), no. 2, 645–660. MR 2510514, DOI https://doi.org/10.2977/prims/1241553132

Additional Information

Alexandru Dimca
Affiliation: Laboratoire J.A. Dieudonné, UMR du CNRS 6621, Université de Nice-Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France
MR Author ID: 58125
Email: Alexandru.DIMCA@unice.fr

Morihiko Saito
Affiliation: RIMS Kyoto University, Kyoto 606-8502 Japan
Email: msaito@kurims.kyoto-u.ac.jp

Received by editor(s): April 3, 2009
Received by editor(s) in revised form: March 22, 2010
Published electronically: April 18, 2011
Additional Notes: This work is partially supported by Kakenhi 19540023 and by ANR-08-BLAN-0317-02 (SEDIGA). Additionally, A. Dimca is grateful to ASSMS, Government College University, Lahore, Pakistan, where part of the work on this paper was done.