Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

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


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: 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.


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

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

DOI: https://doi.org/10.1090/S1056-3911-2011-00572-9
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.

American Mathematical Society