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Number of Jordan blocks of the maximal size for local monodromies

Part of: Singularities

Published online by Cambridge University Press:  11 February 2014

Alexandru Dimca  and
Morihiko Saito
Alexandru Dimca
Affiliation:
Université Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, 06100 Nice, France email Alexandru.DIMCA@unice.fr
Morihiko Saito
Affiliation:
RIMS Kyoto University, Kyoto 606-8502, Japan email msaito@kurims.kyoto-u.ac.jp
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Abstract

We prove formulas for the number of Jordan blocks of the maximal size for local monodromies of one-parameter degenerations of complex algebraic varieties where the bound of the size comes from the monodromy theorem. In the case when the general fibers are smooth and compact, the proof calculates some part of the weight spectral sequence of the limit mixed Hodge structure of Steenbrink. In the singular case, we can prove a similar formula for the monodromy on the cohomology with compact supports, but not on the usual cohomology. We also show that the number can really depend on the position of singular points in the embedded resolution, even in the isolated singularity case, and hence there are no simple combinatorial formulas using the embedded resolution in general.

Keywords

local monodromylimit mixed Hodge structurenearby cycles

MSC classification

Secondary: 32S40: Monodromy; relations with differential equations and $D$-modules
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 3 , March 2014 , pp. 344 - 368
Copyright
© The Author(s) 2013 

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