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Monodromy at Infinity and the Weights of Cohomology

Published online by Cambridge University Press:  04 December 2007

Alexandru Dimca  and
Morihiko Saito
Alexandru Dimca
Affiliation:
Laboratoire d' Analyse et Géométrie, CNRS, Université Bordeaux I, 33405 Talence Cedex, France. e-mail: dimca@math.u-bordeaux.fr
Morihiko Saito
Affiliation:
RIMS Kyoto University, Kyoto 606-8502, Japan. e-mail: msaito@kurims.kyoto-u.ac.jp
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Abstract

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We show that for a polynomial map, the size of the Jordan blocks for the eigenvalue 1 of the monodromy at infinity is bounded by the multiplicity of the reduced divisor at infinity of a good compactification of a general fiber. The existence of such Jordan blocks is related to global invariant cycles of the graded pieces of the weight filtration. These imply some applications to period integrals. We also show that such a Jordan block of size greater than 1 for the graded pieces of the weight filtration is the restriction of a strictly larger Jordan block for the total cohomology group. If there are no singularities at infinity, we have a more precise statement on the monodromy.

Keywords

monodromy at infinitynearby cycleweight
Type
Research Article
Information
Compositio Mathematica , Volume 138 , Issue 1 , August 2003 , pp. 55 - 71
Copyright
© 2003 Kluwer Academic Publishers