Monodromy at Infinity and the Weights of Cohomology

2000/02/25 by Alexandru Dimca, Dimca, Alexandru, Morihiko Saito +1
Mathematics · #32S40 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #math.AG #msc:32S40

paper · pdf · doi:10.48550/arxiv.math/0002214

AMS-TeX, 15 pages; the cohomologically tame case added

openalex publication_date 2000/02/25 · arxiv created 2001/11/29 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Citations

• On the monodromy of complex polynomials

Related

• On the sizes of the Jordan blocks of monodromies at infinity
• The monodromy theorem for compact Kähler manifolds and smooth quasi-projective varieties
• Weight filtration of the limit mixed Hodge structure at infinity for tame polynomials
• Monodromy at infinity of polynomial maps and Newton polyhedra (with Appendix by C. Sabbah)
• Weight-monodromy conjecture over equal characteristic local fields