On the monodromy of complex polynomials

2001/06/01 by Alexandru Dimca, András Némethi · 3 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry and Number Theory

paper · doi:10.1215/s0012-7094-01-10821-1

Abstract

Consider a polynomial function f : ℂn→ℂ with generic fiber F. Let Bf be the bifurcation set of f; hence f induces a smooth locally trivial fibration over ℂ\Bf. Then, for any integer q≥0 and any coefficient ring R, there is an associated monodromy representation ρ(f)_q : π₁(\mathbb C\backslash B_f,\rm pt)→ \rm Aut( H_q(F,R)) in (reduced) homology. Going around a circle in ℂ large enough to contain all of the bifurcation set gives rise to the monodromy operators at infinity, which we denote by M∞(f)q. We show that these monodromy operators at infinity and a certain natural direct sum decomposition of the homology of F in terms of vanishing cycles determine the monodromy representation. The role played by this decomposition is crucial since there are examples of polynomials ℂ2→ℂ having distinct complex monodromy representations but whose monodromy operators at infinity have the same Jordan normal form.

Cited by

• Hodge theory on Alexander invariants -- a survey
• Mixed Hodge Structures on Alexander Modules
• Mixed Hodge structure of affine hypersurfaces
• Monodromy at Infinity and the Weights of Cohomology

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