On the monodromy of complex polynomials

Abstract

Consider a polynomial function f : ℂⁿ→ℂ with generic fiber F. Let B_f be the bifurcation set of f; hence f induces a smooth locally trivial fibration over ℂ\B_f. Then, for any integer q≥0 and any coefficient ring R, there is an associated monodromy representation

$\rho(f)_q : \pi_1(\mathbb {C}\backslash B_f,{\rm pt})\to {\rm Aut}(\tilde {H}_q(F,R))$

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.