On the Monodromy Conjecture for curves on normal surfaces

Abstract

This paper looks at some results concerning the Monodromy Conjecture. The conjecture states that for a nonconstant regular function $f$ on a surface germ $(S,0)$ with $f(0)=0$, if a rational number $s_\circ$ is a pole of the topological zeta function $Z_\mathrm{top}(f,s)$, then $e^{2\pi is_\circ}$ is an eigenvalue of the local monodromy of $f$ at some point of $f^{-1}\!\{0\}$. First we give our own, very elementary and conceptual, proof of the conjecture for the well-known case where the germ $(S,0)$ is nonsingular. This proof is not only included for its simplicity, but also because we will need exactly the same arguments in the second part. There we explain what can, and what cannot, be expected in the singular case.