On a Generalized Monodromy Conjecture for Curves using Differential Forms

Motivic and topological zeta functions are singularity invariants, mainly associated to a function $f$ and a top differential form $\omega$ on a smooth variety. When $\omega$ is the standard form $dx_1\wedge \dots \wedge dx_n$ on affine $n$-space, the monodromy conjecture states that poles of these zeta functions should induce monodromy eigenvalues of $f$. We study natural generalized statements of the monodromy conjecture for functions $f$ on complex surface germs; more precisely on singular surfaces for forms $\omega$ that generalize the standard form, and on the affine plane for forms $\omega$ that are intrinsically associated to $f$. For all cases, we provide counterexamples to the statement. In addition, when the intrinsically associated $\omega$ is given by the generic polar of $f$, we discover a relation between the poles of the zeta functions and the intersection behaviour of the polar curve.