The Topological Monodromy Conjecture for hypersurfaces

Let $f\in\mathbf{C}[x_1,\ldots,x_n]$ be a non-constant polynomial with $f(0)=0$, let $\mu:Y\to\mathbf{C}^n$ be a log resolution of $f^{-1}(0)$, and write

$$Z_{\operatorname{top},f}(s)=\sum_{I\subseteq S}\chi(E_I^{\circ}\cap\mu^{-1}(0))\prod_{i\in I}\frac{1}{a_i s+k_i+1}$$

for its local topological zeta function. For $x\in f^{-1}(0)$, let $M_{f,x}$ be the Milnor fiber of $f$ at $x$, and let $b_f(s)$ be the Bernstein–Sato polynomial of $f$. Topological Monodromy Conjecture. If $c$ is a pole of $Z_{\operatorname{top},f}(s)$, then (a) $\exp(2\pi i c)$ is an eigenvalue of the monodromy action on $H^i(M_{f,x},\mathbf{C})$ for some $i$ and some $x\in f^{-1}(0)$, and (b) $b_f(c)=0$. The conjecture links poles of the topological zeta function with monodromy and Bernstein–Sato data; the paper proves the monodromy assertion for hyperplane arrangements and reduces the Bernstein–Sato assertion in that setting to the conjecture below.