Monodromy Conjecture for log generic polynomials

Abstract

A log generic hypersurface in ${\mathbb{P}}^n$ with respect to a birational modification of ${\mathbb{P}}^n$ is by definition the image of a generic element of a high power of an ample linear series on the modification. A log very-generic hypersurface is defined similarly but restricting to line bundles satisfying a non-resonance condition. Fixing a log resolution of a product $f=f_1\dots f_p$ of polynomials, we show that the monodromy conjecture, relating the motivic zeta function with the complex monodromy, holds for the tuple $(f_1,\dots ,f_p,g)$ and for the product $fg$, if $g$ is log generic. We also show that the stronger version of the monodromy conjecture, relating the motivic zeta function with the Bernstein-Sato ideal, holds for the tuple $(f_1,\dots ,f_p,g)$ and for the product $fg$, if $g$ is log very-generic.

v2: Added 1.9 - 1.11 and 6.3 - 6.7. To appear in Math. Z