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 $$f=f_1\ldots f_p$$ of polynomials, we show that the monodromy conjecture, relating the motivic zeta function with the complex monodromy, holds for the tuple $$(f_1,\ldots ,f_p,g)$$ 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 $$(f_1,\ldots ,f_p,g)$$ and for the product fg, if g is log very-generic. Even the case $$f=1$$ is intricate, the proof depending on nontrivial properties of Bernstein–Sato ideals, and it singles out the class of log (very-) generic hypersurfaces as an interesting class of singularities on its own.