Monodromy conjecture for log generic polynomials

Abstract

A log generic hypersurface in P n with respect to a birational modification of 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 = f1 . . . fp of polynomials, we show that the monodromy conjecture, relating the motivic zeta function with the complex monodromy, holds for the tuple (f1, . . . , fp, 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 (f1, . . . , fp, 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.