$D$-Modules Generated by Rational Powers of Holomorphic Functions

Abstract

We prove some suffcient conditions in order that a root of the Bernstein–Sato polynomial contributes to a difference between certain $D$-modules generated by rational powers of a holomorphic function; for instance, this holds in the case of isolated singularities with semisimple Milnor monodromies. We then construct an example where a root does not contribute to a difference. This also solves an old open problem about the relation between the Milnor monodromy and the exponential of the residue of the Gauss–Manin connection on the saturation of the Brieskorn lattice. This shows that the structure of Brieskorn lattices can be more complicated than one might imagine.