The Jacobian module, the Milnor fiber, and the $D$-module generated by $f^s$

Uli Walther

Published 2015-04-27Version 1

For a germ $f$ on a complex manifold $X$, we introduce a complex derived from the Liouville form acting on logarithmic differential forms, and give an exactness criterion. We use this logarithmic complex to connect properties of the $D$-module generated by $f^s$ to homological data of the Jacobian ideal; specifically in many cases we show that the annihilator of $f^s$ is generated by derivations. Moreover, through local cohomology, we connect the cohomology of the Milnor fiber to the Jacobian module through logarithmic differentials under blow-ups. In particular, we consider (not necessarily reduced) hyperplane arrangements: we prove many cases of the conjecture of Terao on the annihilator of $1/f$; we disprove a corresponding conjecture on the annihilator of $f^s$; we show that the Bernstein--Sato polynomial of an arrangement is not determined by its intersection lattice; we prove that arrangements that satisfy Terao's conjecture fulfill the Strong Monodromy Conjecture, and that this includes as very special cases all arrangements of Coxeter and of crystallographic type, and all multi-arrangements in dimension 3.