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

Abstract

Abstract. 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 Liouville complex to connect properties of the D-module generated by f s to homological data of the Jacobian ideal; specifically we show that for a large class of germs the annihilator of f s is generated by derivations. Through local cohomology, we connect the cohomology of the Milnor fiber to the Jacobian module via logarithmic differentials.In particular, we consider (not necessarily reduced) hyperplane arrangements: we prove a conjecture of Terao on the annihilator of 1/f ; we confirm in many cases a corresponding conjecture on the annihilator of f s but we disprove it in general; we show that the Bernstein-Sato polynomial of an arrangement is not determined by its intersection lattice; we prove that arrangements for which the annihilator of f s is generated by derivations 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.

On fundamental groups of plane curve complements

“…In §1 we give an example of a rational cuspidal curve C such that the corresponding cohomology group H 1 (F ) = 0. This is related to a recent result by U. Walther [14] and the second named author's study of nearly free curves in P 2 [9].…”

Free divisors and rational cuspidal plane curves

“…Using a recent result by U. Walther in [30], we prove the claim (i) in Conjecture 1.1 for all curves of even degree, see Theorem 4.1, which is the our main result. The proof also implies that this conjecture holds for a curve C with an abelian fundamental group π 1 (P 2 \ C) or having a degree a prime power, see Corollary 4.2.…”

“…We have seen that for a reduced curve C : f = 0 the existentce of a resolution (1.1) is equivalent to the vanishing of the S-graded module N(f ) = I f /J f , called the Jacobian module in [30], and the curves satisfying these equivalent properties are called free. The class of curves introduced in this note is defined by imposing the condition that the Jacobian module N(f ) is non-zero, but as small as possible.…”

Computing the monodromy and pole order filtration on Milnor fiber cohomology of plane curves

“…where f s is a generic polynomial of degree d in x, y, z as in Example 7.1. (2) In the case of a line arrangement A : f = 0 in P 2 , the zero set R f is not determined by the combinatorics, see Walther [37] and Saito [32]. In fact, there is a pair of line arrangements A 1 : f 1 = 0 and A 2 : f 2 = 0 of degree d = 9, going back to Ziegler [38], having the same combinatorics but different sets R f and different Hilbert functions for their Milnor algebras M(f 1 ) and M(f 2 ).…”