A duality approach to the symmetry of Bernstein–Sato polynomials of free divisors

Abstract

In this paper we prove that the Bernstein-Sato polynomial of any free divisor for which the D[s]-module D[s]h s admits a Spencer logarithmic resolution satisfies the symmetry property b(−s − 2) = ±b(s). This applies in particular to locally quasi-homogeneous free divisors (for instance, to free hyperplane arrangements), or more generally, to free divisors of linear Jacobian type. We also prove that the Bernstein-Sato polynomial of an integrable logarithmic connection E and of its dual E * with respect to a free divisor of linear Jacobian type are related by the equality b E (s) = ±b E * (−s − 2). Our results are based on the behaviour of the modules D[s]h s and D[s]E[s]h s under duality.

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

“…For reductive linear free divisors, [97,204] discuss symmetry properties of Bernstein-Sato polynomials. In [162] this theme is taken up again, investigating specifically symmetry properties of ρ f when D[s] • f s has a Spencer logarithmic resolution (see [66] for definitions). This covers locally quasi-homogeneous free divisors, and more generally free divisors whose Jacobian is of linear type.…”

Chern classes of logarithmic derivations for free divisors with Jacobian ideal of linear type

“…Later, Calderón-Moreno and Narváez-Macarro studied the LCT problem for integrable logarithmic connections with respect to a free divisor with Jacobian ideal of linear type [CMNM09]. More recently, Narváez-Macarro was able to show divisors with Jacobian ideal of linear type also satisfy LCT [NM15]. In fact, in our previous approach to Question 1.1 [Lia], LCT was the key to allow us to compare the degrees of the classes in (1).…”

Twisted logarithmic complexes of positively weighted homogeneous divisors

“…, x n of X with center p such that D ⊂ X is locally defined by a weighted homogeneous polynomial f p of strictly positive weights w p,i , that is, f p is a linear combination of monomials n i=1 x ν i i with n i=1 w p,i ν i = 1. We say that D is everywhere positively weighted homogeneous if the above condition is satisfied at any p ∈ D. (This is called locally (or strongly) quasi-homogeneous in [CNM96], [CN09], [Na15], etc. )…”

“…So Corollary 3 may be viewed as a partial generalization of [HM98]. Note that the last hypothesis on integral roots cannot be satisfied in the case of free divisors or hyperplane arrangements by [Na15], [Wa05] (or [Sa16a]), and a quasi-isomorphism holds in these cases (see [CNM96], [Ba22]).…”