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

• Main Author: Macarro, Luis Narváez
• Format: Journal Article
• Language: English
• Published: 25-06-2015
• Subjects: Mathematics - Algebraic Geometry
• DOI: 10.48550/arxiv.1201.3594

Advances in Mathematics 281 (2015) 1242-1273 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 $b(-s-2) = \pm b(s)$. 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 $b_{E}(s)=\pm
b_{E^*}(-s-2)$. Our results are based on the behaviour of the modules $D[s]
h^s$ and $D[s] E[s]h^s $ under duality.