{
  "id": "1201.3594",
  "version": "v4",
  "published": "2012-01-17T18:57:44.000Z",
  "updated": "2015-06-25T23:46:11.000Z",
  "title": "A duality approach to the symmetry of Bernstein-Sato polynomials of free divisors",
  "authors": [
    "Luis Narváez Macarro"
  ],
  "comment": "Final version",
  "journal": "Advances in Mathematics 281 (2015) 1242-1273",
  "doi": "10.1016/j.aim.2015.06.012",
  "categories": [
    "math.AG"
  ],
  "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 $b(-s-2) = \\pm b(s)$. 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.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-08-05T18:29:35.000Z",
      "comment": "Some statements and proofs on (logarithmic) Bernstein modules have been corrected and improved. We have added an Appendix with a proof of the commutation of the duality functor with the scalar extension associated with a map of Lie-Rinehart algebras. The open questions have been updated. Minor corrections and improvements. Comments are always welcome",
      "journal": null,
      "doi": null,
      "authors": [
        "Luis Narváez-Macarro"
      ]
    },
    {
      "version": "v4",
      "updated": "2015-06-25T23:46:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14F10",
      "32C38"
    ],
    "keywords": [
      "bernstein-sato polynomial",
      "duality approach",
      "linear jacobian type",
      "spencer logarithmic resolution satisfies",
      "free hyperplane arrangements"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier",
      "journal": "Adv. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.3594N"
    }
  }
}