{
  "id": "1301.7589",
  "version": "v1",
  "published": "2013-01-31T12:25:43.000Z",
  "updated": "2013-01-31T12:25:43.000Z",
  "title": "Asymptotics of a vanishing period : characterization of semi-simplicity",
  "authors": [
    "Daniel Barlet"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:1201.2757",
  "categories": [
    "math.AG",
    "math.CV"
  ],
  "abstract": "In this paper we introduce the word {\\em fresco} to denote a monogenic geometric (a,b)-module. This \"basic object\" (generalized Brieskorn module with one generator) corresponds to the formal germ of the minimal filtered (regular) differential equation. Such an equation is satisfied by a relative de Rham cohomology class at a critical value of a holomorphic function on a smooth complex manifold. In [B.09] the first structure theorems are proved. Then in [B.10] we introduced the notion of {\\em theme} which corresponds in the \\ $[\\lambda]-$primitive case to frescos having a unique Jordan-H{\\\"o}lder sequence (a unique Jordan block for the monodromy). Themes correspond to asymptotic expansion of a given vanishing period, so to an image of a fresco in the module of asymptotic expansions. For a fixed relative de Rham cohomology class (for instance given by a smooth differential form $d-$closed and $df-$closed) each choice of a vanishing cycle in the spectral eigenspace of the monodromy for the eigenvalue \\ $exp(2i\\pi.\\lambda)$ \\ produces a \\ $[\\lambda]-$primitive theme, which is a quotient of the fresco associated to the given relative de Rham class itself. \\\\ We show that for any fresco there exists an {\\em unique} Jordan-H{\\\"o}lder sequence, called the {\\em principal J-H. sequence}, with corresponding quotients giving the opposite of the roots of the Bernstein polynomial in increasing order. We study the semi-simple part of a given fresco and we characterize the semi-simplicity of a fresco by the fact for any given order on the roots of its Bernstein polynomial we may find a J-H. sequence making them appear with this order. Then we construct a numerical invariant, called the \\ $\\beta-$invariant, and we show that it produces numerical criteria in order to give a necessary and sufficient condition on a fresco to be semi-simple. We show that these numerical invariants define a natural algebraic stratification on the set of isomorphism classes of fresco with given fundamental invariants (or equivalently with given roots of the Bernstein polynomial).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-01-31T12:25:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vanishing period",
      "bernstein polynomial",
      "rham cohomology class",
      "semi-simplicity",
      "asymptotic expansion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1301.7589B"
    }
  }
}