Quadratic forms and singularities of genus one or two

Dloussky, Georges

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 20 (2011) no. 1, pp. 15-69.

On étudie les singularités obtenues en contractant le diviseur maximal des surfaces (non kählerienne) qui contiennent des coquilles sphériques globales. Ces singularités sont de genre 1 ou 2, peuvent être $\mathbb{Q}$-Gorenstein, numériquement Gorenstein ou de Gorenstein. On définit une famille de polynômes qui dépendent de la configuration des courbes rationnelles pour calculer les discriminants des formes quadratiques associées à ces singularités. Un invariant topologique multiplicatif, défini à partir des arbres du graphe détermine le coefficient de torsion des 1-formes holomorphes tordues qui ne s’annulent pas sur le complémentaire du point singulier.

We study singularities obtained by the contraction of the maximal divisor in compact (non-kählerian) surfaces which contain global spherical shells. These singularities are of genus 1 or 2, may be $\mathbb{Q}$-Gorenstein, numerically Gorenstein or Gorenstein. A family of polynomials depending on the configuration of the curves computes the discriminants of the quadratic forms of these singularities. We introduce a multiplicative branch topological invariant which determines the twisting coefficient of a non-vanishing holomorphic 1-form on the complement of the singular point.

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