Hodge numbers attached to a polynomial map

López, R. García; Némethi, A.

Annales de l'Institut Fourier, Tome 49 (1999) no. 5, pp. 1547-1579.

Nous associons une structure de Hodge mixte à toute application $f:{ℂ}^n\to ℂ$. Les nombres de Hodge équivariants de cette structure de Hodge mixte sont des invariants de $f$ qui reflètent son comportement à l’infini. Nous les calculons pour une classe générique de polynômes en termes de nombres de Hodge équivariants associés aux singularités isolées d’hypersurface et des nombres de Hodge équivariants des revêtements cycliques de l’espace projectif, ramifiés le long d’une hypersurface. Nous montrons que ces invariants permettent de déterminer des invariants topologiques de $f$ tels que la forme réelle de Seifert à l’infini.

We attach a limit mixed Hodge structure to any polynomial map $f:{ℂ}^n\to ℂ$. The equivariant Hodge numbers of this mixed Hodge structure are invariants of $f$ which reflect its asymptotic behaviour. We compute them for a generic class of polynomials in terms of equivariant Hodge numbers attached to isolated hypersurface singularities and equivariant Hodge numbers of cyclic coverings of projective space branched along a hypersurface. We show how these invariants allow to determine topological invariants of $f$ such as the real Seifert form at infinity.

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