We attach a limit mixed Hodge structure to any polynomial map . The equivariant Hodge numbers of this mixed Hodge structure are invariants of 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 such as the real Seifert form at infinity.
Nous associons une structure de Hodge mixte à toute application . Les nombres de Hodge équivariants de cette structure de Hodge mixte sont des invariants de 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 tels que la forme réelle de Seifert à l’infini.
@article{AIF_1999__49_5_1547_0, author = {L\'opez, R. Garc{\'\i}a and N\'emethi, A.}, title = {Hodge numbers attached to a polynomial map}, journal = {Annales de l'Institut Fourier}, pages = {1547--1579}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {49}, number = {5}, year = {1999}, doi = {10.5802/aif.1729}, mrnumber = {2001i:32045}, zbl = {0944.32029}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.1729/} }
TY - JOUR AU - López, R. García AU - Némethi, A. TI - Hodge numbers attached to a polynomial map JO - Annales de l'Institut Fourier PY - 1999 SP - 1547 EP - 1579 VL - 49 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.1729/ DO - 10.5802/aif.1729 LA - en ID - AIF_1999__49_5_1547_0 ER -
%0 Journal Article %A López, R. García %A Némethi, A. %T Hodge numbers attached to a polynomial map %J Annales de l'Institut Fourier %D 1999 %P 1547-1579 %V 49 %N 5 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.1729/ %R 10.5802/aif.1729 %G en %F AIF_1999__49_5_1547_0
López, R. García; Némethi, A. Hodge numbers attached to a polynomial map. Annales de l'Institut Fourier, Volume 49 (1999) no. 5, pp. 1547-1579. doi : 10.5802/aif.1729. http://archive.numdam.org/articles/10.5802/aif.1729/
[1] Singularités des Applications Différentiable, 2e partie, Éditions Mir Moscou, 1986.
, and ,[2] Polarized mixed Hodge structures and the local monodromy of a variation of Hodge structure, Invent. Math., 67 (1982), 101-115. | EuDML | MR | Zbl
and ,[3] Singularities and Topology of Hypersurfaces, Universitext, Springer Verlag, 1992. | MR | Zbl
,[4] Hodge Numbers of Hypersurfaces, Abh. Math. Sem. Univ. Hamburg, 66 (1996), 377-386. | MR | Zbl
,[5] On the monodromy at infinity of a polynomial map, Compos. Math., 100:205-231, 1996. Appendix by R. García López and J. Steenbrink. | EuDML | Numdam | MR | Zbl
and ,[6] On the monodromy at infinity of a polynomial map, II, Compos. Math., 115 (1999), 1-20. | MR | Zbl
and ,[7] On the periods of certain rational integrals, I, II, Annals of Math., 90 (1987), 460-541. | MR | Zbl
,[8] Hodge numbers for isolated singularities of nondegenerate complete intersections, In Singularities (Oberwolfach, 1996), Progress in Math., 162, pp. 37-60. Birkhäuser, Basel, 1998. | Zbl
,[9] Sur la théorie de Hodge-Deligne, Invent. Math., 90 (1987), 11-76. | EuDML | MR | Zbl
,[10] The real Seifert form and the spectral pairs of isolated hypersurface singularities, Compos. Math., 98 (1995), 23-41. | Numdam | MR | Zbl
,[11] On the Seifert form at infinity associated with polynomial maps, J. Math. Soc. Japan, 51 (1999), 63-70. | MR | Zbl
,[12] The semi-ring structure and the spectral pairs of sesqui-linear forms, Algebra Colloq., 1 (1994), 85-95. | MR | Zbl
,[13] The mixed Hodge structure of a complete intersection with isolated singularity, C.R. Acad. Sci. Paris, t. 321, Série I (1995), 447-452. | MR | Zbl
,[14] Semicontinuity of the spectrum at infinity, preprint. | Zbl
and ,[15] Vanishing homologies and the n variable saddlepoint method, In Proc. Symp. Pure Math., vol. 40 (1983), 319-333. | MR | Zbl
,[16] Hypergeometric periods for a tame polynomial, preprint. | Zbl
,[17] Mixed Hodge modules, Publ. RIMS Kyoto Univ., 26 (1990), 221-333. | MR | Zbl
,[18] On the Mixed Hodge Structure on the Cohomology of the Milnor Fibre, Math. Ann., 271 (1985), 641-665. | MR | Zbl
and ,[19] Variation of Hodge structures : the singularities of the period mapping, Invent. Math., 22 (1973), 211-319. | MR | Zbl
,[20] Limits of Hodge Structures, Inv. Math., 31 (1976), 229-257. | MR | Zbl
,[21] Intersection form for quasi-homogeneous singularities, Compos. Math., 34 (1977), 211-223. | Numdam | MR | Zbl
,[22] Mixed Hodge structure on the vanishing cohomology. In Real and Complex Singularities, Oslo 1977, pages 397-403, Alphen a/d Rhijn, 1977, Sijthoff & Noordhoff. | Zbl
,[23] Mixed Hodge structures associated with isolated singularities, Proc. Symp. Pure Math., vol. 40 (1983), 513-536. | MR | Zbl
,[24] Variation of mixed Hodge structure. I, Invent. Math., 80 (1985), 489-542. | MR | Zbl
and ,[25] Semicontinuity of the singularity spectrum, Invent. Math., 79 (1985), 557-565. | MR | Zbl
,Cited by Sources: