Hodge numbers attached to a polynomial map
Annales de l'Institut Fourier, Tome 49 (1999) no. 5, pp. 1547-1579.

Nous associons une structure de Hodge mixte à toute application f: n . 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: n . 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.

@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://www.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://www.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://www.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, Tome 49 (1999) no. 5, pp. 1547-1579. doi : 10.5802/aif.1729. http://www.numdam.org/articles/10.5802/aif.1729/

[1] V. Arnold, A. Varchenko and S. Goussein-Zadé, Singularités des Applications Différentiable, 2e partie, Éditions Mir Moscou, 1986.

[2] E. Cattani and A. Kaplan, Polarized mixed Hodge structures and the local monodromy of a variation of Hodge structure, Invent. Math., 67 (1982), 101-115. | EuDML | MR | Zbl

[3] A. Dimca, Singularities and Topology of Hypersurfaces, Universitext, Springer Verlag, 1992. | MR | Zbl

[4] A. Dimca, Hodge Numbers of Hypersurfaces, Abh. Math. Sem. Univ. Hamburg, 66 (1996), 377-386. | MR | Zbl

[5] R. García López and A. Némethi, 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

[6] R. García López and A. Némethi, On the monodromy at infinity of a polynomial map, II, Compos. Math., 115 (1999), 1-20. | MR | Zbl

[7] P. Griffiths, On the periods of certain rational integrals, I, II, Annals of Math., 90 (1987), 460-541. | MR | Zbl

[8] H. A. Hamm, 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] V. Navarro Aznar, Sur la théorie de Hodge-Deligne, Invent. Math., 90 (1987), 11-76. | EuDML | MR | Zbl

[10] A. Némethi, The real Seifert form and the spectral pairs of isolated hypersurface singularities, Compos. Math., 98 (1995), 23-41. | Numdam | MR | Zbl

[11] A. Némethi, On the Seifert form at infinity associated with polynomial maps, J. Math. Soc. Japan, 51 (1999), 63-70. | MR | Zbl

[12] A. Némethi, The semi-ring structure and the spectral pairs of sesqui-linear forms, Algebra Colloq., 1 (1994), 85-95. | MR | Zbl

[13] A. Némethi, 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] A. Némethi and C. Sabbah, Semicontinuity of the spectrum at infinity, preprint. | Zbl

[15] F. Pham, Vanishing homologies and the n variable saddlepoint method, In Proc. Symp. Pure Math., vol. 40 (1983), 319-333. | MR | Zbl

[16] C. Sabbah, Hypergeometric periods for a tame polynomial, preprint. | Zbl

[17] M. Saito, Mixed Hodge modules, Publ. RIMS Kyoto Univ., 26 (1990), 221-333. | MR | Zbl

[18] J. Scherk and J.H.M. Steenbrink, On the Mixed Hodge Structure on the Cohomology of the Milnor Fibre, Math. Ann., 271 (1985), 641-665. | MR | Zbl

[19] W. Schmid, Variation of Hodge structures : the singularities of the period mapping, Invent. Math., 22 (1973), 211-319. | MR | Zbl

[20] J.H.M. Steenbrink, Limits of Hodge Structures, Inv. Math., 31 (1976), 229-257. | MR | Zbl

[21] J.H.M. Steenbrink, Intersection form for quasi-homogeneous singularities, Compos. Math., 34 (1977), 211-223. | Numdam | MR | Zbl

[22] J.H.M. Steenbrink, 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] J.H.M. Steenbrink, Mixed Hodge structures associated with isolated singularities, Proc. Symp. Pure Math., vol. 40 (1983), 513-536. | MR | Zbl

[24] J.H.M. Steenbrink and S. Zucker, Variation of mixed Hodge structure. I, Invent. Math., 80 (1985), 489-542. | MR | Zbl

[25] J.H.M. Steenbrink, Semicontinuity of the singularity spectrum, Invent. Math., 79 (1985), 557-565. | MR | Zbl

Cité par Sources :