Hodge numbers attached to a polynomial map
Annales de l'Institut Fourier, Volume 49 (1999) no. 5, p. 1547-1579
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.
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.
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {49},
     number = {5},
     year = {1999},
     pages = {1547-1579},
     doi = {10.5802/aif.1729},
     mrnumber = {2001i:32045},
     zbl = {0944.32029},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/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. https://aif.centre-mersenne.org/item/AIF_1999__49_5_1547_0/

[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. | MR 84a:32046 | Zbl 0516.14005

[3] A. Dimca, Singularities and Topology of Hypersurfaces, Universitext, Springer Verlag, 1992. | MR 94b:32058 | Zbl 0753.57001

[4] A. Dimca, Hodge Numbers of Hypersurfaces, Abh. Math. Sem. Univ. Hamburg, 66 (1996), 377-386. | MR 97h:14013 | Zbl 0879.14018

[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. | Numdam | MR 97g:32047 | Zbl 0855.32016

[6] R. García López and A. Némethi, On the monodromy at infinity of a polynomial map, II, Compos. Math., 115 (1999), 1-20. | MR 2000a:32062 | Zbl 0947.32014

[7] P. Griffiths, On the periods of certain rational integrals, I, II, Annals of Math., 90 (1987), 460-541. | MR 41 #5357 | Zbl 0215.08103

[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 0926.32041

[9] V. Navarro Aznar, Sur la théorie de Hodge-Deligne, Invent. Math., 90 (1987), 11-76. | MR 88j:32037 | Zbl 0639.14002

[10] A. Némethi, The real Seifert form and the spectral pairs of isolated hypersurface singularities, Compos. Math., 98 (1995), 23-41. | Numdam | MR 96i:32036 | Zbl 0851.14015

[11] A. Némethi, On the Seifert form at infinity associated with polynomial maps, J. Math. Soc. Japan, 51 (1999), 63-70. | MR 2000a:32068 | Zbl 0933.32042

[12] A. Némethi, The semi-ring structure and the spectral pairs of sesqui-linear forms, Algebra Colloq., 1 (1994), 85-95. | MR 95a:32058 | Zbl 0814.11022

[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 96i:32035 | Zbl 0861.14007

[14] A. Némethi and C. Sabbah, Semicontinuity of the spectrum at infinity, preprint. | Zbl 0973.32014

[15] F. Pham, Vanishing homologies and the n variable saddlepoint method, In Proc. Symp. Pure Math., vol. 40 (1983), 319-333. | MR 85d:32026 | Zbl 0519.49026

[16] C. Sabbah, Hypergeometric periods for a tame polynomial, preprint. | Zbl 0967.32028

[17] M. Saito, Mixed Hodge modules, Publ. RIMS Kyoto Univ., 26 (1990), 221-333. | MR 91m:14014 | Zbl 0727.14004

[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 87b:32014 | Zbl 0618.14002

[19] W. Schmid, Variation of Hodge structures : the singularities of the period mapping, Invent. Math., 22 (1973), 211-319. | MR 52 #3157 | Zbl 0278.14003

[20] J.H.M. Steenbrink, Limits of Hodge Structures, Inv. Math., 31 (1976), 229-257. | MR 55 #2894 | Zbl 0312.14007

[21] J.H.M. Steenbrink, Intersection form for quasi-homogeneous singularities, Compos. Math., 34 (1977), 211-223. | Numdam | MR 56 #11995 | Zbl 0347.14001

[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 0373.14007

[23] J.H.M. Steenbrink, Mixed Hodge structures associated with isolated singularities, Proc. Symp. Pure Math., vol. 40 (1983), 513-536. | MR 85d:32044 | Zbl 0515.14003

[24] J.H.M. Steenbrink and S. Zucker, Variation of mixed Hodge structure. I, Invent. Math., 80 (1985), 489-542. | MR 87h:32050a | Zbl 0626.14007

[25] J.H.M. Steenbrink, Semicontinuity of the singularity spectrum, Invent. Math., 79 (1985), 557-565. | MR 86h:32033 | Zbl 0568.14021