Hodge numbers attached to a polynomial map

R. García López; A. Némethi

Annales de l'institut Fourier (1999)

  • Volume: 49, Issue: 5, page 1547-1579
  • ISSN: 0373-0956

Abstract

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We attach a limit mixed Hodge structure to any polynomial map f : ℂ n → 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.

How to cite

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López, R. García, and Némethi, A.. "Hodge numbers attached to a polynomial map." Annales de l'institut Fourier 49.5 (1999): 1547-1579. <http://eudml.org/doc/75393>.

@article{López1999,
abstract = {We attach a limit mixed Hodge structure to any polynomial map $f:\{\Bbb C\}^n\rightarrow \{\Bbb C\}$. 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.},
author = {López, R. García, Némethi, A.},
journal = {Annales de l'institut Fourier},
keywords = {mixed Hodge structures; polynomial maps; hypersurface singularities},
language = {eng},
number = {5},
pages = {1547-1579},
publisher = {Association des Annales de l'Institut Fourier},
title = {Hodge numbers attached to a polynomial map},
url = {http://eudml.org/doc/75393},
volume = {49},
year = {1999},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 49
IS - 5
SP - 1547
EP - 1579
AB - We attach a limit mixed Hodge structure to any polynomial map $f:{\Bbb C}^n\rightarrow {\Bbb C}$. 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.
LA - eng
KW - mixed Hodge structures; polynomial maps; hypersurface singularities
UR - http://eudml.org/doc/75393
ER -

References

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  1. [1] V. ARNOLD, A. VARCHENKO and S. GOUSSEIN-ZADÉ, Singularités des Applications Différentiable, 2e partie, Éditions Mir Moscou, 1986. 
  2. [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. Zbl0516.14005MR84a:32046
  3. [3] A. DIMCA, Singularities and Topology of Hypersurfaces, Universitext, Springer Verlag, 1992. Zbl0753.57001MR94b:32058
  4. [4] A. DIMCA, Hodge Numbers of Hypersurfaces, Abh. Math. Sem. Univ. Hamburg, 66 (1996), 377-386. Zbl0879.14018MR97h:14013
  5. [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. Zbl0855.32016MR97g:32047
  6. [6] R. GARCÍA LÓPEZ and A. NÉMETHI, On the monodromy at infinity of a polynomial map, II, Compos. Math., 115 (1999), 1-20. Zbl0947.32014MR2000a:32062
  7. [7] P. GRIFFITHS, On the periods of certain rational integrals, I, II, Annals of Math., 90 (1987), 460-541. Zbl0215.08103MR41 #5357
  8. [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. Zbl0926.32041
  9. [9] V. NAVARRO AZNAR, Sur la théorie de Hodge-Deligne, Invent. Math., 90 (1987), 11-76. Zbl0639.14002MR88j:32037
  10. [10] A. NÉMETHI, The real Seifert form and the spectral pairs of isolated hypersurface singularities, Compos. Math., 98 (1995), 23-41. Zbl0851.14015MR96i:32036
  11. [11] A. NÉMETHI, On the Seifert form at infinity associated with polynomial maps, J. Math. Soc. Japan, 51 (1999), 63-70. Zbl0933.32042MR2000a:32068
  12. [12] A. NÉMETHI, The semi-ring structure and the spectral pairs of sesqui-linear forms, Algebra Colloq., 1 (1994), 85-95. Zbl0814.11022MR95a:32058
  13. [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. Zbl0861.14007MR96i:32035
  14. [14] A. NÉMETHI and C. SABBAH, Semicontinuity of the spectrum at infinity, preprint. Zbl0973.32014
  15. [15] F. PHAM, Vanishing homologies and the n variable saddlepoint method, In Proc. Symp. Pure Math., vol. 40 (1983), 319-333. Zbl0519.49026MR85d:32026
  16. [16] C. SABBAH, Hypergeometric periods for a tame polynomial, preprint. Zbl0967.32028
  17. [17] M. SAITO, Mixed Hodge modules, Publ. RIMS Kyoto Univ., 26 (1990), 221-333. Zbl0727.14004MR91m:14014
  18. [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. Zbl0618.14002MR87b:32014
  19. [19] W. SCHMID, Variation of Hodge structures : the singularities of the period mapping, Invent. Math., 22 (1973), 211-319. Zbl0278.14003MR52 #3157
  20. [20] J.H.M. STEENBRINK, Limits of Hodge Structures, Inv. Math., 31 (1976), 229-257. Zbl0303.14002MR55 #2894
  21. [21] J.H.M. STEENBRINK, Intersection form for quasi-homogeneous singularities, Compos. Math., 34 (1977), 211-223. Zbl0347.14001MR56 #11995
  22. [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. Zbl0373.14007
  23. [23] J.H.M. STEENBRINK, Mixed Hodge structures associated with isolated singularities, Proc. Symp. Pure Math., vol. 40 (1983), 513-536. Zbl0515.14003MR85d:32044
  24. [24] J.H.M. STEENBRINK and S. ZUCKER, Variation of mixed Hodge structure. I, Invent. Math., 80 (1985), 489-542. Zbl0626.14007MR87h:32050a
  25. [25] J.H.M. STEENBRINK, Semicontinuity of the singularity spectrum, Invent. Math., 79 (1985), 557-565. Zbl0568.14021MR86h:32033

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