Mixed Hodge structure of affine hypersurfaces
Annales de l'Institut Fourier, Volume 57 (2007) no. 3, p. 775-801

In this article we give an algorithm which produces a basis of the n-th de Rham cohomology of the affine smooth hypersurface f -1 (t) compatible with the mixed Hodge structure, where f is a polynomial in n+1 variables and satisfies a certain regularity condition at infinity (and hence has isolated singularities). As an application we show that the notion of a Hodge cycle in regular fibers of f is given in terms of the vanishing of integrals of certain polynomial n-forms in n+1 over topological n-cycles on the fibers of f. Since the n-th homology of a regular fiber is generated by vanishing cycles, this leads us to study Abelian integrals over them. Our result generalizes and uses the arguments of J. Steenbrink for quasi-homogeneous polynomials.

Dans cet article nous donnons un algorithme qui produit une base du n-ième groupe de cohomology de De Rham de l’hypersurface affine lisse f -1 (t) compatible avec la structure de Hodge mixte, où f est un polynôme en n+1 variables et satisfait une condition de régularité à l’infini (en particulier, il a des singularités isolées). Comme application nous montrons que la notion de cycle de Hodge dans une fibre régulière de f est donnée par l’annulation des intégrales de certaines n-formes polynomiales dans n+1 sur des n-cycles topologiques dans les fibres de f. Puisque l’homologie de degré n d’une fibre régulière est engendrée par les cycles évanescents, cela conduit à étudier des intégrales abéliennes obtenues en intégrant sur ceux-ci. Notre résultat généralise et utilise les arguments de J. Steenbrink pour les polynômes quasi-homogènes.

DOI : https://doi.org/10.5802/aif.2276
Classification:  14C30,  32S35
Keywords: Mixed Hodge structures of affine varieties, Gauss-Manin connection
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     author = {Movasati, Hossein},
     title = {Mixed Hodge structure of affine hypersurfaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {3},
     year = {2007},
     pages = {775-801},
     doi = {10.5802/aif.2276},
     mrnumber = {2336829},
     zbl = {1123.14007},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2007__57_3_775_0}
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Movasati, Hossein. Mixed Hodge structure of affine hypersurfaces. Annales de l'Institut Fourier, Volume 57 (2007) no. 3, pp. 775-801. doi : 10.5802/aif.2276. http://www.numdam.org/item/AIF_2007__57_3_775_0/

[1] ArnolʼD, V. I.; Guseĭn-Zade, S. M.; Varchenko, A. N. Singularities of differentiable maps. Vol. II, Birkhäuser Boston Inc., Boston, MA, Monographs in Mathematics, Tome 83 (1988) (Monodromy and asymptotics of integrals, Translated from the Russian by Hugh Porteous, Translation revised by the authors and James Montaldi) | MR 966191 | Zbl 0659.58002

[2] Borel, Armand; Haefliger, André La classe d’homologie fondamentale d’un espace analytique, Bull. Soc. Math. France, Tome 89 (1961), pp. 461-513 | Numdam | MR 149503 | Zbl 0102.38502

[3] Brieskorn, Egbert Die Monodromie der isolierten Singularitäten von Hyperflächen, Manuscripta Math., Tome 2 (1970), pp. 103-161 | Article | MR 267607 | Zbl 0186.26101

[4] Chéniot, D. Vanishing cycles in a pencil of hyperplane sections of a non-singular quasi-projective variety, Proc. London Math. Soc. (3), Tome 72 (1996) no. 3, pp. 515-544 | Article | MR 1376767 | Zbl 0851.14003

[5] Deligne, Pierre Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. (1971) no. 40, pp. 5-57 | Article | Numdam | MR 498551 | Zbl 0219.14007

[6] Deligne, Pierre; Milne, James S.; Ogus, Arthur; Shih, Kuang-Yen Hodge cycles, motives, and Shimura varieties, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 900 (1982) | MR 654325 | Zbl 0465.00010

[7] Dimca, Alexandru; Némethi, András On the monodromy of complex polynomials, Duke Math. J., Tome 108 (2001) no. 2, pp. 199-209 | Article | MR 1833390 | Zbl 1020.32022

[8] Dolgachev, Igor Weighted projective varieties, Group actions and vector fields (Vancouver, B.C., 1981), Springer, Berlin (Lecture Notes in Math.) Tome 956 (1982), pp. 34-71 | MR 704986 | Zbl 0516.14014

[9] El Zein, Fouad Théorie de Hodge des cycles évanescents, Ann. Sci. École Norm. Sup. (4), Tome 19 (1986) no. 1, pp. 107-184 | Numdam | MR 860812 | Zbl 0538.14003

[10] Gavrilov, Lubomir Petrov modules and zeros of Abelian integrals, Bull. Sci. Math., Tome 122 (1998) no. 8, pp. 571-584 | Article | MR 1668534 | Zbl 0964.32022

[11] Gavrilov, Lubomir The infinitesimal 16th Hilbert problem in the quadratic case, Invent. Math., Tome 143 (2001) no. 3, pp. 449-497 | Article | MR 1817642 | Zbl 0979.34024

[12] Green, M.; Murre, J.; Voisin, C. Algebraic cycles and Hodge theory, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1594 (1994) (Lectures given at the Second C.I.M.E. Session held in Torino, June 21–29, 1993, Edited by A. Albano and F. Bardelli) | MR 1335238

[13] Greuel, G. M.; Pfister, G.; Schönemann, H. Singular 2.0.4, A computer algebra system for polynomial computations, Centre for Computer Algebra, University of Kaiserslautern (2001) (http://www.singular.uni-kl.de/) | Zbl 0902.14040

[14] Griffiths, Phillip A. On the periods of certain rational integrals. I, II, Ann. of Math. (2) 90 (1969), 460-495; ibid. (2), Tome 90 (1969), pp. 496-541 | MR 260733 | Zbl 0215.08103

[15] Grothendieck, A. On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. (1966) no. 29, pp. 95-103 | Article | Numdam | MR 199194 | Zbl 0145.17602

[16] Kulikov, Vik. S.; Kurchanov, P. F. Complex algebraic varieties: periods of integrals and Hodge structures [ MR1060327 (91k:14010)], Algebraic geometry, III, Springer, Berlin (Encyclopaedia Math. Sci.) Tome 36 (1998), p. 1-217, 263–270 | MR 1602375 | Zbl 0881.14003

[17] Movasati, H.; Reiter, S. Hypergeometric series and Hodge cycles of four dimensional cubic hypersurfaces (to appear in Int. Jou. Number Theory, Vol 2, No 3, 2006) | Zbl 05123418

[18] Movasati, Hossein Calculation of mixed Hodge structures, Gauss-Manin connections and Picard-Fuchs equations (To appear in the proceeding of Sao Carlos Conference at CIRM, Brikhauser. Together with the Library foliation.lib written in Singular and available at http://www.impa.br/~hossein/) | Zbl 1117.14014

[19] Movasati, Hossein Abelian integrals in holomorphic foliations, Rev. Mat. Iberoamericana, Tome 20 (2004) no. 1, pp. 183-204 | MR 2076777 | Zbl 1055.37057

[20] Movasati, Hossein Center conditions: rigidity of logarithmic differential equations, J. Differential Equations, Tome 197 (2004) no. 1, pp. 197-217 | Article | MR 2030154 | Zbl 1049.32033

[21] Movasati, Hossein Relative cohomology with respect to a Lefschetz pencil, J. Reine Angew. Math. (2006) no. 594, pp. 175-199 | Article | MR 2248156 | Zbl 1101.32014

[22] Pink, R. Hodge structures over function fields (preprint 1997)

[23] Sabbah, Claude Hypergeometric period for a tame polynomial, C. R. Acad. Sci. Paris Sér. I Math., Tome 328 (1999) no. 7, pp. 603-608 | Article | MR 1679978 | Zbl 0967.32028

[24] Scherk, J.; Steenbrink, J. H. M. On the mixed Hodge structure on the cohomology of the Milnor fibre, Math. Ann., Tome 271 (1985) no. 4, pp. 641-665 | Article | MR 790119 | Zbl 0618.14002

[25] Schmid, Wilfried Variation of Hodge structure: the singularities of the period mapping, Invent. Math., Tome 22 (1973), pp. 211-319 | Article | MR 382272 | Zbl 0278.14003

[26] Schulze, Mathias Good bases for tame polynomials, J. Symbolic Comput., Tome 39 (2005) no. 1, pp. 103-126 | Article | MR 2168243 | Zbl 1128.32017

[27] Shioda, Tetsuji The Hodge conjecture for Fermat varieties, Math. Ann., Tome 245 (1979) no. 2, pp. 175-184 | Article | MR 552586 | Zbl 0403.14007

[28] Steenbrink, J. H. M. Mixed Hodge structure on the vanishing cohomology, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), Sijthoff and Noordhoff, Alphen aan den Rijn (1977), pp. 525-563 | MR 485870 | Zbl 0373.14007

[29] Steenbrink, Joseph Intersection form for quasi-homogeneous singularities, Compositio Math., Tome 34 (1977) no. 2, pp. 211-223 | Numdam | MR 453735 | Zbl 0347.14001

[30] Steenbrink, Joseph; Zucker, Steven Variation of mixed Hodge structure. I, Invent. Math., Tome 80 (1985) no. 3, pp. 489-542 | Article | MR 791673 | Zbl 0626.14007

[31] Usui, Sampei Effect of automorphisms on variation of Hodge structures, J. Math. Kyoto Univ., Tome 21 (1981) no. 4, pp. 645-672 | MR 637511 | Zbl 0497.14003

[32] Varčenko, A. N. Asymptotic Hodge structure on vanishing cohomology, Izv. Akad. Nauk SSSR Ser. Mat., Tome 45 (1981) no. 3, p. 540-591, 688 | MR 623350 | Zbl 0476.14002

[33] Zucker, Steven The Hodge conjecture for cubic fourfolds, Compositio Math., Tome 34 (1977) no. 2, pp. 199-209 | Numdam | MR 453741 | Zbl 0347.14005