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Hodge type of projective varieties of low degree. (English) Zbl 0784.14003

The paper is devoted to the proof of the following result [conjectured by P. Deligne and A. Dimca in Ann. Sci. Éc. Norm. Supér., IV. Sér. 23, No. 4, 645-656 (1990; Zbl 0743.14028)]: Let \(S\) be a complex projective variety defined in the \(n\)-dimensional projective space by equations of degrees \(d_ 1\geq\cdots\geq d_ r\). Then the Hodge type of \(S\) is at least the integral part of \[ (n-d_ 2-\cdots-d_ r)/d_ 1. \] Two proofs of this fact are given: one based on Nori’s deformation theoretic technique and another inspired from T. Terasoma’s use of the “universal pencil” [cf. Ann. Math., II. Ser. 132, 213-235 (1990; Zbl 0732.14005)].

MSC:

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14C34 Torelli problem

Keywords:

Hodge type
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References:

[1] Deligne, P.: Cohomologie des intersections complètes. In: Deligne, P., Katz, N. (eds.) SGA 7 XI. (Lect. Notes Math., vol. 340) Berlin Heidelberg New York: Springer 1973 · Zbl 0265.14007
[2] Deligne, P., Dimca, A.: Filtrations de Hodge et par l’ordre du pôle pour les hypersurfaces singulières, Ann. Sci. Éc. Norm. Supér.23, 645-665 (1990) · Zbl 0743.14028
[3] Esnault, H.: Hodge type of subvarieties ofP n of small degrees. Math. Ann.288, 549-551 (1990) · Zbl 0755.14006 · doi:10.1007/BF01444548
[4] Nori, M.V: Algebraic cycles and Hodge theoretic connectivity results. (Preprint) · Zbl 0822.14008
[5] Terasoma, T.: Infinitesimal variation of Hodge structures and the weak global Torelli theorem for complete intersections. Ann. Math.132, 213-235 (1990) · Zbl 0732.14005 · doi:10.2307/1971522
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