×

zbMATH — the first resource for mathematics

Hodge classes on self-products of a variety with an automorphism. (English) Zbl 0663.14006
The author studies the Hodge conjecture for a special family of abelian varieties. He constructs first a Hodge structure on self-products of curves with an automorphism, then he constructs algebraic cycles the cohomology classes of which generate this Hodge structure.
He applies his results to prove the Hodge conjecture in the case of the Weil Hodge structure on abelian 4-folds with complex multiplication from the cyclotomic field of cubic roots of unity.
Reviewer: A.Papantonopoulou

MSC:
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14J35 \(4\)-folds
14K05 Algebraic theory of abelian varieties
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] Aoki, N. , On some arithmetic problems related to the Hodge cycles on the Fermat varieties , Math. Ann. 266 (1983) 23-54. · Zbl 0506.14030
[2] Atiyah, M.F. and Bott, R. , A Lefschetz fixed point formula for elliptic differential operators , Bull. of the A.M.S. 72 (1966) 245-250. · Zbl 0151.31801
[3] Arbarello, E. et al., Geometry of Algebraic Curves , Vol. 1, Grundlehren der math. Wissenschaften 267 , Springer, New York (1985). · Zbl 0559.14017
[4] Beauville, A. , Variétés de Prym et Jacobiennes Intermédiares , Ann. Scient. Éc. Norm. Sup., 4e serie, t. 10 (1977) 309-391. · Zbl 0368.14018
[5] Bloch, S. and Srinivas, V. , Remarks on correspondences and algebraic cycles , Amer. J. of Math. 105 (1983) 1235-1254. · Zbl 0525.14003
[6] Bott, R. and Tu, L. , Differential Forms in Algebraic Topology . Springer, New York (1982). · Zbl 0496.55001
[7] Conte, A. and Murre, J.P. , The Hodge conjecture for fourfolds admitting a covering by rational curves , Math. Ann. 238 (1978) 79-88. · Zbl 0373.14006
[8] Deligne, P. , Hodge cycles on Abelian varieties , Notes by J.S. Milne, in ” Hodge cycles, Motives, and Shimura varieties ”, Lect. Notes in Math., 900, Springer (1982). · Zbl 0537.14006
[9] Fulton, W. , Rational equivalence on singular varieties , Publ. Math. I.H.E.S. 45 (1975) 147-167. · Zbl 0332.14002
[10] Gross, B. , On the periods of Abelian integrals and a formula of Chowla and Selberg , Invent. Math. 45 (1978) 193-211. · Zbl 0418.14023
[11] Griffiths, P. and Harris, J. , Principles of Algebraic Geometry . John Wiley and Sons, New York (1978). · Zbl 0408.14001
[12] Gilkey, P. , Lefschetz fixed point formulas and the heat equation , in Partial differential equations and geometry, Proceedings of the Park City Conference , ed. C. Byrnes, Marcel Dekker Lecture Notes in Pure and Applied Mathematics 48 (1979) 91-147. · Zbl 0405.58044
[13] Hodge, W.V.D. , The topological invariants of algebraic varieties , Proc. Int. Cong. Math. 182-192 (1950). · Zbl 0048.41701
[14] Hirzebruch, F. and Zagier, D. , The Atiyah-Singer Theorem and Elementary Number Theory . Publish or Perish, Boston (1974). · Zbl 0288.10001
[15] Kleiman, S.L. , Algebraic cycles and the Weil conjectures , in Dix Exposés sur la Cohomologie des Schémas , North-Holland, Amsterdam, 359-386 (1968). · Zbl 0198.25902
[16] Murty, V.K. , Exceptional Hodge classes on certain Abelian varieties , Math. Ann. 268 (1984) 197-206. · Zbl 0521.14004
[17] Oort, F. , Finite group schemes, local moduli for Abelian varieties, and lifting problems , in Algebraic Geometry , Oslo 1970 (F. Oort, ed.), Wolters-Noordhoff, 233-254 (1972). · Zbl 0239.14018
[18] Ran, Z. , Cycles on Fermat hypersurfaces , Compositio Math. 42 (1980) 121-142. · Zbl 0463.14003
[19] Ribet, K. , Hodge classes on certain types of Abelian varieties , Am. J. Math. 105 (1983) 523-538. · Zbl 0586.14003
[20] Shioda, T. , The Hodge conjecture for Fermat varieties , Math. Ann. 245 (1979) 175-184. · Zbl 0403.14007
[21] Shioda, T. , Algebraic Cycles on Abelian varieties of Fermat type , Math. Ann. 258 (1981) 65-80. · Zbl 0515.14005
[22] Shioda, T. , What is known about the Hodge conjecture ?, in Advanced Studies in Pure Mathematics 1, Algebraic Varieties and Analytic Varities , 55-68 (1983). · Zbl 0527.14010
[23] Berthelot, P. , Grothendieck, A. , Illusie, L. , et al., Théorie des Intersections et Théoreme de Riemann-Roch, SGA 6 1966/67 , Springer Lecture Notes 225 (1971). · Zbl 0218.14001
[24] Shimura, G. , On analytic families of polarized Abelian varieties and automorphic functions , Ann. of Math. 78 (1963) 149-192. · Zbl 0142.05402
[25] Steenbrink, J.H.M. , Mixed Hodge structure on the vanishing cohomology , in Real and Complex Singularities , Oslo 1976, P. Holm (ed.), Sijthoff and Noordhoff, 525-563 (1977). · Zbl 0373.14007
[26] Weil, A. , Abelian varieties and the Hodge ring, in Collected Papers III , (1979) 421-429.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.