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REFINED MOTIVIC DIMENSION OF SOME FERMAT VARIETIES

Part of: Cycles and subschemes

Published online by Cambridge University Press:  25 November 2015

SU-JEONG KANG
SU-JEONG KANG*
Affiliation:
Department of Mathematics and Computer Science, Providence College, Providence, RI 02918, USA email skang2@providence.edu
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Abstract

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Using the inductive structure of a Fermat variety by Shioda and Katsura [‘On Fermat varieties’, Tohoku Math. J. (2) 31(1) (1979), 97–115], we estimate the refined motivic dimension of certain Fermat varieties. As an application of our computation, we present an elementary proof of the generalised Hodge conjecture for those varieties.

Keywords

refined motivic dimensiongeneralised Hodge conjectureFermat varietiesinductive structures

MSC classification

Primary: 14C30: Transcendental methods, Hodge theory , Hodge conjecture
Secondary: 14C25: Algebraic cycles

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 2 , April 2016 , pp. 223 - 230
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Arapura, D., ‘Varieties with very little transcendental cohomology’, in: Motives and Algebraic Cycles, Fields Institute Communications, 56 (American Mathematical Society, Providence, RI, 2009), 1–14.Google Scholar
Deligne, P., ‘Théorie de Hodge. II, III’, Publ. Math. Inst. Hautes Études Sci. 40 (1971), 5–57; 44 (1974), 6–77.CrossRefGoogle Scholar
Deligne, P. and Katz, N. (eds.), ‘Groupes de monodromie en géométrie algébrique. II’, in: Séminaire de Géométrie Algébrique du Bois–Marie 1967–1969 (SGA 7 II), Lecture Notes in Mathematics, 340 (Springer, Berlin–New York, 1973), x+438.Google Scholar
Grothendieck, A., ‘Hodge’s general conjecture is false for trivial reasons’, Topology 8 (1969), 299–303.Google Scholar
Kang, S.-J., ‘Refined motivic dimension’, Canad. Math. Bull. 58(3) (2015), 519–529.CrossRefGoogle Scholar
Lewis, J., A Survey of the Hodge Conjecture, CRM Monograph Series, 10 (American Mathematical Society, Providence, RI, 1999).Google Scholar
Ran, Z., ‘Cycles on Fermat hypersurfaces’, Compositio Math. 42(1) (1980/81), 121–142.Google Scholar
Shioda, T., ‘The Hodge conjecture for Fermat varieties’, Math. Ann. 245(2) (1979), 175–184.CrossRefGoogle Scholar
Shioda, T. and Katsura, T., ‘On Fermat varieties’, Tohoku Math. J. (2) 31(1) (1979), 97–115.CrossRefGoogle Scholar
Steenbrink, J. H. M., ‘Some remarks about the Hodge conjecture’, in: Hodge Theory (Sant Cugat, 1985), Lecture Notes in Mathematics, 1246 (Springer, Berlin, 1987), 165–175.CrossRefGoogle Scholar