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Refined Motivic Dimension

Published online by Cambridge University Press:  20 November 2018

Su-Jeong Kang*
Affiliation:
Department of Mathematics and Computer Science, Providence College, Providence, RI 02918, USA e-mail: [email protected]
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Abstract

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We define a refined motivic dimension for an algebraic variety by modifying the definition of motivic dimension by Arapura. We apply this to check and recheck the generalized Hodge conjecture for certain varieties, such as uniruled, rationally connected varieties and a rational surface fibration.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2015

References

[A] Arapura, D., Varieties with very little transcendental cohomology. In: Motives and algebraic cycles, Fields Inst. Commun., 56, American Mathematical Society, Providence, RI, 2009, pp. 114.Google Scholar
[Al] Arapura, D., TheHodge conjecture for rationally connected fivefolds. arxiv:math/O5O2257Google Scholar
[B] Bloch, S.,Lectures on algebraic cycles. Duke University Mathematics Series, IV, Duke University, Mathematics Department, Durham, NC, 1980.Google Scholar
[BS] Bloch, S. and Srinivas, V., Remarks on correspondences and algebraic cycles. Amer. J. Math. 105(1983), no. 5, 12351253. http://dx.doi.org/10.2307/2374341 Google Scholar
[CM] Conte, A. and Murre, J. P., The Hodge conjecture for fourfolds admitting a covering by rational curves. Math. Ann. 238(1978), no. 1, 7988. http://dx.doi.Org/10.1007/BF01351457 Google Scholar
[D] Deligne, P., Théorème de Lefschetzet critères de dégénérescence de suites spectrales. Inst. Hautes Études Sci. Publ. Math. 35(1968), 259278.Google Scholar
[Dl] Deligne, P., Théorie de Hodge. II, III Inst. Hautes Études Sci. Publ. Math.40(1971 ), 5–57;.44(1974), 577.Google Scholar
[G] Grothendieck, A., Hodge's general conjecture is false for trivial reasons. Topology. 8(1969), 299303. http://dx.doi.Org/10.101 6/0040-9383(69)9001 6-0 Google Scholar
[E] Esnault, H., Varieties over a finite field with trivial Chow group ofO-cycles have a rational point. Invent. Math. 151(2003), no. 1, 187191. http://dx.doi.org/10.1007/s00222-002-0261-8 Google Scholar
[La] Laterveer, R., Algebraic varieties with small Chow groups. J. Math. Kyoto Univ. 38(1998), no. 4, 673694.Google Scholar
[Le] Lewis, J. D., A survey of the Hodge conjecture. Second éd., CRM Monograph Series, 10, American Mathematical Society, Providence, RI, 1999.Google Scholar
[P] Paranjape, K. H., Cohomological and cycle-theoretic connectivity. Ann. of Math.(2). 139(1994), no. 3, 641660. http://dx.doi.org/10.2307/211 8574 Google Scholar
[S] Steenbrink, J. H. M., Some remarks about the Hodge conjecture. In: Hodge theory (SantCugat, 1985), Lecture Notes in Math., 1246, Springer, Berlin, 1987, pp. 165175.Google Scholar
[V] Voisin, C., Hodge theory and complex algebraic geometry. I. Cambridge Studies in Advanced Mathematics, 76, Cambridge University Press, Cambridge, 2007.Google Scholar