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CHOW GROUPS, CHOW COHOMOLOGY, AND LINEAR VARIETIES

Part of: Special varieties Cycles and subschemes (Co)homology theory

Published online by Cambridge University Press:  24 June 2014

BURT TOTARO
BURT TOTARO*
Affiliation:
UCLA Mathematics Department, Box 951555, Los Angeles, CA 90095-1555, USA; [email protected]

Abstract

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We compute the Chow groups and the Fulton–MacPherson operational Chow cohomology ring for a class of singular rational varieties including toric varieties. The computation is closely related to the weight filtration on the ordinary cohomology of these varieties. We use the computation to answer one of the open problems about operational Chow cohomology: it does not have a natural map to ordinary cohomology.

MSC classification

Secondary: 14C15: (Equivariant) Chow groups and rings; motives 14F42: Motivic cohomology; motivic homotopy theory 14M20: Rational and unirational varieties
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e17
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© The Author 2014

References

Atiyah, M., K-Theory (W.A. Benjamin, 1967).Google Scholar
Barbieri Viale, L. and Srinivas, V., ‘The Néron–Severi group and the mixed Hodge structure on H 2 ’, J. Reine Angew. Math. 450 (1994), 37–42.Google Scholar
Bloch, S., ‘Algebraic cycles and higher K-theory’, Adv. Math. 61 (1986), 267–304.CrossRefGoogle Scholar
Bloch, S., ‘The moving lemma for higher Chow groups’, J. Algebraic Geom. 3 (1994), 537–568.Google Scholar
Bloch, S., Gillet, H. and Soulé, C., ‘Non-archimedean Arakelov theory’, J. Algebraic Geom. 4 (1995), 427–485.Google Scholar
Deligne, P., ‘Poids dans la cohomologie des variétés algébriques’, inProceedings of the International Congress of Mathematicians (Vancouver, 1974), Canad. Math. Congress, vol. 1 (1975), 79–85.Google Scholar
Dugger, D. and Isaksen, D., ‘Motivic cell structures’, Algebr. Geom. Topol. 5 (2005), 615–652.CrossRefGoogle Scholar
Esnault, H. and Viehweg, E., Lectures on Vanishing Theorems (Birkhäuser, 1992).CrossRefGoogle Scholar
Faltings, G. and Chai, C.-L., Degeneration of Abelian Varieties (Springer, 1990).CrossRefGoogle Scholar
Franz, M., ‘The integral cohomology of toric manifolds’, Tr. Mat. Inst. Steklova 252 (2006), 61–70. English translation in Proc. Steklov Inst. Math. 2006, no. 1 (252), 53–62.Google Scholar
Fulton, W., Introduction to Toric Varieties (Princeton, 1993).CrossRefGoogle Scholar
Fulton, W. and MacPherson, R., ‘Categorical framework for the study of singular spaces’, Mem. Amer. Math. Soc. 243 (1981).Google Scholar
Fulton, W., MacPherson, R., Sottile, F. and Sturmfels, B., ‘Intersection theory on spherical varieties’, J. Algebraic Geom. 4 (1995), 181–193.Google Scholar
Huber, A. and Kahn, B., ‘The slice filtration and mixed Tate motives’, Compos. Math. 142 (2006), 907–936.CrossRefGoogle Scholar
Jannsen, U., ‘Deligne homology, Hodge D-conjecture, and motives’, inBeilinson’s Conjectures on Special Values of L-Functions (eds Rapoport, M., Schappacher, N. and Schneider, P.), (Academic Press, 1988), 305–372.CrossRefGoogle Scholar
Jannsen, U., Mixed Motives and Algebraic K-Theory, Lecture Notes in Mathematics, vol. 1400 (Springer, 1990).CrossRefGoogle Scholar
Joshua, R., ‘Algebraic K-theory and higher Chow groups of linear varieties’, Math. Proc. Cambridge Philos. Soc. 130 (2001), 37–60.CrossRefGoogle Scholar
Kelly, S., ‘Triangulated categories of motives in positive characteristic’. arXiv:1305.5349.Google Scholar
Kimura, S., ‘Fractional intersection and bivariant theory’, Comm. Algebra 20 (1992), 285–302.Google Scholar
Kollár, J. and Mori, S., ‘Classification of three-dimensional flips’, J. Amer. Math. Soc. 5 (1992), 533–703.CrossRefGoogle Scholar
Levine, M., ‘Techniques of localization in the theory of algebraic cycles’, J. Algebraic Geom. 10 (2001), 299–363.Google Scholar
Mazza, C., Voevodsky, V. and Weibel, C., Lecture Notes on Motivic Cohomology (Amer. Math. Soc., 2006).Google Scholar
Payne, S., ‘Equivariant Chow cohomology of toric varieties’, Math. Res. Lett. 13 (2006), 29–41.CrossRefGoogle Scholar
Rosenlicht, M., ‘Questions of rationality for solvable algebraic groups over nonperfect fields’, Annali di Mat. 61 (1963), 97–120.CrossRefGoogle Scholar
Steenbrink, J. and Zucker, S., ‘Variations of mixed Hodge structure I’, Invent. Math. 80 (1985), 489–542.CrossRefGoogle Scholar
Voevodsky, V., ‘Triangulated categories of motives over a field’, inCycles, Transfers, and Motivic Homology Theories (Princeton, 2000), 188–238.Google Scholar
Voevodsky, V., ‘Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic’, Int. Math. Res. Notices (7) (2002).CrossRefGoogle Scholar