Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Real regulators on self-products of $ K3$ surfaces

Authors: Xi Chen and James D. Lewis
Journal: J. Algebraic Geom. 20 (2011), 101-125
Published electronically: October 7, 2009
MathSciNet review: 2729276
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Abstract | References | Additional Information

Abstract: Based on a novel application of an archimedean type pairing to the geometry and deformation theory of $ K3$ surfaces, we construct a regulator indecomposable $ K_1$-class on a self-product of a $ K3$ surface. In the Appendix, we explain how this pairing is a special instance of a general pairing on precycles in the equivalence relation defining Bloch's higher Chow groups.

References [Enhancements On Off] (What's this?)

  • [B] S. Bloch, Algebraic cycles and higher $ K$-theory, Advances in Math. 61 (1986), 267-304. -, The moving lemma for higher Chow groups, J. Algebraic Geom. 3(3) (1994), 493-535. MR 852815 (88f:18010)
  • [C1] X. Chen, Rational Curves on $ K3$ Surfaces, J. Algebraic Geom. 8 (1999), 245-278. Also preprint math.AG/9804075. MR 1675158 (2000d:14057)
  • [C2] -, A simple proof that rational curves on $ K3$ are nodal, Math. Ann. 324 (2002), no. 1, 71-104. MR 1931759 (2003k:14047)
  • [C-L1] X. Chen and J. D. Lewis, Noether-Lefschetz for $ K_{1}$ of a certain class of surfaces, Bol. Soc. Mat. Mexicana 10(3) (2004), 29-41. MR 2072000 (2005i:14010)
  • [C-L2] -, The Hodge- $ {\mathcal D}$-conjecture for $ K3$ and Abelian surfaces, J. Algebraic Geom. 14 (2005), 213-240. MR 2123228 (2005m:14008)
  • [C-L3] -, The real regulator for a product of $ K3$ surfaces, in Mirror Symmetry V, Proceedings of the BIRS conference in Banff, Alberta (Edited by S.-T. Yau, N. Yui and J. D. Lewis), AMS/IP Studies in Advanced Mathematics, Volume 38 (2006), 271-283. MR 2282963 (2007k:14004)
  • [El-V] P. Elbaz-Vincent, A short introduction to higher Chow groups, in Transcendental Aspects of Algebraic Cycles, Proceedings of the Grenoble Summer School, 2001, Edited by S. Müller-Stach and C. Peters, London Mathematical Society Lecture Note Series 313, Cambridge University Press (2004), 171-196. MR 2077769 (2005g:14012)
  • [K] M. Kerr, Geometric construction of Regulator currents with applications to algebraic cycles, Thesis, Princeton University (2003).
  • [KLM] M. Kerr, J. D. Lewis, S. Müller-Stach, The Abel-Jacobi map for higher Chow groups, Compositio Math. 142 (2006), 374-396. MR 2218900 (2007b:14007)
  • [L1] J. D. Lewis, Real regulators on Milnor complexes, $ K$-Theory 25 (2002), 277-298. MR 1909870 (2003d:19004)
  • [L2] -, A note on indecomposable motivic cohomology classes, J. reine angew. Math. 485 (1997), 161-172. MR 1442192 (98i:14008)

Additional Information

Xi Chen
Affiliation: 632 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

James D. Lewis
Affiliation: 632 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

Received by editor(s): July 18, 2008
Received by editor(s) in revised form: November 17, 2008
Published electronically: October 7, 2009
Additional Notes: Both authors were partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.

American Mathematical Society