Real regulators on self-products of $K3$ surfaces
Authors:
Xi Chen and James D. Lewis
Journal:
J. Algebraic Geom. 20 (2011), 101-125
DOI:
https://doi.org/10.1090/S1056-3911-09-00525-6
Published electronically:
October 7, 2009
MathSciNet review:
2729276
Full-text PDF
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
- Spencer Bloch, Algebraic cycles and higher $K$-theory, Adv. in Math. 61 (1986), no. 3, 267–304. MR 852815, DOI https://doi.org/10.1016/0001-8708%2886%2990081-2
- Xi Chen, Rational curves on $K3$ surfaces, J. Algebraic Geom. 8 (1999), no. 2, 245–278. MR 1675158
- Xi Chen, A simple proof that rational curves on $K3$ are nodal, Math. Ann. 324 (2002), no. 1, 71–104. MR 1931759, DOI https://doi.org/10.1007/s00208-002-0329-1
- Xi Chen and James D. Lewis, Noether-Lefschetz for $K_1$ of a certain class of surfaces, Bol. Soc. Mat. Mexicana (3) 10 (2004), no. 1, 29–41. MR 2072000
- Xi Chen and James D. Lewis, The Hodge-$\scr D$-conjecture for $K3$ and abelian surfaces, J. Algebraic Geom. 14 (2005), no. 2, 213–240. MR 2123228, DOI https://doi.org/10.1090/S1056-3911-04-00390-X
- Xi Chen and James D. Lewis, The real regulator for a product of $K3$ surfaces, Mirror symmetry. V, AMS/IP Stud. Adv. Math., vol. 38, Amer. Math. Soc., Providence, RI, 2006, pp. 271–283. MR 2282963
- Philippe Elbaz-Vincent, A short introduction to higher Chow groups, Transcendental aspects of algebraic cycles, London Math. Soc. Lecture Note Ser., vol. 313, Cambridge Univ. Press, Cambridge, 2004, pp. 171–196. MR 2077769, DOI https://doi.org/10.1017/CBO9780511734984.006
- M. Kerr, Geometric construction of Regulator currents with applications to algebraic cycles, Thesis, Princeton University (2003).
- Matt Kerr, James D. Lewis, and Stefan Müller-Stach, The Abel-Jacobi map for higher Chow groups, Compos. Math. 142 (2006), no. 2, 374–396. MR 2218900, DOI https://doi.org/10.1112/S0010437X05001867
- James D. Lewis, Real regulators on Milnor complexes, $K$-Theory 25 (2002), no. 3, 277–298. MR 1909870, DOI https://doi.org/10.1023/A%3A1015696107010
- James D. Lewis, A note on indecomposable motivic cohomology classes, J. Reine Angew. Math. 485 (1997), 161–172. MR 1442192, DOI https://doi.org/10.1515/crll.1997.485.161
References
- 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)
- X. Chen, Rational Curves on $K3$ Surfaces, J. Algebraic Geom. 8 (1999), 245-278. Also preprint math.AG/9804075. MR 1675158 (2000d:14057)
- ---, A simple proof that rational curves on $K3$ are nodal, Math. Ann. 324 (2002), no. 1, 71-104. MR 1931759 (2003k:14047)
- 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)
- ---, The Hodge-${\mathcal D}$-conjecture for $K3$ and Abelian surfaces, J. Algebraic Geom. 14 (2005), 213-240. MR 2123228 (2005m:14008)
- ---, 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)
- 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)
- M. Kerr, Geometric construction of Regulator currents with applications to algebraic cycles, Thesis, Princeton University (2003).
- 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)
- J. D. Lewis, Real regulators on Milnor complexes, $K$-Theory 25 (2002), 277-298. MR 1909870 (2003d:19004)
- ---, 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
Email:
xichen@math.ualberta.ca
James D. Lewis
Affiliation:
632 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
MR Author ID:
204180
Email:
lewisjd@ualberta.ca
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.
