Arithmetic normal functions and filtrations on Chow groups
Author:
James D. Lewis
Journal:
Proc. Amer. Math. Soc. 140 (2012), 26632670
MSC (2010):
Primary 14C25; Secondary 14C30, 14C35
Published electronically:
December 27, 2011
MathSciNet review:
2910754
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Abstract: Let be a smooth projective variety, and let be the higher Chow group defined by Bloch. Saito and Asakura defined a descending candidate BlochBeilinson filtration , using the language of mixed Hodge modules. Another more geometrically defined filtration is constructed by Kerr and Lewis in terms of germs of normal functions. We show that under the assumptions (i) where is defined over , and (ii) the general Hodge conjecture, that coincides with the aforementioned geometric filtration. More specifically, it is characterized in terms of germs of reduced arithmetic normal functions.
 [A]
Donu
Arapura, The Leray spectral sequence is motivic, Invent. Math.
160 (2005), no. 3, 567–589. MR 2178703
(2006m:14025), 10.1007/s002220040416x
 [As]
Masanori
Asakura, Motives and algebraic de Rham cohomology, The
arithmetic and geometry of algebraic cycles (Banff, AB, 1998) CRM Proc.
Lecture Notes, vol. 24, Amer. Math. Soc., Providence, RI, 2000,
pp. 133–154. MR 1736879
(2001c:14041)
 [Be]
A.
A. Be&ibreve;linson, Notes on absolute Hodge cohomology,
Applications of algebraic 𝐾theory to algebraic geometry and number
theory, Part I, II (Boulder, Colo., 1983) Contemp. Math., vol. 55,
Amer. Math. Soc., Providence, RI, 1986, pp. 35–68. MR 862628
(87m:14019), 10.1090/conm/055.1/862628
 [BrZ]
JeanLuc
Brylinski and Steven
Zucker, An overview of recent advances in Hodge theory,
Several complex variables, VI, Encyclopaedia Math. Sci., vol. 69,
Springer, Berlin, 1990, pp. 39–142. MR 1095090
(91m:14010)
 [B]
Spencer
Bloch, Algebraic cycles and higher 𝐾theory, Adv. in
Math. 61 (1986), no. 3, 267–304. MR 852815
(88f:18010), 10.1016/00018708(86)900812
 [Ca]
James
A. Carlson, Extensions of mixed Hodge structures,
Journées de Géometrie Algébrique d’Angers,
Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff,
Alphen aan den Rijn—Germantown, Md., 1980, pp. 107–127. MR 605338
(82g:14013)
 [Ja]
Uwe
Jannsen, Mixed motives and algebraic 𝐾theory, Lecture
Notes in Mathematics, vol. 1400, SpringerVerlag, Berlin, 1990. With
appendices by S. Bloch and C. Schoen. MR 1043451
(91g:14008)
 [KL]
Matt
Kerr and James
D. Lewis, The AbelJacobi map for higher Chow groups. II,
Invent. Math. 170 (2007), no. 2, 355–420. MR 2342640
(2008j:14005), 10.1007/s002220070066x
 [Lew1]
James
D. Lewis, A filtration on the Chow groups of a complex projective
variety, Compositio Math. 128 (2001), no. 3,
299–322. MR 1858339
(2002h:14003), 10.1023/A:1011882030468
 [Lew2]
James
D. Lewis, A survey of the Hodge conjecture, 2nd ed., CRM
Monograph Series, vol. 10, American Mathematical Society, Providence,
RI, 1999. Appendix B by B.\ Brent Gordon. MR 1683216
(2000a:14010)
 [LS]
James
D. Lewis and Shuji
Saito, Algebraic cycles and MumfordGriffiths invariants,
Amer. J. Math. 129 (2007), no. 6, 1449–1499. MR 2369886
(2008j:14014), 10.1353/ajm.2007.0046
 [SS]
Shuji
Saito, Motives and filtrations on Chow groups. II, The
arithmetic and geometry of algebraic cycles (Banff, AB, 1998) NATO Sci.
Ser. C Math. Phys. Sci., vol. 548, Kluwer Acad. Publ., Dordrecht,
2000, pp. 321–346. MR 1744952
(2001d:14025)
 [A]
 D. Arapura, The Leray spectral sequence is motivic, Inventiones Math. 160 (2005), 567589. MR 2178703 (2006m:14025)
 [As]
 M. Asakura, Motives and algebraic de Rham cohomology, in ``The Arithmetic and Geometry of Algebraic Cycles'' (Banff), CRM Proc. Lect. Notes 24, AMS, 2000, 133154. MR 1736879 (2001c:14041)
 [Be]
 A. Beilinson, Notes on absolute Hodge cohomology, Contemp. Math., Volume 55, Part I, Amer. Math. Soc. (1986), 3568. MR 862628 (87m:14019)
 [BrZ]
 J.L. Brylinski and S. Zucker, An overview of recent advances in Hodge theory, in Several complex variables, VI, Encyclopedia Math. Sci. 69, Springer, Berlin (1990), 39142. MR 1095090 (91m:14010)
 [B]
 S. Bloch, Algebraic cycles and higher theory, Advances in Math. 61 (1986), 267304. MR 852815 (88f:18010)
 [Ca]
 J. Carlson, Extensions of mixed Hodge structures, Journées de geométrie algébrique d'Angers, 1979, Sijthoff & Noordhof (1980), pp. 107127. MR 605338 (82g:14013)
 [Ja]
 U. Jannsen, Mixed Motives and Algebraic Theory, Lecture Notes in Mathematics 1400, SpringerVerlag, Berlin (1990). MR 1043451 (91g:14008)
 [KL]
 M. Kerr, J. D. Lewis, The AbelJacobi map for higher Chow groups II, Invent. Math. 170 (2) (2007), 355420. MR 2342640 (2008j:14005)
 [Lew1]
 J. D. Lewis, A filtration on the Chow groups of a complex projective variety, Compositio Math. 128 (2001), 299322. MR 1858339 (2002h:14003)
 [Lew2]
 J. D. Lewis, A survey of the Hodge conjecture, Second edition. Appendix B by B. Brent Gordon. CRM Monograph Series 10. American Mathematical Society, Providence, RI, 1999. MR 1683216 (2000a:14010)
 [LS]
 J. D. Lewis, S. Saito, Algebraic cycles and MumfordGriffiths invariants, Amer. J. Math. 129 (2007), no. 6, 14491499. MR 2369886 (2008j:14014)
 [SS]
 S. Saito, Motives and filtrations on Chow groups, II, in ``The Arithmetic and Geometry of Algebraic Cycles'', Proceedings of the CRM Summer School, June 719, 1998, Banff, Alberta, Canada (Editors: B. Gordon, J. Lewis, S. MüllerStach, S. Saito and N. Yui), NATO Science Series 548 (2000), 321346, Kluwer Academic Publishers. MR 1744952 (2001d:14025)
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Additional Information
James D. Lewis
Affiliation:
Department of Mathematical and Statistical Sciences, 632 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
Email:
lewisjd@ualberta.ca
DOI:
http://dx.doi.org/10.1090/S000299392011111304
PII:
S 00029939(2011)111304
Keywords:
BlochBeilinson filtration,
arithmetic normal function,
Chow group
Received by editor(s):
March 28, 2010
Received by editor(s) in revised form:
March 16, 2011
Published electronically:
December 27, 2011
Additional Notes:
The author was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.
Communicated by:
Lev Borisov
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
