Arithmetic normal functions and filtrations on Chow groups

Author:
James D. Lewis

Journal:
Proc. Amer. Math. Soc. **140** (2012), 2663-2670

MSC (2010):
Primary 14C25; Secondary 14C30, 14C35

Published electronically:
December 27, 2011

MathSciNet review:
2910754

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a smooth projective variety, and let be the higher Chow group defined by Bloch. Saito and Asakura defined a descending candidate Bloch-Beilinson 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**, 10.1007/s00222-004-0416-x**[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****[Be]**A. A. Beĭlinson,*Notes on absolute Hodge cohomology*, Applications of algebraic 𝐾-theory 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**, 10.1090/conm/055.1/862628**[Br-Z]**Jean-Luc 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****[B]**Spencer Bloch,*Algebraic cycles and higher 𝐾-theory*, Adv. in Math.**61**(1986), no. 3, 267–304. MR**852815**, 10.1016/0001-8708(86)90081-2**[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****[Ja]**Uwe Jannsen,*Mixed motives and algebraic 𝐾-theory*, Lecture Notes in Mathematics, vol. 1400, Springer-Verlag, Berlin, 1990. With appendices by S. Bloch and C. Schoen. MR**1043451****[K-L]**Matt Kerr and James D. Lewis,*The Abel-Jacobi map for higher Chow groups. II*, Invent. Math.**170**(2007), no. 2, 355–420. MR**2342640**, 10.1007/s00222-007-0066-x**[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**, 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****[L-S]**James D. Lewis and Shuji Saito,*Algebraic cycles and Mumford-Griffiths invariants*, Amer. J. Math.**129**(2007), no. 6, 1449–1499. MR**2369886**, 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**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
14C25,
14C30,
14C35

Retrieve articles in all journals with MSC (2010): 14C25, 14C30, 14C35

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/S0002-9939-2011-11130-4

Keywords:
Bloch-Beilinson 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.