### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 118, 12 pages      arXiv:1805.04646      https://doi.org/10.3842/SIGMA.2018.118
Contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui

### Integral Regulators for Higher Chow Complexes

Muxi Li
University of Science and Technology, Hefei, Anhui, P.R. China

Received May 12, 2018, in final form October 31, 2018; Published online November 03, 2018

Abstract
Building on Kerr, Lewis and Müller-Stach's work on the rational regulator, we prove the existence of an integral regulator on higher Chow complexes and give an explicit expression. This puts firm ground under some earlier results and speculations on the torsion in higher cycle groups by Kerr-Lewis-Müller-Stach, Petras, and Kerr-Yang.

Key words: integral regulator; higher Chow groups; algebraic cycles; Abel-Jacobi map.

pdf (416 kb)   tex (18 kb)

References

1. Bloch S., Algebraic cycles and higher $K$-theory, Adv. Math. 61 (1986), 267-304.
2. Bloch S., Algebraic cycles and the Beǐlinson conjectures, in The Lefschetz Centennial Conference, Part I (Mexico City, 1984), Contemp. Math., Vol. 58, Amer. Math. Soc., Providence, RI, 1986, 65-79.
3. Bloch S., Some notes on elementary properties of higher chow groups, including functoriality properties and cubical chow groups, Preprint, available at http://www.math.uchicago.edu/~bloch/publications.html.
4. Kerr M., Lewis J.D., Müller-Stach S., The Abel-Jacobi map for higher Chow groups, Compos. Math. 142 (2006), 374-396, math.AG/0409116.
5. Kerr M., Lewis J.D., The Abel-Jacobi map for higher Chow groups. II, Invent. Math. 170 (2007), 355-420, math.AG/0611333.
6. Kerr M., Li M., Two applications of the integral regulator, arXiv:1809.04114.
7. Kerr M., Yang Y., An explicit basis for the rational higher Chow groups of abelian number fields, Ann. K-Theory 3 (2018), 173-191, arXiv:1608.07477.
8. Lion J.-M., Rolin J.-P., Théorème de préparation pour les fonctions logarithmico-exponentielles, Ann. Inst. Fourier (Grenoble) 47 (1997), 859-884.
9. Nowak K.J., Flat morphisms between regular varieties, Univ. Iagel. Acta Math. 35 (1997), 243-246.
10. Petras O., Functional equations of the dilogarithm in motivic cohomology, J. Number Theory 129 (2009), 2346-2368, arXiv:0712.3987.
11. Weißschuh T., A commutative regulator map into Deligne-Beilinson cohomology, Manuscripta Math. 152 (2017), 281-315, arXiv:1410.4686.