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RELATIVE CYCLES WITH MODULI AND REGULATOR MAPS

Part of: Cycles and subschemes (Co)homology theory

Published online by Cambridge University Press:  02 November 2017

Federico Binda Open the ORCID record for Federico Binda [Opens in a new window]  and
Shuji Saito
Federico Binda
Affiliation:
Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann Strasse 9, 45127 Essen, Germany Fakultät für Mathematik, Universität Regensburg, 93040, Regensburg, Germany ([email protected])
Shuji Saito
Affiliation:
Interactive Research Center of Science, Graduate School of Science and Engineering, Tokyo Institute of Technology, Ookayama, Meguro, Tokyo 152-8551, Japan ([email protected])
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Abstract

Let $\overline{X}$ be a separated scheme of finite type over a field $k$ and $D$ a non-reduced effective Cartier divisor on it. We attach to the pair $(\overline{X},D)$ a cycle complex with modulus, those homotopy groups – called higher Chow groups with modulus – generalize additive higher Chow groups of Bloch–Esnault, Rülling, Park and Krishna–Levine, and that sheafified on $\overline{X}_{\text{Zar}}$ gives a candidate definition for a relative motivic complex of the pair, that we compute in weight $1$. When $\overline{X}$ is smooth over $k$ and $D$ is such that $D_{\text{red}}$ is a normal crossing divisor, we construct a fundamental class in the cohomology of relative differentials for a cycle satisfying the modulus condition, refining El Zein’s explicit construction of the fundamental class of a cycle. This is used to define a natural regulator map from the relative motivic complex of $(\overline{X},D)$ to the relative de Rham complex. When $\overline{X}$ is defined over $\mathbb{C}$, the same method leads to the construction of a regulator map to a relative version of Deligne cohomology, generalizing Bloch’s regulator from higher Chow groups. Finally, when $\overline{X}$ is moreover connected and proper over $\mathbb{C}$, we use relative Deligne cohomology to define relative intermediate Jacobians with modulus $J_{\overline{X}|D}^{r}$ of the pair $(\overline{X},D)$. For $r=\dim \overline{X}$, we show that $J_{\overline{X}|D}^{r}$ is the universal regular quotient of the Chow group of $0$-cycles with modulus.

Keywords

algebraic geometryK-theory

MSC classification

Primary: 14C25: Algebraic cycles
Secondary: 14C30: Transcendental methods, Hodge theory , Hodge conjecture 14F42: Motivic cohomology; motivic homotopy theory 14F43: Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 18 , Issue 6 , November 2019 , pp. 1233 - 1293
Copyright
© Cambridge University Press 2017 

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