RELATIVE CYCLES WITH MODULI AND REGULATOR MAPS

• Published in: Journal of the Institute of Mathematics of Jussieu Vol. 18; no. 6; pp. 1233 - 1293
• Main Authors: Binda, Federico, Saito, Shuji
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.11.2019
• Subjects: Homology; Jacobians; Quotients; Zein; Primary 14C25; 14F43; 14F42; algebraic geometry; K-theory; Secondary 14C30
• ISSN: 1474-7480; 1475-3030
• DOI: 10.1017/S1474748017000391

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.