RELATIVE CYCLES WITH MODULI AND REGULATOR MAPS

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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.