Relative cycles with moduli and regulator maps

Let (Formula presented.) be a separated scheme of finite type over a field (Formula presented.) and (Formula presented.) a non-reduced effective Cartier divisor on it. We attach to the pair (Formula presented.) 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 (Formula presented.) gives a candidate definition for a relative motivic complex of the pair, that we compute in weight (Formula presented.). When (Formula presented.) is smooth over (Formula presented.) and (Formula presented.) is such that (Formula presented.) is a normal crossing divisor, we construct a fundamental class in the cohomology of relativedifferentials 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 (Formula presented.) to the relative de Rham complex. When (Formula presented.) is defined over (Formula presented.), 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 (Formula presented.) is moreover connected and proper over (Formula presented.), we use relative Deligne cohomology to define relative intermediate Jacobians with modulus (Formula presented.) of the pair (Formula presented.). For (Formula presented.), we show that (Formula presented.) is the universal regular quotient of the Chow group of (Formula presented.)-cycles with modulus.