Zusammenfassung
Let (X) over bar 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 ((X) over bar, D) a cycle complex with modulus, those homotopy groups - called higher Chow groups with modulus - generalize additive higher Chow groups of Bloch-Esnault, Riffling, Park and Krishna-Levine, and that sheafified on (X) over bar (Zar) gives a ...
Zusammenfassung
Let (X) over bar 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 ((X) over bar, D) a cycle complex with modulus, those homotopy groups - called higher Chow groups with modulus - generalize additive higher Chow groups of Bloch-Esnault, Riffling, Park and Krishna-Levine, and that sheafified on (X) over bar (Zar) gives a candidate definition for a relative motivic complex of the pair, that we compute in weight 1. When (X) over bar is smooth over k and D is such that D-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 ((X) over bar, D) to the relative de Rham complex. When (X) over bar is defined over 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 (X) over bar is moreover connected and proper over C, we use relative Deligne cohomology to define relative intermediate Jacobians with modulus J((X) over bar vertical bar D)(r) of the pair ((X) over bar, D). For r = dim (X) over bar, we show that J((X) over bar vertical bar D)(r) is the universal regular quotient of the Chow group of 0-cycles with modulus.
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