RELATIVE CYCLES WITH MODULI AND REGULATOR MAPS

<jats:p>Let<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748017000391_inline1" xlink:type="simple" /><jats:tex-math>$\overline{X}$</jats:tex-math></jats:alternatives></jats:inline-formula>be a separated scheme of finite type over a field<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748017000391_inline2" xlink:type="simple" /><jats:tex-math>$k$</jats:tex-math></jats:alternatives></jats:inline-formula>and<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748017000391_inline3" xlink:type="simple" /><jats:tex-math>$D$</jats:tex-math></jats:alternatives></jats:inline-formula>a non-reduced effective Cartier divisor on it. We attach to the pair<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748017000391_inline4" xlink:type="simple" /><jats:tex-math>$(\overline{X},D)$</jats:tex-math></jats:alternatives></jats:inline-formula>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<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748017000391_inline5" xlink:type="simple" /><jats:tex-math>$\overline{X}_{\text{Zar}}$</jats:tex-math></jats:alternatives></jats:inline-formula>gives a candidate definition for a relative motivic complex of the pair, that we compute in weight<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748017000391_inline6" xlink:type="simple" /><jats:tex-math>$1$</jats:tex-math></jats:alternatives></jats:inline-formula>. When<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748017000391_inline7" xlink:type="simple" /><jats:tex-math>$\overline{X}$</jats:tex-math></jats:alternatives></jats:inline-formula>is smooth over<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748017000391_inline8" xlink:type="simple" /><jats:tex-math>$k$</jats:tex-math></jats:alternatives></jats:inline-formula>and<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748017000391_inline9" xlink:type="simple" /><jats:tex-math>$D$</jats:tex-math></jats:alternatives></jats:inline-formula>is such that<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748017000391_inline10" xlink:type="simple" /><jats:tex-math>$D_{\text{red}}$</jats:tex-math></jats:alternatives></jats:inline-formula>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<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748017000391_inline11" xlink:type="simple" /><jats:tex-math>$(\overline{X},D)$</jats:tex-math></jats:alternatives></jats:inline-formula>to the relative de Rham complex. When<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748017000391_inline12" xlink:type="simple" /><jats:tex-math>$\overline{X}$</jats:tex-math></jats:alternatives></jats:inline-formula>is defined over<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748017000391_inline13" xlink:type="simple" /><jats:tex-math>$\mathbb{C}$</jats:tex-math></jats:alternatives></jats:inline-formula>, 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<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748017000391_inline14" xlink:type="simple" /><jats:tex-math>$\overline{X}$</jats:tex-math></jats:alternatives></jats:inline-formula>is moreover connected and proper over<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748017000391_inline15" xlink:type="simple" /><jats:tex-math>$\mathbb{C}$</jats:tex-math></jats:alternatives></jats:inline-formula>, we use relative Deligne cohomology to define relative intermediate Jacobians with modulus<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748017000391_inline16" xlink:type="simple" /><jats:tex-math>$J_{\overline{X}|D}^{r}$</jats:tex-math></jats:alternatives></jats:inline-formula>of the pair<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748017000391_inline17" xlink:type="simple" /><jats:tex-math>$(\overline{X},D)$</jats:tex-math></jats:alternatives></jats:inline-formula>. For<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748017000391_inline18" xlink:type="simple" /><jats:tex-math>$r=\dim \overline{X}$</jats:tex-math></jats:alternatives></jats:inline-formula>, we show that<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748017000391_inline19" xlink:type="simple" /><jats:tex-math>$J_{\overline{X}|D}^{r}$</jats:tex-math></jats:alternatives></jats:inline-formula>is the universal regular quotient of the Chow group of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748017000391_inline20" xlink:type="simple" /><jats:tex-math>$0$</jats:tex-math></jats:alternatives></jats:inline-formula>-cycles with modulus.</jats:p>