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Relative cycles with moduli and regulator maps. (English) Zbl 1439.14038

Summary: 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 S. Bloch and H. Esnault [Ann. Sci. Éc. Norm. Supér. (4) 36, No. 3, 463–477 (2003; Zbl 1100.14014)], J. Park [Am. J. Math. 131, No. 1, 257–276 (2009; Zbl 1176.14001)] and A. Krishna and M. Levine [J. Reine Angew. Math. 619, 75–140 (2008; Zbl 1158.14009)], and that sheafified on \(\overline{X}_{\mathrm{Zar}}\) 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 \(D_{\mathrm{red}}\) 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.

MSC:

14C25 Algebraic cycles
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14F42 Motivic cohomology; motivic homotopy theory
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
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