Federico Binda

In this talk we will survey some recent results involving the so-called (higher) Chow groups with modulus for a pair $(X,D)$, where $X$ is a smooth variety over a field $k$ and $D$ is a (possibly non reduced) effective Cartier divisor on $X$, generalizing Bloch-Esnault additive Chow groups studied by Rülling, Park and Krishna as well as the Chow groups of zero cycles with modulus introduced by Kerz and Saito. These groups are a modified version of Bloch's higher Chow groups and are related to several non homotopy invariant objects.

We will sketch the construction over $\mathbb{C}$ of a regulator map to a relative version of Deligne cohomology, providing an Abel-Jacobi map to some intermediate Jacobians with additive part. Such Abel-Jacobi map satisfies in the case of zero-cycles a universal property that is the analogue of the one proved by Esnault-Srinivas-Viehweg for the Chow group of 0-cycles of a singular variety. This is a joint work with Shuji Saito.