# Regulator Maps for Higher Chow Groups via Current Transforms

We  explain  the construction  of an explicit regulator map  at the level of complexes:
$\operatorname{Reg} \colon {CH_\Delta^{p}(X, n)} \longrightarrow H^{2p-n}_{\mathscr{D}}(X;\mathbb{Z}(p)),$
from the higher Chow groups of a smooth complex algebraic variety $$X$$,  in their simplicial formulation with $$\mathbb{Z}$$ coefficients, into integral Deligne-Beilinson cohomology.

We start  by   using a suitably  defined \emph{equidimensional cycles} subcomplex  $$\mathcal{Z}^p_{\Delta, \text{eq}}(X,*)$$
of Bloch's higher Chow complex $$\mathcal{Z}^p_\Delta(X,*)$$ to compute the higher Chow groups.  This relies on Suslin's \emph{generic equidimensionality} results.

Next, we use algebraic correspondences to introduce transform operations on a fairly general class of currents. Then we combine these transforms with basic properties of equidimensional cycles to construct a map of complexes
$\operatorname{Reg} \colon \mathcal{Z}^p_{\Delta, \text{eq}}(X,*) \to \mathbb{Z}(p)_{\mathscr{D}}(X),$
where $$\mathbb{Z}(p)_{\mathscr{D}}(X)$$ is a complex of currents yielding Deligne-Beilinson cohomology.

Start Date
Venue
Room 1.09
End Date

## Speaker

Pedro Ferreira dos Santos

## Speaker's Institution

Instituto Superior Técnico / CAMGSD

## Area

Geometry and Topology