#### DOCUMENTA MATHEMATICA, Extra Volume: Kazuya Kato's Fiftieth Birthday (2003), 131-155

Spencer Bloch and Hélène Esnault

A notion of additive dilogarithm for a field $k$ is introduced, based on the $K$-theory and higher Chow groups of the affine line relative to $2(0)$. Analogues of the $K_2$-regulator, the polylogarithm Lie algebra, and the $\ell$-adic realization of the dilogarithm motive are discussed. The higher Chow groups of $0$-cycles in this theory are identified with the Kähler differential forms $\Omega^*_k$. It is hoped that these results will serve as a guide in developing a theory of contravariant motivic cohomology with modulus, modelled on the generalized Jacobians of Rosenlicht and Serre.