Infinitesimal Bloch regulator

Abstract

The aim of the paper is to define an infinitesimal analog of the Bloch regulator, which attaches to a pair of meromorphic functions on a Riemann surface, a line bundle with connection on the punctured surface. In the infinitesimal context, we consider a pair (X,X_) of schemes over a field of characteristic 0, such that the regular scheme X_ is defined in X by a square-zero sheaf of ideals which is locally free on X_. We propose a definition of the weight two motivic cohomology of X based on the Bloch group, which is defined in terms of the functional equation of the dilogarithm. The analog of the Bloch regulator is a map from a subspace of the infinitesimal part of HM 2(X,Q(2)) to the first cohomology group of the Zariski sheaf associated to an André-Quillen homology group. Using Goodwillie's theorem, we deduce that this map is an isomorphism, which is an infinitesimal analog of the injectivity conjecture for the Bloch regulator.