A regulator map for singular varieties (Q1823276)

For an algebraic variety X over \({\mathbb{C}}\), we construct a Zariski sheaf \({\mathcal H}^ n(n)\) and a regulator \(\rho\) from the Zariski sheaf \({\mathcal K}^ M_{nX}\) of Milnor K-theory to it. If X is smooth, \({\mathcal H}^ n(n)\) is the Deligne-Beilinson sheaf, and \(\rho\) is the regulator constructed by S. Bloch (on smooth curves). Otherwise, \({\mathcal H}^ n(n)\) and \(\rho\) are functorial etc... A motivation for this construction is to understand the image of \(K_ 2({\mathcal O}_{X,x})\) in \(K_ 2({\mathbb{C}}(X))\) at a singular point x. As \({\mathcal H}^ n(n)\) maps to some differential forms, the cohomology of \(\rho\) gives an estimation for it, generalizing the work of V. Srinivas and A. Collino. The method consists of a careful study of the vanishing behavior of the Kähler differentials of X along the exceptional locus of some desingularization, of the introduction - following A. Beilinson - of the ``subsheaf'' of it with logarithmic poles at infinity, and of the technique of forgetting this growth in order to compute the cohomology of \(\rho\).