THE EXISTENCE OF HIGHER LOGARITHMS

In this paper we establish the existence of all higher logarithms as D eligne cohomology classes in a sense slightly weaker than that of [13, Sect. 12], but in a sense that should be strong enough for defining C hern classes on the algebraic K-theory of complex algebraic varieties. In particular, for each integer p greater than or equal to 1, we cons truct a multivalued holomorphic function on a Zariski open subset of t he self dual grassmannian of p-planes in C-2p which satisfies a canoni cal 2p + 1 term functional equation. The key new technical ingredient is the construction of a topology on the generic part of each Grassman nian which is coarser than the Zariski topology and where each open co ntains another which is both a K(pi, 1) and a rational K(pi, 1).