  For an integer n>2 we define a polylogarithm, which is a holomorphic function
on the universal abelian cover of C-{0,1} defined modulo (2 pi i)^n/(n-1)!. We
use the formal properties of its functional relations to define groups lifting
Goncharov's Bloch groups of a field F, and show that they fit into a complex
lifting Goncharov's Bloch complex. When F=C we show that the imaginary part
(when n is even) or real part (when n is odd) of the holomorphic polylogarithm
agrees with a real valued polylogarithm on the first cohomology group of the
lifted Bloch complex. When n=2, this group is Neumann's extended Bloch group.
Goncharov's complex conjecturally computes the rational motivic cohomology of
F, and one may speculate whether the lifted complex computes the integral
motivic cohomology. Finally, we construct a lift of Goncharov's real valued map
on the 5th homology of SL(3,C) to a complex valued map. The lift makes use of
the cluster ensemble structure on the Grassmannian Gr(3,6).
