# Holomorphic polylogarithms and Bloch complexes

## TL;DR

This paper introduces holomorphic polylogarithms on universal covers, constructs lifted Bloch complexes, and explores their connections to motivic cohomology and cluster structures, advancing the understanding of polylogarithm relations and algebraic K-theory.

## Contribution

It defines new holomorphic polylogarithms and associated lifted Bloch complexes, linking them to motivic cohomology and cluster algebra structures, extending Goncharov's framework.

## Key findings

- Holomorphic polylogarithms are defined on universal abelian covers.
- Lifted Bloch complexes fit into a complex extending Goncharov's Bloch complex.
- Constructed a lift of Goncharov's map on the 5th homology of SL(3,C).

## Abstract

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).