Complexes of exact Hermitian cubes and the Zagier conjecture (Q1423609)

In this paper, based on the theory of exact Hermitian cubes and their Bott-Chern forms, the author presents a new approach to show the Zagier conjecture on the regulator map for the higher rational algebraic \(K\)-group of a number field. The author constructs a series of nice elements \(L_m(z)\) of the complex of exact Hermitian cubes on \(\mathbb A_{\mathbb Q}'-\{0,1\}\) and shows that the differential form associated to \(L_m(z)\), which is an extension of the Bott-Chern form by S. Wang, is expressed by Levin's polylogarithm. Using the element \(L_m(z)\), he succeeds in constructing a map from the higher Bloch group to the higher rational \(K\)-group of a number field whose composition with the regulator is given by the polylogarithm. This confirms the Zagier conjecture.