Holomorphic polylogarithms and Bloch complexes

Abstract

Abstract For an integer n ≥ 2 {n\geq 2} we define a polylogarithm ℒ ^ n {\widehat{\mathcal{L}}_{n}} , which is a holomorphic function on the universal abelian cover of ℂ ∖ { 0 , 1 } {\mathbb{C}\setminus\{0,1\}} defined modulo ( 2 ⁢ π ⁢ i ) n / ( n - 1 ) ! {(2\pi i)^{n}/(n-1)!} . We use the formal properties of its functional relations to define groups ℬ ^ k ⁢ ( F ^ ) {\widehat{\mathcal{B}}_{k}(\widehat{F})} lifting Goncharov’s Bloch groups ℬ k ⁢ ( F ) {\mathcal{B}_{k}(F)} of a field F, and show that they fit into a complex Γ ^ ⁢ ( F , n ) {\widehat{\Gamma}(F,n)} lifting Goncharov’s Bloch complex Γ ⁢ ( F , n ) {\Gamma(F,n)} . When F = ℂ {F=\mathbb{C}} , we show that the imaginary part (when n is even) or real part (when n is odd) of ℒ ^ n {\widehat{\mathcal{L}}_{n}} agrees with a real single valued polylogarithm ℒ n {\mathcal{L}_{n}} on the group H 1 ⁢ ( Γ ^ ⁢ ( ℂ , n ) ) {H^{1}(\widehat{\Gamma}(\mathbb{C},n))} . When n = 2 {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 extended complex computes the integral motivic cohomology. Finally, we use ℒ ^ 3 {\widehat{\mathcal{L}}_{3}} to construct a lift of Goncharov’s map H 5 ⁢ ( SL ⁡ ( 3 , ℂ ) ) → ℝ {H_{5}(\operatorname{SL}(3,\mathbb{C}))\to\mathbb{R}} to a complex valued map whose real part agrees with that of Goncharov. The lift makes use of the cluster ensemble structure on the Grassmannian Gr ⁡ ( 3 , 6 ) {\operatorname{Gr}(3,6)} .