Euler complexes and geometry of modular varieties
  There is a mysterious connection between the multiple polylogarithms at N-th roots of unity and modular varieties. In this paper we "explain" it in the simplest case of the double logarithm.   We introduce an Euler complex data on modular curves. It includes a length two complex on every modular curve. Their second cohomology groups recover the Beilinson-Kato Euler system in K_2 of modular curves. We show that the above connection in the double logarithm case is provided by the specialization at a cusp of the Euler complex data on the modular curve Y_1(N).   Furthermore, specializing the Euler complexes at CM points we find new examples of the connection with geometry of modular varieties, this time hyperbolic 3-folds. 
A.B. Goncharov