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In a joint work [9] with Kazuya Kato and Chikara Nakayama, log higher Albanese manifolds was constructed as an application of log mixed Hodge theory with group action. In this framework, we describe a work of Deligne in [3] on some nilpotent quotients of the fundamental group of the projective line minus three points, where polylogarithms appear. As a result, we have $q$-expansions of higher Albanese maps at boundary points, i.e., log higher Albanese maps over the boundary.

A description of a result of Deligne by log higher Albanese map