A construction of the polylogarithm motive

TL;DR

This paper constructs the polylogarithm motive explicitly as a relative cohomology motive associated with a hypersurface complement in affine space over the punctured projective line.

Contribution

It provides an explicit geometric construction of the polylogarithm motive as a relative cohomology motive, extending previous abstract existence results.

Findings

Constructed the polylogarithm motive explicitly as a relative cohomology motive.

Connected the motive to the complement of a specific hypersurface in affine space.

Confirmed the existence of the polylogarithm motive in the motivic and Hodge settings.

Abstract

Classical polylogarithms give rise to a variation of mixed Hodge-Tate structures on the punctured projective line $S=\mathbb{P}^1\setminus \{0, 1, \infty\}$, which is an extension of the symmetric power of the Kummer variation by a trivial variation. By results of Beilinson-Deligne, Huber-Wildeshaus, and Ayoub, this polylogarithm variation has a lift to the category of mixed Tate motives over $S$, whose existence is proved by computing the corresponding space of extensions in both the motivic and the Hodge settings. In this paper, we construct the polylogarithm motive as an explicit relative cohomology motive, namely that of the complement of the hypersurface $\{1-zt_1\cdots t_n=0\}$ in affine space $\mathbb{A}^n_S$ relative to the union of the hyperplanes $\{t_i=0\}$ and $\{t_i=1\}$.