A construction of the polylogarithm motive

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-Wildesha...

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Bibliographic Details
Published in:arXiv.org 2023-05
Main Authors: Dupont, Clément, Fresán, Javier
Format: Article
Language:English
Subjects:
Homology
Hyperplanes
Hyperspaces
Online Access:Get full text
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Summary: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 \(\{t_i=1\}\).
ISSN:2331-8422