A construction of the polylogarithm motive
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866913162789388288 |
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| author | Dupont, Clément Fresán, Javier |
| author_facet | Dupont, Clément Fresán, Javier |
| contents | 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 $\{t_i=1\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_00789 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A construction of the polylogarithm motive Dupont, Clément Fresán, Javier Algebraic Geometry K-Theory and Homology Number Theory 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 $\{t_i=1\}$. |
| title | A construction of the polylogarithm motive |
| topic | Algebraic Geometry K-Theory and Homology Number Theory |
| url | https://arxiv.org/abs/2305.00789 |