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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2009.00533 |
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| _version_ | 1866929330454528000 |
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| author | Kahn, Bruno Liu, with an appendix by Qing |
| author_facet | Kahn, Bruno Liu, with an appendix by Qing |
| contents | For a $d$-dimensional smooth projective variety $X$ over the function field of a smooth variety $B$ over a field $k$ and for $i\ge 0$, we define a subgroup $CH^i(X)^{(0)}$ of $CH^i(X)$ and construct a "refined height pairing" \[CH^i(X)^{(0)}\times CH^{d+1-i}(X)^{(0)}\to CH^1(B)\] in the category of abelian groups modulo isogeny. For $i=1,d$, $CH^i(X)^{(0)}$ is the group of cycles numerically equivalent to $0$. This pairing relates to pairings defined by P. Schneider and A. Beilinson if $B$ is a curve, to a refined height defined by L. Moret-Bailly when $X$ is an abelian variety, and to a pairing with values in $H^2(B_{\bar k},\mathbf{Q}_l(1))$ defined by D. Rössler and T. Szamuely in general. We study it in detail when $i=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_00533 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Refined height pairing Kahn, Bruno Liu, with an appendix by Qing Algebraic Geometry For a $d$-dimensional smooth projective variety $X$ over the function field of a smooth variety $B$ over a field $k$ and for $i\ge 0$, we define a subgroup $CH^i(X)^{(0)}$ of $CH^i(X)$ and construct a "refined height pairing" \[CH^i(X)^{(0)}\times CH^{d+1-i}(X)^{(0)}\to CH^1(B)\] in the category of abelian groups modulo isogeny. For $i=1,d$, $CH^i(X)^{(0)}$ is the group of cycles numerically equivalent to $0$. This pairing relates to pairings defined by P. Schneider and A. Beilinson if $B$ is a curve, to a refined height defined by L. Moret-Bailly when $X$ is an abelian variety, and to a pairing with values in $H^2(B_{\bar k},\mathbf{Q}_l(1))$ defined by D. Rössler and T. Szamuely in general. We study it in detail when $i=1$. |
| title | Refined height pairing |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2009.00533 |