Integral aspects of Fourier duality for abelian varieties
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arXiv
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| Autori principali: | , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913421331529728 |
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| author | Hasan, Junaid Hassan, Hazem Lin, Milton Manivel, Marcella McBeath, Lily Moonen, Ben |
| author_facet | Hasan, Junaid Hassan, Hazem Lin, Milton Manivel, Marcella McBeath, Lily Moonen, Ben |
| contents | We prove several results about integral versions of Fourier duality for abelian schemes, making use of Pappas's work on integral Grothendieck-Riemann-Roch. If $S$ is smooth quasi-projective of dimension $d$ over a field and $π\colon X\to S$ is a $g$-dimensional abelian scheme, we prove, under very mild assumptions on $X/S$, that all classical results about Fourier duality, including the existence of a Beauville decomposition, are valid for the Chow ring $\mathrm{CH}(X;Λ)$ with coefficients in the ring $Λ= \mathbb{Z}[1/(2g+d+1)!]$. If $X$ admits a polarization $θ$ of degree $ν(θ)^2$ we further construct an $\mathfrak{sl}_2$-action on $\mathrm{CH}(X;Λ_θ)$ with $Λ_θ= Λ[1/ν(θ)]$, and we show that $\mathrm{CH}(X;Λ_θ)$ is a sum of copies of the symmetric powers $\mathrm{Sym}^n(\mathrm{St})$ of the $2$-dimensional standard representation, for $n=0,\ldots,g$. For an abelian variety over an algebraically closed field, we use our results to produce torsion classes in $\mathrm{CH}^i(X;Λ_θ)$ for every $i\in \{1,\ldots,g\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06184 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Integral aspects of Fourier duality for abelian varieties Hasan, Junaid Hassan, Hazem Lin, Milton Manivel, Marcella McBeath, Lily Moonen, Ben Algebraic Geometry 14C15, 14K05 We prove several results about integral versions of Fourier duality for abelian schemes, making use of Pappas's work on integral Grothendieck-Riemann-Roch. If $S$ is smooth quasi-projective of dimension $d$ over a field and $π\colon X\to S$ is a $g$-dimensional abelian scheme, we prove, under very mild assumptions on $X/S$, that all classical results about Fourier duality, including the existence of a Beauville decomposition, are valid for the Chow ring $\mathrm{CH}(X;Λ)$ with coefficients in the ring $Λ= \mathbb{Z}[1/(2g+d+1)!]$. If $X$ admits a polarization $θ$ of degree $ν(θ)^2$ we further construct an $\mathfrak{sl}_2$-action on $\mathrm{CH}(X;Λ_θ)$ with $Λ_θ= Λ[1/ν(θ)]$, and we show that $\mathrm{CH}(X;Λ_θ)$ is a sum of copies of the symmetric powers $\mathrm{Sym}^n(\mathrm{St})$ of the $2$-dimensional standard representation, for $n=0,\ldots,g$. For an abelian variety over an algebraically closed field, we use our results to produce torsion classes in $\mathrm{CH}^i(X;Λ_θ)$ for every $i\in \{1,\ldots,g\}$. |
| title | Integral aspects of Fourier duality for abelian varieties |
| topic | Algebraic Geometry 14C15, 14K05 |
| url | https://arxiv.org/abs/2407.06184 |