On the Picard number and the extension degree of period matrices of complex tori
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908550845956096 |
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| author | Auffarth, Robert Franco, Jorge Duque |
| author_facet | Auffarth, Robert Franco, Jorge Duque |
| contents | The rank $ρ$ of the Néron-Severi group of a complex torus $X$ of dimension $g$ satisfies $0\leqρ\leq g^2=h^{1,1}.$ The degree $\mathfrak{d}$ of the extension field generated over $\mathbb{Q}$ by the entries of a period matrix of $X$ imposes constraints on its Picard number $ρ$ and, consequently, on the structure of $X$. In this paper, we show that when $\mathfrak{d}$ is $2$, $3$, or $4$, the Picard number $ρ$ is necessarily large. Moreover, for an abelian variety $X$ of dimension $g$ with $\mathfrak{d}=3,$ we establish a structure-type result: $X$ must be isogenous to $E^g$, where $E$ is an elliptic curve without complex multiplication. In this case, the Picard number satisfies $ρ(X)=\frac{g(g+1)}{2}.$ As a byproduct, we obtain that if $\mathfrak{d}$ is odd, then $ρ(X)\leq\frac{g(g+1)}{2}.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_21642 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Picard number and the extension degree of period matrices of complex tori Auffarth, Robert Franco, Jorge Duque Algebraic Geometry Complex Variables 14K20, 14K22, 32G20, 11G10 The rank $ρ$ of the Néron-Severi group of a complex torus $X$ of dimension $g$ satisfies $0\leqρ\leq g^2=h^{1,1}.$ The degree $\mathfrak{d}$ of the extension field generated over $\mathbb{Q}$ by the entries of a period matrix of $X$ imposes constraints on its Picard number $ρ$ and, consequently, on the structure of $X$. In this paper, we show that when $\mathfrak{d}$ is $2$, $3$, or $4$, the Picard number $ρ$ is necessarily large. Moreover, for an abelian variety $X$ of dimension $g$ with $\mathfrak{d}=3,$ we establish a structure-type result: $X$ must be isogenous to $E^g$, where $E$ is an elliptic curve without complex multiplication. In this case, the Picard number satisfies $ρ(X)=\frac{g(g+1)}{2}.$ As a byproduct, we obtain that if $\mathfrak{d}$ is odd, then $ρ(X)\leq\frac{g(g+1)}{2}.$ |
| title | On the Picard number and the extension degree of period matrices of complex tori |
| topic | Algebraic Geometry Complex Variables 14K20, 14K22, 32G20, 11G10 |
| url | https://arxiv.org/abs/2503.21642 |