On a noncommutative deformation of holomorphic line bundles on complex tori and the SYZ transform
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915842499805184 |
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| author | Kobayashi, Kazushi |
| author_facet | Kobayashi, Kazushi |
| contents | By regarding a given $n$-dimensional complex torus $X^n$ as the trivial torus fibration $X^n \to \mathbb{R}^n/\mathbb{Z}^n$, we can obtain a mirror dual complexified symplectic torus $\check{X}^n$ based on the SYZ construction. In the middle 2000s, as a part of the study on noncommutative deformations of $X^n$, Kajiura examined the noncommutative complex torus $X_θ^n$ obtained via the (real) nonformal deformation quantization of $X^n \to \mathbb{R}^n/\mathbb{Z}^n$ by a Poisson bivector $θ$ defined along the fibers. In particular, he constructed the noncommutative deformations $L_θ \to X_θ^n$ of holomorphic line bundles on $X^n$ and a curved dg-category consisting of them. On the other hand, associated to this noncommutative deformation, we can construct a non-trivial deformation of the trivial holomorphic line bundle on $X^n$ by twisting it with a suitable isomorphism. In this paper, from this point of view, we extend the construction of $L_θ$ to the more general setting. Moreover, we also consider objects defined on a mirror partner of $X_θ^n$ which are mirror dual to such extended noncommutative objects. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_07239 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On a noncommutative deformation of holomorphic line bundles on complex tori and the SYZ transform Kobayashi, Kazushi Differential Geometry High Energy Physics - Theory 14F08, 14J33, 58B34, 53D37, 53C08 By regarding a given $n$-dimensional complex torus $X^n$ as the trivial torus fibration $X^n \to \mathbb{R}^n/\mathbb{Z}^n$, we can obtain a mirror dual complexified symplectic torus $\check{X}^n$ based on the SYZ construction. In the middle 2000s, as a part of the study on noncommutative deformations of $X^n$, Kajiura examined the noncommutative complex torus $X_θ^n$ obtained via the (real) nonformal deformation quantization of $X^n \to \mathbb{R}^n/\mathbb{Z}^n$ by a Poisson bivector $θ$ defined along the fibers. In particular, he constructed the noncommutative deformations $L_θ \to X_θ^n$ of holomorphic line bundles on $X^n$ and a curved dg-category consisting of them. On the other hand, associated to this noncommutative deformation, we can construct a non-trivial deformation of the trivial holomorphic line bundle on $X^n$ by twisting it with a suitable isomorphism. In this paper, from this point of view, we extend the construction of $L_θ$ to the more general setting. Moreover, we also consider objects defined on a mirror partner of $X_θ^n$ which are mirror dual to such extended noncommutative objects. |
| title | On a noncommutative deformation of holomorphic line bundles on complex tori and the SYZ transform |
| topic | Differential Geometry High Energy Physics - Theory 14F08, 14J33, 58B34, 53D37, 53C08 |
| url | https://arxiv.org/abs/2603.07239 |