On a noncommutative deformation of holomorphic line bundles on complex tori and the SYZ transform

Abstract

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 $X_{\theta}^n$ obtained via the (real) nonformal deformation quantization of $X^n \to \mathbb{R}^n/\mathbb{Z}^n$ by a Poisson bivector $\theta$ defined along the fibers. In particular, he constructed the noncommutative deformations $L_{\theta} \to X_{\theta}^n$ 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 $L_{\theta}$ to the more general setting. Moreover, we also consider objects defined on a mirror partner of $X_{\theta}^n$ which are mirror dual to such extended noncommutative objects.