Searcharxiv⌕ Search

arXiv subjects

Kazushi Kobayashi

Publications and source records attributed to Kazushi Kobayashi.

7 recordsLinked to original sources
arXiv2026

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

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.

math.DG↗
arXiv2024

On a B-field transform of generalized complex structures over complex tori

Let $(X^n,\check{X}^n)$ be a mirror pair of an $n$-dimensional complex torus $X^n$ and its mirror partner $\check{X}^n$. Then, by SYZ transform, we can construct a holomorphic line bundle with an integrable connection from each pair of a Lagrangian section of $\check{X}^n\to \mathbb{R}^n/\mathbb{Z}^n$ and a unitary local system along it, and those holomorphic line bundles with integrable connections forms a dg-category $DG_{X^n}$. In this paper, we focus on a certain B-field transform of the generalized complex structure induced from the complex structure on $X^n$, and interpret it as the deformation $X_{\mathcal{G}}^n$ of $X^n$ by a flat gerbe $\mathcal{G}$. Moreover, we construct the deformation of $DG_{X^n}$ associated to the deformation from $X^n$ to $X_{\mathcal{G}}^n$, and also discuss the homological mirror symmetry between $X_{\mathcal{G}}^n$ and its mirror partner on the object level.

math.DG↗
arXiv2023

A gerby deformation of complex tori and the homological mirror symmetry

Let $(X,\check{X})$ be a mirror pair of a complex torus $X$ and its mirror partner $\check{X}$. This mirror pair is described as the trivial special Lagrangian torus fibrations $X\rightarrow B$ and $\check{X}\rightarrow B$ on the same base space $B$ by SYZ construction. Then, we can associate a holomorphic line bundle $E(s,\mathcal{L})\rightarrow X$ to a pair $(s,\mathcal{L})$ of a Lagrangian section $s$ of $\check{X}\rightarrow B$ and a unitary local system $\mathcal{L}$ along it. In this paper, we first construct the deformation $X_{\mathcal{G}}$ of $X$ by a certain flat gerbe $\mathcal{G}$ and its mirror partner $\check{X}_{\mathcal{G}}$ from the mirror pair $(X,\check{X})$, and discuss deformations of objects $E(s,\mathcal{L})$ and $(s,\mathcal{L})$ over the deformed mirror pair $(X_{\mathcal{G}},\check{X}_{\mathcal{G}})$.

math.DG↗
arXiv2021

Geometric interpretation for exact triangles consisting of projectively flat bundles on higher dimensional complex tori

Let $(X^n, \check{X}^n)$ be a mirror pair of an $n$-dimensional complex torus $X^n$ and its mirror partner $\check{X}^n$. Then, a simple projectively flat bundle $E(L,\mathcal{L})\rightarrow X^n$ is constructed from each affine Lagrangian submanifold $L$ in $\check{X}^n$ with a unitary local system $\mathcal{L} \rightarrow L$. In this paper, we first interpret these simple projectively flat bundles $E(L,\mathcal{L})$ in the language of factors of automorphy. Furthermore, we give a geometric interpretation for exact triangles consisting of three simple projectively flat bundles $E(L,\mathcal{L})$ and their shifts by focusing on the dimension of intersections of the corresponding affine Lagrangian submanifolds $L$. Finally, as an application of this geometric interpretation, we discuss whether such an exact triangle on $X^n$ ($n \geq 2$) is obtained as the pullback of an exact triangle on $X^1$ by a suitable holomorphic projection $X^n \rightarrow X^1$.

math.DG↗
arXiv2021

Remarks on the homological mirror symmetry for tori

Let us consider an $n$-dimensional complex torus $T^{2n}_{J=T}:=\mathbb{C}^n/2π(\mathbb{Z}^n \oplus T\mathbb{Z}^n)$. Here, $T$ is a complex matrix of order $n$ whose imaginary part is positive definite. In particular, when we consider the case $n=1$, the complexified symplectic form of a mirror partner of $T^2_{J=T}$ is defined by using $-\frac{1}{T}$ or $T$. However, if we assume $n \geq 2$ and that $T$ is a singular matrix, we can not define a mirror partner of $T^{2n}_{J=T}$ as a natural generalization of the case $n=1$ to the higher dimensional case. In this paper, we propose a way to avoid this problem, and discuss the homological mirror symmetry.

math.DG↗
arXiv2020

The bijectivity of mirror functors on tori

By the SYZ construction, a mirror pair $(X,\check{X})$ of a complex torus $X$ and a mirror partner $\check{X}$ of the complex torus $X$ is described as the special Lagrangian torus fibrations $X \rightarrow B$ and $\check{X} \rightarrow B$ on the same base space $B$. Then, by the SYZ transform, we can construct a simple projectively flat bundle on $X$ from each affine Lagrangian multi section of $\check{X} \rightarrow B$ with a unitary local system along it. However, there are ambiguities of the choices of transition functions of it, and this causes difficulties when we try to construct a functor between the symplectic geometric category and the complex geometric category. In this paper, we prove that there exists a bijection between the set of the isomorphism classes of their objects by solving this problem.

math.DG↗