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

Kazushi Kobayashi

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 $X_θ^n$ 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.

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

Abstract

By regarding a given -dimensional complex torus as the trivial torus fibration , we can obtain a mirror dual complexified symplectic torus based on the SYZ construction. In the middle 2000s, as a part of the study on noncommutative deformations of , Kajiura examined the noncommutative complex torus obtained via the (real) nonformal deformation quantization of by a Poisson bivector defined along the fibers. In particular, he constructed the noncommutative deformations of holomorphic line bundles on 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 by twisting it with a suitable isomorphism. In this paper, from this point of view, we extend the construction of to the more general setting. Moreover, we also consider objects defined on a mirror partner of which are mirror dual to such extended noncommutative objects.
Paper Structure (27 sections, 18 theorems, 315 equations)

This paper contains 27 sections, 18 theorems, 315 equations.

Table of Contents

  1. Introduction
  2. Background
  3. The homological mirror symmetry and the SYZ construction
  4. A problem on noncommutative deformations of $X^n$
  5. Motivation
  6. Other related works
  7. Main results and the plan of this paper
  8. Notations
  9. Preliminaries
  10. Complex tori and their mirror partners
  11. Holomorphic line bundles on $X^n$
  12. Affine Lagrangian submanifolds in $\check{X}^n$ with unitary local systems
  13. The deformation of holomorphic line bundles associated to a noncommutative deformation of $X^n$
  14. A noncommutative deformation of $E_{(A,p,q)}^{\nabla}$
  15. Ambiguity of a noncommutative deformation of $E_{(A,p,q)}^{\nabla}$
  16. ...and 12 more sections

Key Result

Proposition 2.2

For a given triple $(A,p,q)\in M(n;\mathbb{Z})\times \mathbb{R}^n\times \mathbb{R}^n$, the connection $\nabla_{(A,p,q)}$ is an integrable connection on $E_A \to X^n$ if and only if

Theorems & Definitions (34)

  • Definition 2.1: steven
  • Proposition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • ...and 24 more