October 2023 A gerby deformation of complex tori and the homological mirror symmetry
Kazushi Kobayashi
Author Affiliations +
Kodai Math. J. 46(3): 291-323 (October 2023). DOI: 10.2996/kmj46304

Abstract

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 $(X_{\mathcal{G}},\check{X}_{\mathcal{G}})$.

Funding Statement

This work was supported by JSPS KAKENHI Grant Number 21H04994.

Acknowledgment

I am grateful to Hiroshige Kajiura for helpful comments. I also would like to thank Manabu Akaho for telling me the notion of deformed Hermitian-Yang-Mills connections. Finally, I am grateful to the referee for reading this paper carefully.

Citation

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Kazushi Kobayashi. "A gerby deformation of complex tori and the homological mirror symmetry." Kodai Math. J. 46 (3) 291 - 323, October 2023. https://doi.org/10.2996/kmj46304

Information

Received: 8 November 2022; Revised: 1 March 2023; Published: October 2023
First available in Project Euclid: 3 October 2023

MathSciNet: MR4649986
Digital Object Identifier: 10.2996/kmj46304

Subjects:
Primary: 14J33 , 53C08 , 53D37
Secondary: 14F08 , 53D18

Keywords: gerbe , homological mirror symmetry , Torus

Rights: Copyright © 2023 Tokyo Institute of Technology, Department of Mathematics

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Vol.46 • No. 3 • October 2023
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