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Transactions of the American Mathematical Society

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SYZ transforms for immersed Lagrangian multisections


Authors: Kwokwai Chan and Yat-Hin Suen
Journal: Trans. Amer. Math. Soc. 372 (2019), 5747-5780
MSC (2010): Primary 53D37; Secondary 53D40, 53D12
DOI: https://doi.org/10.1090/tran/7757
Published electronically: May 20, 2019
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Abstract: In this paper, we study the geometry of the SYZ transform on a semiflat Lagrangian torus fibration. Our starting point is an investigation on the relation between Lagrangian surgery of a pair of straight lines in a symplectic 2-torus and the extension of holomorphic vector bundles over the mirror elliptic curve, via the SYZ transform for immersed Lagrangian multisections defined by Arinkin and Joyce [Fukaya category and Fourier transform, AMS/IP Stud. Adv. Math., Amer. Math. Soc., Providence, RI, 2001] and Leung, Yau, and Zaslow [Adv. Theor. Math. Phys. 4 (2000), no. 6, 1319-1341]. This study leads us to a new notion of equivalence between objects in the immersed Fukaya category of a general compact symplectic manifold $ (M, \omega )$, under which the immersed Floer cohomology is invariant; in particular, this provides an answer to a question of Akaho and Joyce [J. Differential Geom. 86 (2010), no. 3, 831-500, Question 13.15]. Furthermore, if $ M$ admits a Lagrangian torus fibration over an integral affine manifold, we prove, under some additional assumptions, that this new equivalence is mirror to an isomorphism between holomorphic vector bundles over the dual torus fibration via the SYZ transform.


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Additional Information

Kwokwai Chan
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
Email: kwchan@math.cuhk.edu.hk

Yat-Hin Suen
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
Address at time of publication: Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang 37673, Republic of Korea
Email: yhsuen@ibs.re.kr

DOI: https://doi.org/10.1090/tran/7757
Received by editor(s): April 16, 2018
Received by editor(s) in revised form: October 23, 2018
Published electronically: May 20, 2019
Additional Notes: The work of the first author described in this paper was substantially supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CUHK14302015 $&$ CUHK14302617).
The work of the second author was supported by IBS-R003-D1.
Article copyright: © Copyright 2019 American Mathematical Society