SYZ transforms for immersed Lagrangian multisections
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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, 1319โ1341]. 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, 831โ500, 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.References
- Mohammed Abouzaid, Homogeneous coordinate rings and mirror symmetry for toric varieties, Geom. Topol. 10 (2006), 1097โ1156. [Paging previously given as 1097โ1157]. MR 2240909, DOI 10.2140/gt.2006.10.1097
- Mohammed Abouzaid, On the Fukaya categories of higher genus surfaces, Adv. Math. 217 (2008), no.ย 3, 1192โ1235. MR 2383898, DOI 10.1016/j.aim.2007.08.011
- Mohammed Abouzaid, Morse homology, tropical geometry, and homological mirror symmetry for toric varieties, Selecta Math. (N.S.) 15 (2009), no.ย 2, 189โ270. MR 2529936, DOI 10.1007/s00029-009-0492-2
- Manabu Akaho and Dominic Joyce, Immersed Lagrangian Floer theory, J. Differential Geom. 86 (2010), no.ย 3, 381โ500. MR 2785840
- D. Arinkin and A. Polishchuk, Fukaya category and Fourier transform, Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999) AMS/IP Stud. Adv. Math., vol. 23, Amer. Math. Soc., Providence, RI, 2001, pp.ย 261โ274. MR 1876073, DOI 10.1090/amsip/023/11
- M. F. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 (1957), 414โ452. MR 131423, DOI 10.1112/plms/s3-7.1.414
- Denis Auroux, A beginnerโs introduction to Fukaya categories, Contact and symplectic topology, Bolyai Soc. Math. Stud., vol. 26, Jรกnos Bolyai Math. Soc., Budapest, 2014, pp.ย 85โ136. MR 3220941, DOI 10.1007/978-3-319-02036-5_{3}
- E. Ballico and B. Russo, Exact sequences of semistable vector bundles on algebraic curves, Bull. London Math. Soc. 32 (2000), no.ย 5, 537โ546. MR 1767706, DOI 10.1112/S0024609300007232
- U. Bruzzo, G. Marelli, and F. Pioli, A Fourier transform for sheaves on real tori. I. The equivalence $\textrm {Sky}(T)\simeq \textrm {Loc}(\hat T)$, J. Geom. Phys. 39 (2001), no.ย 2, 174โ182. MR 1844832, DOI 10.1016/S0393-0440(01)00009-2
- U. Bruzzo, G. Marelli, and F. Pioli, A Fourier transform for sheaves on real tori. II. Relative theory, J. Geom. Phys. 41 (2002), no.ย 4, 312โ329. MR 1888468, DOI 10.1016/S0393-0440(01)00065-1
- Philip Candelas, Xenia C. de la Ossa, Paul S. Green, and Linda Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), no.ย 1, 21โ74. MR 1115626, DOI 10.1016/0550-3213(91)90292-6
- Kwokwai Chan, Holomorphic line bundles on projective toric manifolds from Lagrangian sections of their mirrors by SYZ transformations, Int. Math. Res. Not. IMRN 24 (2009), 4686โ4708. MR 2564372, DOI 10.1093/imrn/rnp105
- Kwokwai Chan, Homological mirror symmetry for $A_n$-resolutions as a $T$-duality, J. Lond. Math. Soc. (2) 87 (2013), no.ย 1, 204โ222. MR 3022713, DOI 10.1112/jlms/jds048
- Kwokwai Chan and Naichung Conan Leung, Mirror symmetry for toric Fano manifolds via SYZ transformations, Adv. Math. 223 (2010), no.ย 3, 797โ839. MR 2565550, DOI 10.1016/j.aim.2009.09.009
- Kwokwai Chan and Naichung Conan Leung, On SYZ mirror transformations, New developments in algebraic geometry, integrable systems and mirror symmetry (RIMS, Kyoto, 2008) Adv. Stud. Pure Math., vol. 59, Math. Soc. Japan, Tokyo, 2010, pp.ย 1โ30. MR 2683205, DOI 10.2969/aspm/05910001
- Kwokwai Chan and Naichung Conan Leung, Matrix factorizations from SYZ transformations, Advances in geometric analysis, Adv. Lect. Math. (ALM), vol. 21, Int. Press, Somerville, MA, 2012, pp.ย 203โ224. MR 3077258
- K. Chan, D. Pomerleano, and K. Ueda, Lagrangian sections on mirrors of toric Calabi-Yau 3-folds, preprint (2016), arXiv:1602.07075.
- Kwokwai Chan, Daniel Pomerleano, and Kazushi Ueda, Lagrangian torus fibrations and homological mirror symmetry for the conifold, Comm. Math. Phys. 341 (2016), no.ย 1, 135โ178. MR 3439224, DOI 10.1007/s00220-015-2477-7
- Kwokwai Chan and Kazushi Ueda, Dual torus fibrations and homological mirror symmetry for $A_n$-singlarities, Commun. Number Theory Phys. 7 (2013), no.ย 2, 361โ396. MR 3164868, DOI 10.4310/CNTP.2013.v7.n2.a5
- Jingyi Chen, Lagrangian sections and holomorphic $\textrm {U}(1)$-connections, Pacific J. Math. 203 (2002), no.ย 1, 139โ160. MR 1895929, DOI 10.2140/pjm.2002.203.139
- B. Fang, Central charges of T-dual branes for toric varieties, preprint (2016), arXiv:1611.05153.
- Bohan Fang, Homological mirror symmetry is $T$-duality for $\Bbb P^n$, Commun. Number Theory Phys. 2 (2008), no.ย 4, 719โ742. MR 2492197, DOI 10.4310/CNTP.2008.v2.n4.a2
- Bohan Fang, Chiu-Chu Melissa Liu, David Treumann, and Eric Zaslow, The coherent-constructible correspondence and homological mirror symmetry for toric varieties, Geometry and analysis. No. 2, Adv. Lect. Math. (ALM), vol. 18, Int. Press, Somerville, MA, 2011, pp.ย 3โ37. MR 2882439
- Bohan Fang, Chiu-Chu Melissa Liu, David Treumann, and Eric Zaslow, T-duality and homological mirror symmetry for toric varieties, Adv. Math. 229 (2012), no.ย 3, 1875โ1911. MR 2871160, DOI 10.1016/j.aim.2011.10.022
- Kenji Fukaya, Mirror symmetry of abelian varieties and multi-theta functions, J. Algebraic Geom. 11 (2002), no.ย 3, 393โ512. MR 1894935, DOI 10.1090/S1056-3911-02-00329-6
- Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Lagrangian intersection Floer theory: anomaly and obstruction. Part I, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009. MR 2553465, DOI 10.1090/amsip/046.1
- Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Lagrangian intersection Floer theory: anomaly and obstruction. Part II, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009. MR 2548482, DOI 10.1090/amsip/046.2
- Mikhael Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 1986. MR 864505, DOI 10.1007/978-3-662-02267-2
- Oleksandr Iena, Vector bundles on elliptic curves and factors of automorphy, Rend. Istit. Mat. Univ. Trieste 43 (2011), 61โ94. MR 2933124
- Kazushi Kobayashi, On exact triangles consisting of stable vector bundles on tori, Differential Geom. Appl. 53 (2017), 268โ292. MR 3679186, DOI 10.1016/j.difgeo.2017.06.005
- Maxim Kontsevich, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zรผrich, 1994) Birkhรคuser, Basel, 1995, pp.ย 120โ139. MR 1403918
- J. Alexander Lees, On the classification of Lagrange immersions, Duke Math. J. 43 (1976), no.ย 2, 217โ224. MR 410764
- Naichung Conan Leung, Mirror symmetry without corrections, Comm. Anal. Geom. 13 (2005), no.ย 2, 287โ331. MR 2154821, DOI 10.4310/CAG.2005.v13.n2.a2
- Naichung Conan Leung, Shing-Tung Yau, and Eric Zaslow, From special Lagrangian to Hermitian-Yang-Mills via Fourier-Mukai transform, Adv. Theor. Math. Phys. 4 (2000), no.ย 6, 1319โ1341. MR 1894858, DOI 10.4310/ATMP.2000.v4.n6.a5
- Barbara Russo and Montserrat Teixidor i Bigas, On a conjecture of Lange, J. Algebraic Geom. 8 (1999), no.ย 3, 483โ496. MR 1689352
- Andrew Strominger, Shing-Tung Yau, and Eric Zaslow, Mirror symmetry is $T$-duality, Nuclear Phys. B 479 (1996), no.ย 1-2, 243โ259. MR 1429831, DOI 10.1016/0550-3213(96)00434-8
- R. P. Thomas, Moment maps, monodromy and mirror manifolds, Symplectic geometry and mirror symmetry (Seoul, 2000) World Sci. Publ., River Edge, NJ, 2001, pp.ย 467โ498. MR 1882337, DOI 10.1142/9789812799821_{0}013
Additional Information
- Kwokwai Chan
- Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
- MR Author ID: 821162
- 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
- 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. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 5747-5780
- MSC (2010): Primary 53D37; Secondary 53D40, 53D12
- DOI: https://doi.org/10.1090/tran/7757
- MathSciNet review: 4014293