2018
DOI: 10.48550/arxiv.1805.11738
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Immersed two-spheres and SYZ with Application to Grassmannians

Abstract: We develop a Floer theoretical gluing technique and apply it to deal with the most generic singular fiber in the SYZ program, namely the product of a torus with the immersed two-sphere with a single nodal self-intersection. As an application, we construct immersed Lagrangians in Gr(2, C n ) and OG(1, C 5 ) and derive their SYZ mirrors. It recovers the Lie theoretical mirrors constructed by Rietsch. It also gives an effective way to compute stable disks (with non-trivial obstructions) bounded by immersed Lagran… Show more

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Cited by 13 publications
(18 citation statements)
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“…The disk potential function of OG(1, C 5 ) was computed in [HKL19]. They have used the Lefschetz fibration model in order to construct a singular Lagrangian and to compute open Gromov-Witten invariants by the wall-crossing.…”
Section: Relation With Other Workmentioning
confidence: 99%
“…The disk potential function of OG(1, C 5 ) was computed in [HKL19]. They have used the Lefschetz fibration model in order to construct a singular Lagrangian and to compute open Gromov-Witten invariants by the wall-crossing.…”
Section: Relation With Other Workmentioning
confidence: 99%
“…Recently, the above speculations have been realized by the work of Hong, Kim and Lau [12], at least for the case of Gr(2, n) (and also for OG (1,5)). The idea is that, by deforming the Gelfand-Cetlin fibration to the special Lagrangian torus fibration ρ, 1 the non-torus fibers of the Gelfand-Cetlin fibration, which are known to be non-trivial objects in the Fukaya category [20], are deformed to certain immersed Lagrangians.…”
Section: Introductionmentioning
confidence: 86%
“…What Hong, Kim and Lau showed in [12] was that deformations of the immersed Lagrangian above (a local model of which was given in [6]) produces the final missing chart in Marsh-Rietsch's mirror of Gr (2, n). This completes the SYZ construction, namely, by applying SYZ (or family Floer theory) to regular as well as singular fibers of the special Lagrangian torus fibration ρ, one can recover Marsh-Rietsch's mirror.…”
Section: Introductionmentioning
confidence: 97%
“…Moreover, gluing between the local mirror charts based on Fukaya isomorphisms was developed in [9]. Applying to singular fibers, it gives a canonical (partial) compactification of the SYZ mirror by gluing the local mirror charts of singular fibers with those of regular tori [27].…”
Section: Introductionmentioning
confidence: 99%
“…On the other hand, in the Fano situation of this paper, we can use the method in [9,27] to construct a C-valued mirror. The special Lagrangian fibration on P 2 \ E [12] has three singular fibers which are nodal tori.…”
Section: Introductionmentioning
confidence: 99%