On the reconstruction problem in mirror symmetry

Tu, Junwu

doi:10.1016/j.aim.2014.02.005

Cited by 40 publications

(50 citation statements)

References 12 publications

(26 reference statements)

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“…Holomorphic discs are the building blocks of the Fukaya category [12,33,15]. Holomorphic discs also play the role of "quantum corrections" in the reconstruction of mirror manifolds [13,35,1,14,48]. More precisely, the "quantum corrections" arise from counting holomorphic discs with boundaries on torus fibers of an SYZ fibration (Strominger-Yau-Zaslow [46]).…”

Section: Introductionmentioning

confidence: 99%

Enumeration of holomorphic cylinders in log Calabi–Yau surfaces. I

2016

Math. Ann.

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Abstract. We define the counting of holomorphic cylinders in log CalabiYau surfaces. Although we start with a complex log Calabi-Yau surface, the counting is achieved by applying methods from non-archimedean geometry. This gives rise to new geometric invariants. Moreover, we prove that the counting satisfies a property of symmetry. Explicit calculations are given for a del Pezzo surface in detail, which verify the conjectured wall-crossing formula for the focusfocus singularity. Our holomorphic cylinders are expected to give a geometric understanding of the combinatorial notion of broken line by Gross, Hacking, Keel and Siebert. Our tools include Berkovich spaces, tropical geometry, GromovWitten theory and the GAGA theorem for non-archimedean analytic stacks.

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Section: Introductionmentioning

confidence: 99%

Enumeration of holomorphic cylinders in log Calabi–Yau surfaces. I

2016

Math. Ann.

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“…The unobstructedness of fibers of Lagrangian fibrations is needed for the family Floer cohomology approach to proving homological mirror symmetry initiated by Fukaya [9] and further developed in recent work of J. Tu [28] and M. Abouzaid [1,2].…”

Section: 3mentioning

confidence: 99%

Involutions, obstructions and mirror symmetry

Solomon

2020

Advances in Mathematics

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Consider a Maslov zero Lagrangian submanifold diffeomorphic to a Lie group on which an anti-symplectic involution acts by the inverse map of the group. We show that the Fukaya A ∞ endomorphism algebra of such a Lagrangian is quasi-isomorphic to its de Rham cohomology tensored with the Novikov field. In particular, it is unobstructed, formal, and its Floer and de Rham cohomologies coincide. Our result implies that the smooth fibers of a large class of singular Lagrangian fibrations are unobstructed and their Floer and de Rham cohomologies coincide. This is a step in the SYZ and family Floer cohomology approaches to mirror symmetry.More generally, our result continues to hold if the Lagrangian has cohomology the free graded algebra on a graded vector space V concentrated in odd degree, and the antisymplectic involution acts on the cohomology of the Lagrangian by the induced map of negative the identity on V. It suffices for the Maslov class to vanish modulo 4.

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“…We will follow the idea of family Floer homology [5] [6] (see also [27]) and review the construction of the open Gromov-Witten invariants on elliptic K3 surface defined in [18]. Let X → B be an elliptic K3 surface and let L u denote the fibre over u ∈ B.…”

Section: Open Gromov-witten Invariants On K3 Surfacesmentioning

confidence: 99%