Cyclic symmetry and adic convergence in Lagrangian Floer theory

Fukaya, Kenji

doi:10.1215/0023608x-2010-004

Cited by 113 publications

(327 citation statements)

References 12 publications

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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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“…Such a theory would require extensive reworking of the foundations. On the other hand, if successful, it might allow substantial simplifications and extensions of the main arguments in this paper, bringing them in line with existing ideas about convergence in Floer cohomology [31,33].…”

Section: Working Definition 125 a Class [G] ∈ Hhmentioning

confidence: 77%

“…To make a more abstract version of the argument, we consider families of objects in the Fukaya category whose deformation is driven by the class [dq]. The idea of introducing families of Lagrangian submanifolds into Floer cohomology theory is due to Fukaya [31,33]. Generally speaking, it shows much potential for leading to fundamental insights as well as applications, but also encounters considerable foundational difficulties.…”

Section: Introductionmentioning

confidence: 99%

Abstract Analogues of Flux as Symplectic Invariants

Seidel¹

2014

Mémoires Soc. Math. France

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“…(See [6] Section 4.3.) In [10,11,5] we enhanced it to one over Λ R 0,nov coefficients in case R = R or R = C. We will discuss this enhancement in the case when R is a Dedekind domain or a finite field in Subsection 15.3.…”

Section: 2mentioning

confidence: 99%

Lagrangian floer Theory over integers: spherically positive symplectic manifolds

Fukaya

Ohta

et al. 2013

Pure and Applied Mathematics Quarterly

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Abstract. In this paper we study the Lagrangian Floer theory over Z or Z 2 . Under an appropriate assumption on ambient symplectic manifold, we show that the whole story of Lagrangian Floer theory in [6], [7] can be developed over Z 2 coefficients, and over Z coefficients when Lagrangian submanifolds are relatively spin. The main technical tools used for the construction are the notion of the sheaf of groups, and stratification and compatibility of the normal cones applied to the Kuranishi structure of the moduli space of pseudoholomorphic discs.

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Cyclic symmetry and adic convergence in Lagrangian Floer theory

Cited by 113 publications

References 12 publications

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

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

Abstract Analogues of Flux as Symplectic Invariants

Lagrangian floer Theory over integers: spherically positive symplectic manifolds

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