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Kenji Fukaya

Publications and source records attributed to Kenji Fukaya.

At least 19 recordsLinked to original sources
arXiv2024

Corrigendum of "Construction of Kuranishi structures on the moduli spaces of pseudo holomorphic disks I, Surveys in Differential Geometry XXII (2018), 133-190"

This is a corrigendum of Lemma 9.1 of the paper [FOOO3] in the title. This lemma is not correct as pointed out by A. Daemi and a referee of the paper [DF]. The corrigendum does not affect the applications of this lemma in [FOOO3] and other papers and exactly the same proofs as therein apply if one replaces the statement of [FOOO3,Lemma 9.1] by Lemma 2 of the present note.

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arXiv2022

Monotone Lagrangian Floer theory in smooth divisor complements: I

In this paper, Floer homology for Lagrangian submanifolds in an open symplectic manifold given as the complement of a smooth divisor is discussed. The main new feature of this construction is that we do not make any assumption on positivity or negativity of the divisor. To achieve this goal, we use a compactification of the moduli space of pseudo-holomorphic discs into the divisor complement satisfying Lagrangian boundary condition that is stronger than the stable map compactification and is inspired by the compactifications that are used in relative Gromov--Witten theory. This is the first of a series of three papers, this compactification is introduced and some of its fundamental properties as a topological space, essential for the definition of Lagrangian Floer homology, are established.

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arXiv2022

Monotone Lagrangian Floer theory in smooth divisor complements: II

In the first part of the present series of papers, we studied the moduli spaces of holomorphic discs and strips into an open symplectic manifold, isomorphic to the complement of a smooth divisor in a closed symplectic manifold. In particular, we introduced a compactification of this moduli space, which is called the RGW compactification. The goal of this paper is to show that the RGW compactifications admit Kuranishi structures. This result provides the crucial ingredient for the main construction of this series of papers: Floer homology for monotone Lagrangians in a smooth divisor complement.

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arXiv2022

Construction of a linear K-system in Hamiltonian Floer theory

The notion of linear K-system is introduced by the present authors as an abstract model arising from the structure of compactified moduli spaces of solutions to Floer's equation in the book [FOOO14]. The purpose of the present article is to provide a geometric realization of the linear K-system associated with solutions to Floer's equation in the Morse-Bott setting. Immediate consequences (when combined with the abstract theory from [FOOO14]) are construction of Floer cohomology for periodic Hamiltonian system on general compact symplectic manifold without any restriction, and construction of an isomorphism over the Novikov ring between the Floer cohomology and the singular cohomology of the underlying symplectic manifold. The present article utilizes various analytical results on pseudo-holomorphic curves established in our earlier papers and books. However the paper itself is geometric in nature, and does not presume the readers' prior knowledge much on Kuranishi structure and its construction but assumes only the elementary part of thereof, and results from [FOOO11] and [FOOO,Chapter 8] on its construction, and the standard knowledge on Hamiltonian Floer theory. We explain the general procedure of the construction of a linear K-system by explaining in detail the inductive steps of ensuring the compatibility conditions for the system of Kuranishi structures leading to a linear K-system for the case of Hamiltonian Floer theory. A certain minor errors are corrected in this version. (to appear in the special volume dedicated to Claude Viterbo of Journal of Fixed Point Theory and Applications.)

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arXiv2021

Lagrangians, SO(3)-instantons and mixed equation

The mixed equation, defined as a combination of the anti-self-duality equation in gauge theory and Cauchy-Riemann equation in symplectic geometry, is studied. In particular, regularity and Fredholm properties are established for the solutions of this equation, and it is shown that the moduli spaces of solutions to the mixed equation satisfy a compactness property which combines Uhlenbeck and Gormov compactness theorems. The results of this paper are used in a sequel to study the Atiyah-Floer conjecture.

math.DG↗
arXiv2021

Gromov-Hausdorff distance between filtered $A_{\infty}$ categories 1: Lagrangian Floer theory

In this paper we introduce and study a distance, Gromov-Hausdorff distance, which measures how two filtered A $A_{\infty}$ categories are far away each other. In symplectic geometry the author associated a filtered$A_{\infty}$ category, Fukaya category, to a finite set of Lagrangian submanifolds. The Gromov-Hausdorff distance then gives a new invariant of a finite set of Lagrangian submanifolds. One can estimate it by the Hofer distance of Hamiltonian diffeomorphisms needed to send one Lagrangain submanifold to the other. A motivation to introduce Gromov-Hausdorff distance is to obtain a certain completion of Fukaya category. If we have a sequence of sets of Lagrangian submanifolds, which is a Cauchy sequence in the sense of Hofer metric, then the associated filtered A infinity categories also form a Cauchy sequence in Gromov-Hausdorff distance. In this paper we develop a theory to obtain an inductive limit of such a sequence of filtered $A_{\infty}$ categories. In other words, we give an affirmative answer to [Fu5] Conjecture 15.34.

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arXiv2020

Lie groupoid, deformation of unstable curve, and construction of equivariant Kuranishi charts

In this paper we give detailed construction of $G$-equivariant Kuranishi chart of moduli spaces of pseudo-holomorphic curves to a symplectic manifold with $G$-action, for an arbitrary compact Lie group $G$. The proof is based on the deformation theory of {\it unstable} marked curves using the language of Lie groupoid (which is {\it not} necessary etale) and the Riemannnian center of mass technique. This proof is actually similar to [FOn,Sections 13 and 15] except the usage of the language of Lie groupoid makes the argument more transparent. This version correct some errors of the previous version especially those pointed out by the referee.

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arXiv2018

Construction of Kuranishi structures on the moduli spaces of pseudo holomorphic disks: II

This is the second of a series of two articles in which we provide detailed and self-contained account of the construction of a system of Kuranishi structures on the moduli spaces of pseudo holomorphic disks. Using the notion of obstruction bundle data introduced in [FOOO8], we give a systematic way of constructing a system of Kuranishi structures on the moduli spaces of pseudo holomorphic disks which are compatible at the boundary and corners. More specifically, it defines a tree like K-system in the sense of [FOOO6, Definition 21.9]. The method given in this paper does not only simplify the description of the constructions in the earlier literature, but also is designed to provide a systematic utility tool for the construction of a system of Kuranishi structures in the future research. We also establish its uniqueness.

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arXiv2017

Construction of Kuranishi structures on the moduli spaces of pseudo holomorphic disks: I

This is the first of two articles in which we provide detailed and self-contained account of the construction of a system of Kuranishi structures on the moduli spaces of pseudo holomorphic disks, using the exponential decay estimate given in [FOOO7]. This article completes the construction of a Kuranishi structure of a single moduli space. This article is an improved version of [FOOO4, Part 4] and its mathematical content is taken from our earlier writing [FOn,FOOO2,FOOO4,FOOO7]. The version 2 contains a correction of typological errors in the statement of Lemma 10.11 and the statement of Proposition 10.4. The proof of them are not changed. (Note these places were written in the same way as version 1 in the version published in Surveys in Differential Geometry.)

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arXiv2017

Atiyah-Floer Conjecture: a Formulation, a Strategy to Prove and Generalizations

Around 1988, Floer introduced two important theories: instanton Floer homology as invariants of 3-manifolds and Lagrangian Floer homology as invariants of pairs of Lagrangians in symplectic manifolds. Soon after that, Atiyah conjectured that the two theories should be related to each other and Lagrangian Floer homology of certain Lagrangians in the moduli space of flat connections on Riemann surfaces should recover instanton Floer homology. However, the space of flat connections on a Riemann surface is singular and the first step to address this conjecture is to make sense of Lagrangian Floer homology on this space. In this note, we formulate a possible approach to resolve this issue. A strategy to construct the desired isomorphism in the Atiyah-Floer conjecture is also sketched. We also use the language of A infty-categories to state generalizations of the Atiyah-Floer conjecture.

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arXiv2017

Unobstructed Immersed Lagrangian Correspondence and Filtered $A_{\infty}$ Functor

In this paper, we 'construct' a 2-functor from the unobstructed immersed Weinstein category to the category of all filtered $A_{\infty}$ categories. We consider arbitrary (compact) symplectic manifolds and its arbitrary (relatively spin) immersed Lagrangian submanifolds. The filtered $A_{\infty}$ category associated to $(X,\omega)$ is defined by using Lagrangian Floer theory in such generality, see Akaho-Joyce (2010) and Fukaya-Oh-Ohta-Ono (2009). The morphism of unobstructed immersed Weinstein category (from $(X_1,\omega_1)$ to $(X_2,\omega_2)$) is by definition a pair of an immersed Lagrangian submanifold of the direct product and its bounding cochain (in the sense of Akaho-Joyce (2010) and Fukaya-Oh-Ohta-Ono (2009)). Such a morphism transforms an (immersed) Lagrangian submanifold of $(X_1,\omega_1)$ to one of $(X_2,\omega_2)$. The key new result proved in this paper shows that this geometric transformation preserves unobstructedness of the Lagrangian Floer theory. Thus, this paper generalizes earlier results by Wehrheim-Woodward and Mau's-Wehrheim-Woodward so that it works in complete generality in the compact case. The main idea of the proofs are based on Lekili-Lipyanskiy's Y diagram and a lemma from homological algebra, together with systematic use of Yoneda functor. In other words, the proofs are based on a different idea from those which are studied by Bottmann-Mau's-Wehrheim-Woodward, where strip shrinking and figure 8 bubble plays the central role.

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arXiv2017

Kuranishi structure, Pseudo-holomorphic curve, and virtual fundamental chain: Part 2

This article is the second part of the article we promised to write at the end of Section 1 of [FOOO15] (arXiv:1209.4410). (Part I appeared in [Part I] (arXiv:1503.07631).) We discuss the foundation of the virtual fundamental chain and cycle technique, especially its version that appeared in [FOn] and also in Section A1, Section 7.5 [FOOO4], Section 12 [FOOO7], [Fu2]. This article is independent of our earlier writing [FOOO15]. We also do not assume that the readers have any knowledge on the pseudo-holomorphic curve. In this second part, we consider a system of spaces with Kuranishi structures (abbreviated as a K-system) and its simultaneous perturbations.

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arXiv2017

Categorification of invariants in gauge theory and sypmplectic geometry

This is a mixture of survey article and research anouncement. We discuss Instanton Floer homology for 3 manifolds with boundary. We also discuss a categorification of the Lagrangian Floer theory using the unobstructed immersed Lagrangian correspondence as a morphism in the category of symplectic manifolds. During the year 1998-2012, those problems have been studied emphasising the ideas from analysis such as degeneration and adiabatic limit (Instanton Floer homology) and strip shrinking (Lagrangian correspondence). Recently we found that replacing those analytic approach by a combination of cobordism type argument and homological algebra, we can resolve various difficulties in the analytic approach. It thus solves various problems and also simplify many of the proofs.

math.GT↗
arXiv2017

Gromov-Witten theory via Kuranishi structures

In this expository manuscript, we review the construction of Gromov-Witten virtual fundamental class via FOOO's theory of Kuranishi structures for moduli spaces of pseudo-holomorphic maps defined on closed Riemann surfaces. We consider constraints coming from the ambient space and Deligne-Mumford moduli, called primary insertions, as well as intrinsic classes such as $ψ$-classes and Hodge classes.

math.SG↗
arXiv2017

Spectral invariants with bulk, quasimorphisms and Lagrangian Floer theory

In this paper we first develop various enhancements of the theory of spectral invariants of Hamiltonian Floer homology and of Entovi-Polterovich theory of spectral symplectic quasi-states and quasimorphisms by incorporating \emph{bulk deformations}, i.e., deformations by ambient cycles of symplectic manifolds, of the Floer homology and quantum cohomology. Essentially the same kind of construction is independently carried out by Usher [Us4] in a slightly less general context. Then we explore various applications of these enhancements to the symplectic topology, especially new construction of symplectic quasi-states, quasimorphisms and new Lagrangian intersection results on toric manifolds. The most novel part of this paper is to use open-closed Gromov-Witten theory (operator $\frak q$ in [FOOO1] and its variant involving closed orbits of periodic Hamiltonian system) to connect spectral invariants (with bulk deformation), symplectic quasi-states, quasimorphism to the Lagrangian Floer theory (with bulk deformation). We use this open-closed Gromov-Witten theory to produce new examples. Especially using the calculation of Lagrangian Floer homology with bulk deformation in [FOOO3,FOOO4], we produce examples of compact toric manifolds $(M,ω)$ which admits uncountably many independent quasimorphisms $\widetilde{\operatorname{Ham}}(M,ω) \to \mathbb R$. We also obtain a new intersection result of Lagrangian submanifolds on $S^2 \times S^2$ discovered in [FOOO6]. Many of these applications were announced in [FOOO3,FOOO4,FOOO6].

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