Lagrangian Floer theory on compact toric manifolds I

The present authors introduced the notion of \emph{weakly unobstructed} Lagrangian submanifolds and constructed their \emph{potential function} $\mathfrak{PO}$ purely in terms of $A$-model data in [FOOO2]. In this paper, we carry out explicit calculations involving $\mathfrak{PO}$ on toric manifolds and study the relationship between this class of Lagrangian submanifolds with the earlier work of Givental [Gi1] which advocates that quantum cohomology ring is isomorphic to the Jacobian ring of a certain function, called the Landau-Ginzburg superpotential. Combining this study with the results from [FOOO2], we also apply the study to various examples to illustrate its implications to symplectic topology of Lagrangian fibers of toric manifolds. In particular we relate it to Hamiltonian displacement property of Lagrangian fibers and to Entov-Polterovich's symplectic quasi-states.Comment: 84 pages, submitted version ; more examples and new results added, exposition polished, minor typos corrected; v3) to appear in Duke Math.J., Example 10.19 modified, citations from the book [FOOO2,3] updated accoding to the final version of [FOOO3] to be publishe

Local system and smooth correspondence in de rham theory with twisted coefficients

Let ℒ be a local system, i.e., a flat vector bundle, on a manifold M. We denote by (Ω •(M; ℒ) = Γ (M; ∧ •T∗M⊗ ℒ), d= dℒ) the de Rham complex with coefficients in ℒ. We recall some basic operations on the de Rham complex with twisted coefficients. © Springer Nature Singapore Pte Ltd 2020.11Nscopu

Covering space of effective orbifolds and k-spaces

We first define the notion of a covering space of an orbifold. Let U1, U2 be orbifolds and let π : U1 → U2 be a continuous map between their underlying topological spaces. © Springer Nature Singapore Pte Ltd 2020.11Nscopu

Stokes’ formula

In this chapter, we state and prove Stokes’ formula. We first discuss the notion of a boundary or a corner of an orbifold and of a Kuranishi structure in more detail. The discussion below is a detailed version of [FOOO4, the last paragraph of page 762]. See also [Jo1, page 11]. [Jo3] gives a systematic account on this issue. © Springer Nature Singapore Pte Ltd 2020.11Nscopu

Cf-perturbations and integration along the fiber (pushout)

As we mentioned in the Introduction, we study systems of K-spaces (K-systems) so that the boundary of each of its members is described by a fiber product of other members. We will obtain an algebraic structure on certain cochain complexes which realize the homology groups of certain spaces. They are the spaces over which we take fiber products between members of the system of K-spaces. © Springer Nature Singapore Pte Ltd 2020.11Nscopu

Construction of good coordinate systems

In this chapter we prove Theorem 3.35 together with various addenda and variants. © Springer Nature Singapore Pte Ltd 2020.11Nscopu

Multivalued perturbations

We next define the notion of multivalued perturbations associated to a given good coordinate system. © Springer Nature Singapore Pte Ltd 2020.11Nscopu

Construction of cf-perturbations

In this chapter, we prove Theorem 7.51, that is, the existence of CF-perturbations with respect to which a given weakly submersive map becomes strongly submersive. We also prove its relative version, Proposition 7.59. For this purpose we use the language of sheaf theory to prove Proposition 12.2 for a single Kuranishi chart and Theorem 12.24 for a general case. © Springer Nature Singapore Pte Ltd 2020.11Nscopu

Kuranishi structures and good coordinate systems

In this chapter we define the notions of a Kuranishi structure and of a good coordinate system. We also study embedding between them, which describes a relation among those structures. © Springer Nature Singapore Pte Ltd 2020.11Nscopu