相关论文: Holomorphic disks and the disk potential for a fib…

We develop Floer theory of Lagrangian torus fibers in compact symplectic toric orbifolds. We first classify holomorphic orbi-discs with boundary on Lagrangian torus fibers. We show that there exists a class of basic discs such that we have…

Let $L$ be a compact oriented Lagrangian surface in a K\"ahler surface endowed with a complete Riemannian metric (compatible with the symplectic structure and the complex structure) with bounded sectional curvatures and a positive lower…

Let n>3, and let L be a Lagrangian embedding of an n-disk into the cotangent bundle of n-dimensional Euclidean space that agrees with the cotangent fiber over a non-zero point x outside a compact set. Assume that L is disjoint from the…

For the sake of hyperk{\"a}hler SYZ conjecture, finding holomorphic Lagrangian fibrations becomes an important issue. Toric hyperk{\"a}hler manifolds are real dimension $4n$ non-compact hyperk{\"a}hler manifolds which are quaternion analog…

We derive a Floer theoretical SYZ mirror for an equilateral and generic polygon space. The disk potential function of the monotone torus fiber of the caterpillar bending system is calculated by computing non-trivial open Gromov--Witten…

Let $G$ be a connected compact Lie group and let $\mathbb{G}$ be its complexification. In this paper, we establish a correspondence between the moduli spaces of holomorphic disks bounded by a $G$-invariant Lagrangian submanifold $L…

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…

We compute the ring structure of Floer cohomology groups of Lagrangian torus fibers in some toric Fano manifolds continuing the study of \cite{CO}. Related $\AI$-formulas hold for transversal choice of chains. Two different computations are…

The purpose of this paper is to establish several new results about the Hodge theory of Lagrangian fibrations on (not necessarily compact) holomorphic symplectic manifolds. Let $M$ be a holomorphic symplectic manifold of dimension $2n$ that…

We present an unobstructedness criterion for Lagrangian threefolds $L\subset X^A$ using the $H_1(L)$-class associated with the boundary of a pseudoholomorphic disk. As an application, let $X^A\to Q$ be a Lagrangian torus fibration whose…

The purpose of this paper is to carry out a classical construction of a non-constant holomorphic disk with boundary on (the suspension of) a Lagrangian submanifold in $\mathbb{R}^{2 n}$ in the case the Lagrangian is the lift of a…

We build a bridge between Floer theory on open symplectic manifolds and the enumerative geometry of holomorphic disks inside their Fano compactifications, by detecting elements in symplectic cohomology which are mirror to Landau-Ginzburg…

A hypercomplex manifold is a manifold equipped with a triple of complex structures satisfying the quaternionic relations. A holomorphic Lagrangian variety on a hypercomplex manifold with trivial canonical bundle is a holomorphic subvariety…

This is a continuation of part I in the series of the papers on Lagrangian Floer theory on toric manifolds. Using the deformations of Floer cohomology by the ambient cycles, which we call bulk deformations, we find a continuum of…

We suggest a general framework for compactifing quasi-projective Lagrangian fibrations of geometric origin by holomorphic symplectic varieties. This framework includes a compactification criterion, which we then apply to various fibrations…

For a large class of toric domains in $\mathbb{R}^4$ we determine which product Lagrangian tori can be mapped into the domain by a Hamiltonian diffeomorphism. In other words, we compute the Hamiltonian shape invariant of these toric…

Given a Lagrangian submanifold $L$ in a symplectic manifold $X$, the homological Lagrangian monodromy group $\mathcal{H}_L$ describes how Hamiltonian diffeomorphisms of $X$ preserving $L$ setwise act on $H_*(L)$. We begin a systematic study…

We define an equivariant Lagrangian Floer theory for Lagrangian torus fibers in a compact symplectic toric manifold equipped with a subtorus action. We show that the set of all Lagrangian torus fibers with weak bounding cochain data whose…

This paper studies the self-Floer theory of a monotone Lagrangian submanifold $L$ of a symplectic manifold $X$ in the presence of various kinds of symmetry. First we suppose $L$ is $K$-homogeneous and compute the image of low codimension…

We extend Floer theory for monotone Lagrangians to allow coefficients in local systems of arbitrary rank. Unlike the rank 1 case, this is often obstructed by Maslov 2 discs. We study exactly what the obstruction is and define some natural…