Monotone Lagrangian Floer theory in smooth divisor complements: I

Abstract

In this paper, we discuss Floer homology of Lagrangian submanifolds in an open symplectic manifold given as the complement of a smooth divisor. Firstly, a compactification of moduli spaces of holomorphic strips in a smooth divisor complement is introduced. Next, this compactification is used to define Lagrangian Floer homology of two Lagrangians in the divisor complement. The main new feature of this paper is that we do not make any assumption on positivity or negativity of the divisor.

Monotone Lagrangian Floer theory in smooth divisor complements: II

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In [DF2], the first part of the present paper, we study the moduli spaces of holomorphic discs and strips into an open symplectic, isomorphic to the complement of a smooth divisor in a closed symplectic manifold. In particular, we introduce a compactification of this moduli space, which is called the relative Gromov-Witten compactification.

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“…In [DF2], the authors studied Lagrangian intersection Floer homology of a pair of monotone Lagrangians in an open symplectic manifold, which is isomorphic to a divisor complement. At the heart of the construction of [DF2], there is a compactification of the moduli spaces of holomorphic discs and strips satisfying Lagrangian boundary condition. The main purpose of this sequel to [DF2] is to show that this compactification, called the RGW compactification, admits a Kuranishi structure.…”

“…At the heart of the construction of [DF2], there is a compactification of the moduli spaces of holomorphic discs and strips satisfying Lagrangian boundary condition. The main purpose of this sequel to [DF2] is to show that this compactification, called the RGW compactification, admits a Kuranishi structure. The virtual count of the elements of these Kuranishi spaces is used in [DF2] to define the desired Lagrangian Floer homology.…”

“…The main purpose of this sequel to [DF2] is to show that this compactification, called the RGW compactification, admits a Kuranishi structure. The virtual count of the elements of these Kuranishi spaces is used in [DF2] to define the desired Lagrangian Floer homology.…”

“…For any β ∈ Π 2 (X; L i ) with β ∩ D = 0, let M reg k+1 (L i ; β) be the moduli space of J-holomorphic disks of homology class β with k + 1 boundary marked points and Lagrangian boundary condition associated to L i . In [DF2], we introduced the RGW compactification M RGW k+1 (L i ; β) of M reg k+1 (L i ; β). (See [DF2, Section 3] for the definition of this moduli space.…”