Monotone Lagrangian Floer theory in smooth divisor complements: I

TL;DR

This paper develops a Floer homology framework for Lagrangian submanifolds in symplectic manifolds with smooth divisors, introducing a novel compactification method that does not rely on positivity assumptions.

Contribution

It introduces a new compactification of moduli spaces for pseudo-holomorphic discs in divisor complements, enabling Floer homology without positivity constraints.

Findings

Established fundamental properties of the new compactification.

Defined Floer homology in the divisor complement setting.

Paved the way for further developments in Floer theory in open symplectic manifolds.

Abstract

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.