Monotone Lagrangian Floer theory in smooth divisor complements: I

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.