In 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. The goal of this paper is to show that the RGW compactifications admit Kuranishi structures which are compatible with each other. This result provides the remaining ingredient for the main construction of the first part: Floer homology for monotone Lagrangians in a smooth divisor complement.
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