Maurer–Cartan description via domain-dependent neck stretching and divisor complements

Establish that when there exists a symplectic crossing divisor V ⊂ X that carries the symplectic form and satisfies D ⊂ X \ V, there is a Maurer–Cartan element x in the intrinsic symplectic cochains SC^*_{\theta}(D) (well-defined up to gauge equivalence) such that the ambient symplectic cochain complex SC^*_M(D) is quasi-isomorphic to the twist of SC^*_{\theta}(D) by x.

Background

Beyond S^1-equivariant constructions, the authors propose a domain-dependent neck-stretching framework tailored to settings where D lies in the complement of a symplectic crossing divisor V ⊂ X. In this scenario, deformation data are expected to arise directly in SC^*_{\theta}(D), the intrinsic complex defined via a local Liouville primitive θ.

The conjecture asserts that the ambient deformation SC^*_M(D) can be obtained by twisting SC^*_{\theta}(D) by a Maurer–Cartan element x extracted from holomorphic planes in X\V. This formulation is closely related to recent approaches by Alami–Borman–Sheridan on producing Maurer–Cartan elements from compactifications and would generalize their methods to this broader setting.

A resolution would yield a uniform mechanism to encode ambient effects in divisor complement geometries, bridging SFT-type neck stretching with intrinsic Floer-theoretic structures.