Lagrangian Floer Homology and Mirror Symmetry for Fano Manifolds with Divisor Complements

We investigate the structural correspondence between the Lagrangian Floer homology of a Fano manifold X relative to an anticanonical divisor D |-KX| and the singularity category of its mirror Landau-Ginzburg model (Y, W). By analyzing the A-structure of the Fukaya category F(X D) enriched by bounding cochains, we establish a rigorous isomorphism between the quantum-corrected Floer cohomology rings and the Jacobian rings of the superpotential W. We specifically address the wall-crossing phenomena of Maslov index 0 discs in the SYZ fibration, demonstrating that the disc instanton corrections to the mirror complex structure precisely match the mutation of the superpotential. Furthermore, we prove a spectral sequence convergence relating the relative symplectic cohomology SH*(X, D) to the Hochschild homology of the mirror category of matrix factorizations, thereby verifying a relative version of the Homological Mirror Symmetry conjecture for log-Calabi-Yau pairs.