Homological Mirror Symmetry and Fukaya Categories for Calabi-Yau Hypersurfaces in Toric Varieties

This monograph establishes a rigorous framework for Homological Mirror Symmetry (HMS) in the context of Calabi-Yau hypersurfaces embedded within smooth Fano toric varieties. We construct the derived Fukaya category F(Y) of the mirror Landau-Ginzburg model, explicitly analyzing the A-structures arising from Lagrangian intersection Floer theory. By invoking the tropical degeneration of the mirror manifold, we identify a split-generator of the Fukaya category consisting of an admissible Lagrangian section of the SYZ fibration. We subsequently prove that the A-endomorphism algebra of this generator is quasi-isomorphic to the differential graded algebra governing the derived category of coherent sheaves Db(Coh(X)) on the dual Calabi-Yau hypersurface X. This confirms Kontsevich's conjecture for this class of varieties, extending results by Sheridan and Abouzaid to higher-dimensional toric ambient spaces with non-trivial obstruction terms.