Homological mirror symmetry for Batyrev mirrors (general quasi-equivalence)

Establish a quasi-equivalence of Λ-linear, ℤ-graded A∞ categories Perf(X_{t,}, ω_)^bc ≃ D^b_{dg}Coh(X^*_{b()}), for Calabi–Yau hypersurfaces X_{t,} constructed from dual reflexive polytopes with Σ^* simplicial, where b() ∈ 𝔸^P_Λ satisfies val(b()_p) = _p for all p ∈ P. This asserts the full homological mirror symmetry equivalence between the relative Fukaya category of X_{t,} (with bounding cochains) and the dg-derived category of coherent sheaves on the Batyrev mirror X^*_{b()} determined by the mirror map.

Background

The paper studies homological mirror symmetry (HMS) for Batyrev mirror pairs of Calabi–Yau hypersurfaces in toric varieties. It constructs the A-side (relative Fukaya category of the symplectic hypersurface X_{t,}) and the B-side (derived category of coherent sheaves on the mirror hypersurface X^*_r), and formulates the expected HMS equivalence in terms of a mirror map b().

While the main theorem proves a fully faithful embedding and split-generation in many cases (under smoothness of Σ^*, MPCS and connectedness conditions, and characteristic assumptions), the general HMS quasi-equivalence is presented as a conjecture in the simplicial case without those additional hypotheses. The equivalence is expected to hold for the Fukaya category with bounding cochains and the dg-derived category on the B-side, with the mirror map’s valuation constraints tying A-side deformations to B-side parameters.