Homological mirror symmetry for Batyrev mirrors (simplicial Σ*)

Establish a quasi-equivalence between the idempotent-completed pre-triangulated closure Perf(X_{t,}, ω_)^bc of the Fukaya category of the symplectic Calabi–Yau hypersurface X_{t,} ⊂ Y (with Kähler form ω_) and the dg derived category of coherent sheaves D^b_{dg}Coh(X^*_{b()}) on the mirror hypersurface X^*_{b()} ⊂ Y^* over the Novikov field Λ, in the setting of Batyrev mirror pairs constructed from dual reflexive polytopes (Δ, Δ*), assuming the dual fan Σ* is simplicial. The mirror parameter b() ∈ A^P_Λ should satisfy val(b()_p) = ε_p for each lattice point p in P (the set of lattice points on codimension ≥ 2 faces of Δ*), where ε_p are the A-side Kähler weights governing the deformation.

Background

The paper studies homological mirror symmetry (HMS) for Calabi–Yau hypersurfaces arising from Batyrev mirror pairs associated to dual reflexive polytopes. On the A-side, the authors construct a relative/Fukaya-type category Perf(X_{t,}, ω)^bc for a symplectic Calabi–Yau hypersurface X{t,} inside a toric variety Y equipped with a Kähler form ω_. On the B-side, they consider the derived dg category of coherent sheaves on the mirror hypersurface X^*_r in the dual toric variety Y^*, defined over a Novikov field Λ with parameters r indexed by lattice points p in P.

They formulate a precise HMS conjecture asserting a Λ-linear ℤ-graded A_∞ quasi-equivalence between these two categories under the assumption that the dual fan Σ* is simplicial, with the mirror map b(): A^P_Λ → parameters constrained by a valuation condition val(b()_p) = ε_p matching the A-side Kähler weights. The main theorem of the paper proves HMS under stronger hypotheses (e.g., smooth Σ*, MPCS and connectedness conditions), leaving the conjectural statement above as the general open case.