Mirror of the complement by union of toric divisors and commutative diagram

Establish that the union of toric divisors Z = ∪α Zα inside the toric variety Y, viewed as the zero locus W−1(0) of the superpotential W in the toric Landau–Ginzburg model (Y, W), is mirror to the complement (C*)^n \ H of the hypersurface H ⊂ (C*)^n defined by a Laurent polynomial, and construct the commutative diagram of functors relating the wrapped Fukaya category of (C*)^n \ H and the algebraic categories on the B-side, with ρ the restriction functor and p the quotient projection functor.

Background

In the Fano setting, hypersurfaces H in (C*)^n defined by Laurent polynomials are studied via wrapped Fukaya categories on the A-side and categories of singularities of a toric Landau–Ginzburg model (Y, W) on the B-side. The text explains that WFuk(H) is expected to be equivalent to D^b_sg(Z) for Z = W−1(0) and then formulates a conjecture strengthening this perspective by identifying a mirror for the complement (C*)^n \ H.

The conjecture further specifies a functorial relationship via a commutative diagram involving restriction to a neighborhood of H and a projection functor, encoding how Lagrangian data at infinity and on the divisor side should match under mirror symmetry.