Identification of W(Φ) with the expected divisor complement in the Gross–Siebert general fiber

Establish that the Weinstein manifold \mathbf{W}(Φ) constructed from a closed fanifold Φ is the expected divisor complement in the general fiber of the corresponding Gross–Siebert toric degeneration; i.e., prove that \mathbf{W}(Φ) coincides with the mirror of the complement of the toric boundary divisor in the smoothed fiber.

Background

The author proves that line bundles on toric buildings associated to closed fanifolds Φ map under mirror symmetry to local systems on compact Lagrangians diffeomorphic to |Φ|, yielding “enough Lagrangian spheres” in the Calabi–Yau case.

To advance toward mirror symmetry for the Gross–Siebert general fiber, one key issue is the geometric identification of the constructed Weinstein space \mathbf{W}(Φ) with the expected divisor complement in the smoothed fiber of the toric degeneration. This identification has not yet been established.