Log HMS surface data with distinguished complex structure

A log HMS surface datum is a smooth rational projective surface $Y$ over $\mathbb{C}$ with a nodal anticanonical divisor $D\in \mathrm{\mid }-K_Y\mathrm{\mid }$ as in Log Calabi–Yau pair, toric degeneration and scattering data, together with a chosen toric model and the source’s distinguished complex-structure data. A toric model includes a smooth projective toric pair $(\bar{Y},\bar{D})$, a sequence of interior blow-ups of points on the toric boundary components that leaves the boundary unchanged, and the corner blow-ups and total transforms specifying $(Y,D)$. For the distinguished structure, on each toric boundary component choose its torus coordinate ${\mathbb{C}}^{\ast }$ from the fixed torus action and place the selected interior blow-up point at $-1\in {\mathbb{C}}^{\ast }$, using the corresponding point on successive strict transforms when iterating. The selected model and blow-up sequence are part of the datum.

This records the hypotheses used by Hacking–Keating; it does not assert existence or uniqueness of a toric model, nor does it produce a mirror fibration or categorical equivalence.

Log Calabi–Yau pair, toric degeneration and scattering data

Toric fan and Laurent potential data for the line-bundle case