Toric degenerations of Calabi--Yau complete intersections and metric SYZ conjecture

We consider a toric degeneration $\mathcal{X}$ of Calabi–Yau complete intersections of Batyrev–Borisov in the Gross–Siebert program. For the toric degeneration $\mathcal{X}$, we study the real Monge–Ampère equation corresponding to the non-archimedean Monge–Ampère equation that yields the non-archimedean Calabi–Yau metric. Our main theorem describes the real Monge–Ampère equation in terms of tropical geometry and proves the metric SYZ conjecture for the toric degeneration $\mathcal{X}$ supposing the existence of its solution.