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

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