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

TL;DR

This paper links toric degenerations of Calabi--Yau complete intersections with the real Monge--Ampère equation, providing a geometric approach to the metric SYZ conjecture within the Gross--Siebert program.

Contribution

It expresses the non-archimedean Monge--Ampère equation via tropical geometry and proves the metric SYZ conjecture assuming the existence of solutions.

Findings

Describes the real Monge--Ampère equation in terms of tropical geometry.

Proves the metric SYZ conjecture for the considered toric degeneration.

Connects non-archimedean and tropical geometric frameworks.

Abstract

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--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 $\mathcal{X}$ supposing the existence of its solution.