Metric SYZ conjecture for certain toric Fano hypersurfaces

Abstract

We prove the metric version of the SYZ conjecture for a class of Calabi-Yau hypersurfaces inside toric Fano manifolds, by solving a variational problem whose minimizer may be interpreted as a global solution of the real Monge-Amp\`ere equation on certain polytopes. This does not rely on discrete symmetry.