Geometry of Twisted Kähler–Einstein Metrics and Collapsing

We prove that the twisted Kähler–Einstein metrics that arise on the base of certain holomorphic fiber space with Calabi–Yau fibers have conical-type singularities along the discriminant locus. These fiber spaces arise naturally when studying the collapsing of Ricci-flat Kähler metrics on Calabi–Yau manifolds, and of the Kähler–Ricci flow on compact Kähler manifolds with semiample canonical bundle and intermediate Kodaira dimension. Our results allow us to understand their collapsed Gromov–Hausdorff limits when the base is smooth and the discriminant has simple normal crossings.