Gromov-Hausdorff limits of collapsing Calabi-Yau fibrations

TL;DR

We address the problem of describing Gromov-Hausdorff limits of collapsing Calabi–Yau metrics along a holomorphic fibration by placing the base into a general normal Kähler framework and proving that the GH limit is the metric completion of the base with respect to a canonical metric. The authors develop a general theorem: if $(X,\omega)$ satisfies structural conditions (1)-(3) and its regularized completion $\hat X$ is an $RCD(K,2n)$ space, then $\hat X$ is homeomorphic to $X$ and the singular sets have controlled Hausdorff dimension bounds; they prove these using Stein covers, Hörmander $L^2$ estimates, and holomorphic chart techniques, together with an analysis of regular vs singular tangent cones. They then apply this to collapsing Calabi–Yau fibrations, showing that the GH limit of $(M,\omega_t)$ is the metric completion of $(X^{\circ},\omega_{can})$ and that the associated renormalized measure is a multiple of $\omega_{can}^n$, yielding an $RCD(0,2n)$ space and confirming Tosatti’s conjectures in general. The results provide a precise geometric and measure-theoretic description of the limit, including codimension bounds on the discriminant, and connect the collapsing behavior to the RCD framework. They also discuss a related setting via the La Nave–Tian continuity method, where analogous conclusions hold.

Abstract

We study Calabi-Yau metrics on a projective manifold in Kähler classes converging to a semiample class given by a fibration. We show that the Gromov-Hausdorff limit of the metrics is homeomorphic to the base of the fibration and in addition the discriminant locus has Hausdorff codimension at least 2. This resolves conjectures of Tosatti.

Theorems & Definitions (22)

• Conjecture 1
• Theorem 2
• Theorem 3
• Conjecture 4
• Theorem 5
• Theorem 6
• Proposition 7
• proof
• Proposition 8
• proof