On collapsing Calabi-Yau fibrations

Abstract

We develop some techniques to study the adiabatic limiting behaviour of Calabi-Yau metrics on the total space of a fibration, and obtain strong control near the singular fibres by imposing restrictions on the singularity types. We prove a uniform lower bound on the metric up to the singular fibre, under fairly general hypotheses. Assuming a result in pluripotential theory, we prove a uniform fibre diameter bound for a Lefschetz K3 fibred Calabi-Yau 3-fold, which reduces the study of the collapsing metric to a locally non-collapsed situation, and we identify the Gromov-Hausdorff limit of the rescaled neighbourhood of the singular fibre.

arXiv comment32 pages

References in corpus (1)

• The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits

Cited by in corpus (5)

• Collapsing hyperkähler manifolds
• A "boundedness implies convergence" principle and its applications to collapsing estimates in Kähler geometry
• Collapsing Calabi-Yau fibrations and uniform diameter bounds
• Global higher order estimates for collapsing Calabi-Yau metrics on elliptic K3 surfaces
• Kahler-Ricci flow on blowups along submanifolds