Collapsing hyperkähler manifolds
arXiv:1705.03299 · doi:10.24033/asens.2433
Abstract
Given a projective hyperkahler manifold with a holomorphic Lagrangian fibration, we prove that hyperkahler metrics with volume of the torus fibers shrinking to zero collapse in the Gromov-Hausdorff sense (and smoothly away from the singular fibers) to a compact metric space which is a half-dimensional special Kahler manifold outside a singular set of real Hausdorff codimension 2 and is homeomorphic to the base projective space.
39 pages; final version to appear in Ann. Sci. Ec. Norm. Super
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- A "boundedness implies convergence" principle and its applications to collapsing estimates in Kähler geometry
- Smooth asymptotics for collapsing Calabi-Yau metrics
- Gromov-Hausdorff limits of flat Riemannian surfaces
- Special Kähler geometry and holomorphic Lagrangian fibrations
- Collapsing K3 Surfaces and Moduli Compactification
- Ricci-flat metrics on Calabi-Yau manifolds
- On the collapsing of Calabi-Yau manifolds and Kähler-Ricci flows