Collapsing hyperk\"ahler manifolds
Differential Geometry
2020-07-02 v5 Mathematical Physics
Algebraic Geometry
Metric Geometry
math.MP
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.
Cite
@article{arxiv.1705.03299,
title = {Collapsing hyperk\"ahler manifolds},
author = {Valentino Tosatti and Yuguang Zhang},
journal= {arXiv preprint arXiv:1705.03299},
year = {2020}
}
Comments
39 pages; final version to appear in Ann. Sci. Ec. Norm. Super