@article{82a0f9c229d148cfadbf8fadd31a27ae,
title = "Collapsing hyperk{\"A}hler manifolds",
abstract = "Given a projective hyperk{\"a}hler manifold with a holomorphic Lagrangian fibration, we prove that hyperk{\"a}hler 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 K{\"a}hler manifold outside a singular set of real Hausdorff codimension 2, and is homeomorphic to the base projective space.",
author = "Valentino Tosatti and Yuguang Zhang",
note = "Funding Information: We thank H.-J. Hein, M. Popa, J. Song and G. Sz{\'e}kelyhidi for useful discussions, and Y.S. Zhang and the referees for comments. Part of this work was done during the first-named author{\textquoteright}s visits to the Yau Mathematical Sciences Center at Tsinghua University in Beijing and to the Institut Henri Poincar{\'e} in Paris (supported by a Chaire Poincar{\'e} at IHP funded by the Clay Mathematics Institute) and during the second-named author{\textquoteright}s visit to the Department of Mathematics at Northwestern University, which we would like to thank for the hospitality. The first-named author was partially supported by NSF grant DMS-1610278. Publisher Copyright: {\textcopyright} 2020 Soci{\'e}t{\'e} Math{\'e}matique de France.",
year = "2020",
doi = "10.24033/asens.2433",
language = "English (US)",
volume = "53",
pages = "751--786",
journal = "Annales Scientifiques de l'Ecole Normale Superieure",
issn = "0012-9593",
publisher = "Societe Mathematique de France",
number = "3",
}