Collapsing hyperkÄhler manifolds

Valentino Tosatti; Yuguang Zhang

doi:10.24033/asens.2433

Collapsing hyperkÄhler manifolds

Valentino Tosatti, Yuguang Zhang

Research output: Contribution to journal › Article › peer-review

10 Citations (Scopus)

Abstract

Given a projective hyperkähler manifold with a holomorphic Lagrangian fibration, we prove that hyperkähler 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 Kähler manifold outside a singular set of real Hausdorff codimension 2, and is homeomorphic to the base projective space.

Original language	English
Pages (from-to)	751-786
Number of pages	36
Journal	Annales Scientifiques de l'Ecole Normale Superieure
Volume	53
Issue number	3
DOIs	https://doi.org/10.24033/asens.2433
Publication status	Published - 2020
Externally published	Yes

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10.24033/asens.2433

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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",

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year = "2020",

doi = "10.24033/asens.2433",

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AU - Tosatti, Valentino

AU - Zhang, Yuguang

PY - 2020

Y1 - 2020

N2 - Given a projective hyperkähler manifold with a holomorphic Lagrangian fibration, we prove that hyperkähler 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 Kähler manifold outside a singular set of real Hausdorff codimension 2, and is homeomorphic to the base projective space.

AB - Given a projective hyperkähler manifold with a holomorphic Lagrangian fibration, we prove that hyperkähler 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 Kähler manifold outside a singular set of real Hausdorff codimension 2, and is homeomorphic to the base projective space.

UR - https://www.scopus.com/pages/publications/85090657349

U2 - 10.24033/asens.2433

DO - 10.24033/asens.2433

M3 - Article

AN - SCOPUS:85090657349

SN - 0012-9593

VL - 53

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JO - Annales Scientifiques de l'Ecole Normale Superieure

JF - Annales Scientifiques de l'Ecole Normale Superieure

IS - 3

ER -

Collapsing hyperkÄhler manifolds

Abstract

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