Base manifolds for Lagrangian fibrations on hyperkähler manifolds
type: Item
instance of: scholarly article
zbMATH Open document ID: 1307.53059
author: Daniel Greb
author: Christian Lehn
publication date: 2014-11-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1303.3919
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 32Q15
Mathematics Subject Classification ID: 53D12
Mathematics Subject Classification ID: 53C26
zbMATH DE Number: 6369079
zbMATH Keywords: hyper-Kähler manifolds
zbMATH Keywords: Lagrangian fibration
zbMATH Keywords: base of a Lagragian fibration
MaRDI profile type: Publication
OpenAlex ID: W3098715264
arXiv ID: 1303.3919
DOI: 10.1093/IMRN/RNT133
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published in: IMRN. International Mathematics Research Notices
title: Base manifolds for Lagrangian fibrations on hyperkähler manifolds
review text: A compact Kähler manifold \(X\) is called hyper-Kähler or irreducible holomorphic symplectic if it is simply-connected, and if \(H^0(X, \Omega^2_X)=\mathbb{C} \sigma \) where \(\sigma\) is an everywhere nondegenerate 2-form. A fibration on \(X\) is a (proper) surjective holomorphic map \( f : X \to B,\, 0<\dim B<\dim X\). It is called Lagrangian if every irreducible component of every fibre is a Lagrangian subvariety with respect to the holomorphic symplectic form \(\sigma\). The following theorem is proven.NEWLINENEWLINELet \(f:X \to B\) be a holomorphic fibration of a compact hyper-Kähler manifold onto a smooth complex manifold. Then the fibration is Lagrangian and \(B\) is biholomorphic to the projective space \(P^n\).NEWLINENEWLINEThis is a generalization of a result by J-M. Hwang, who assumed moreover that \(X\) is a projective manifold.
reviewed by: Dmitri Alekseevsky
description: scientific article; zbMATH DE number 6369079