Greb, D.; Lehn, C.; Rollenske, S. Lagrangian fibrations on hyperkähler fourfolds. (English. Russian original) Zbl 1292.53033 Izv. Math. 78, No. 1, 22-33 (2014); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 78, No. 1, 25-36 (2014). A. Beauville posed the following question (see [Springer Proc. Math. 8, 49–63 (2011; Zbl 1231.32012)]): Let \(X\) be a hyper-Kähler manifold and \(L\) a Lagrangian torus in \(X\). Is \(L\) a fiber of a (meromorphic) Lagrangian fibration \(f: X \rightarrow B\)? In this paper the authors answer the strong form of Beauville’s question for hyper-Kähler fourfolds, i.e., in the settings above \(L\) is a fiber of a holomorphic Lagrangian fibration \(f: X \rightarrow B\).The authors have positively answered Beauville’s question in their previous work [D. Greb et al., Ann. Sci. Éc. Norm. Supér. (4) 46, No. 3, 375–403 (2013; Zbl 1281.32016)] in the case where \(X\) is not projective. In the four-dimensional projective case their approach is based on a detailed study of the deformation theory of \(L\) in \(X\). Reviewer: Ljudmila Kamenova (New York) Cited in 4 Documents MSC: 53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry 14D06 Fibrations, degenerations in algebraic geometry 14E30 Minimal model program (Mori theory, extremal rays) 32G10 Deformations of submanifolds and subspaces 32G05 Deformations of complex structures Keywords:hyper-Kähler manifold; Lagrangian fibration Citations:Zbl 1231.32012; Zbl 1281.32016 PDFBibTeX XMLCite \textit{D. Greb} et al., Izv. Math. 78, No. 1, 22--33 (2014; Zbl 1292.53033); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 78, No. 1, 25--36 (2014) Full Text: DOI arXiv