Lagrangian fibrations on symplectic fourfolds
A \textit{symplectic variety} is a variety \(M\) of dimension \(2n\) endowed with a closed, non-degenerate, holomorphic 2-form \(\Omega\); a subvariety \(N\subset M\) is \textit{Lagrangian} if the restriction of \(\Omega\) to the smooth part of \(N\) is identically zero. A fibration \(f\colon M\to X\) from a symplectic variety to a normal variety is a \textit{Lagrangian fibration} if every component of every fiber of \(f\) is a Lagrangian subvariety of \(M\). The prototype of such a fibration is provided by an elliptic \(K3\) surface. A simply connected complex projective manifold \(M\) is \textit{irreducible holomorphic symplectic (IHS)} if \(M\) has a symplectic 2-form \(\Omega\) and \(H^0(M,\Omega^2_M)=\mathbb{C}\Omega\). By a result of \textit{D. Matsushita} [Topology 38, No. 1, 79--83 (1999; Zbl 0932.32027)], if \(f\colon M\to X\) is a fibration from a complex projective IHS manifold \(M\) of dimension \(2n\) to a normal variety \(X\) with \(0<\dim X<2n\), then \(f\) is a Lagrangian fibration and \(X\) is \(\mathbb{Q}\)-factorial klt Fano variety of dimension \(n\) with Picard number 1. It is conjectured that the base space of such a fibration is isomorphic to \(\mathbb{P}^n\), even under the milder assumption that the total space is a simply connected Kähler IHS manifold. \textit{J.-M. Hwang} proved [Invent. Math. 174, No. 3, 625--644 (2008; Zbl 1161.14029)] that the conjecture is true if the base space is smooth. The conjecture is known to hold also when the IHS manifold \(M\) is deformation equivalent to \(K3^{[n]}\) or to generalized Kummer. In fact, if the base is smooth, the conjecture holds even under the weaker assumption that the total space is a simply connected Kähler IHS manifold, by work of \textit{D. Greb} and \textit{C. Lehn} [Int. Math. Res. Not. 2014, No. 19, 5483--5487 (2014; Zbl 1307.53059)]. In this paper, the author studies the conjecture in the context of fourfolds. His main result is the following: {Theorem}. Let \(f\colon M\to X\) be a Lagrangian fibration from a complex projective IHS manifold of dimension 4 to a normal surface \(X\). Then either \(X\cong \mathbb{P}^2\) or to \(X\cong S^n(E_8)\). Here \(S^n(E_8)\) is the unique Fano surface with exactly one Du Val singularity of type \(E_8\). Recently, the case \(X\cong S^n(E_8)\) was excluded by \textit{D. Huybrechts} and \textit{C. Xu} [``Lagrangian fibrations of hyperkähler fourfolds'', J. Inst. Math. Jussieu (to appear)] and \textit{F. Bogomolov} and \textit{N. Kurnosov} [``Lagrangian fibrations for IHS fourfolds'' Preprint, \url{arXiv:1810.11011}], thus confirming the conjecture for fourfolds.
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