A finiteness theorem for Lagrangian fibrations (Q2804212)

Let \(X\) be a complex projective manifold of dimension \(2n\) that is irreducible symplectic, i.e. there exists a nowhere degenerate holomorphic two-form \(\sigma\) such that \(H^0(X, \Omega_X^2) = \mathbb C \sigma\). Suppose that \(X\) admits a fibration \(\pi: X \rightarrow B\) onto some projective manifold \(B\). By a theorem of \textit{D. Matsushita} [Topology 38, No. 1, 79--83 (1999; Zbl 0932.32027)], the fibration \(\pi\) is Lagrangian, in particular the general fibre is an abelian variety \(A\). Moreover by a theorem of \textit{J.-M. Hwang} [Invent. Math. 174, No. 3, 625--644 (2008; Zbl 1161.14029)], the manifold \(B\) is a projective space \(\mathbb P^n\).NEWLINENEWLINEThe goal of the paper under review is to show that, for a fixed polarisation type of the general fibre, there are only finitely many deformation families of irreducible symplectic manifolds having such a Lagrangian fibration. More precisely, fix positive integers \(d_1, d_2, \ldots, d_n\) such that \(d_1 | d_2 | \cdots | d_n\) and suppose that there exists a very ample line bundle \(L\) on \(X\) such that the restriction to a general fibre \(A\) is a polarisation of type \((d_1, \ldots, d_n)\). For technical reasons, suppose also that \(\pi\) has a global section and the \textit{general} singular fibres are rank one semi-stable degenerations. If \(\pi\) has maximal variation, the author proves that the Lagrangian fibration \(X \rightarrow \mathbb P^n\) belongs to a finite set of deformation families.NEWLINENEWLINEThe conditions on the general singular fibres imply that \(\pi\) induces a rational map \(\bar \phi\) from \(\mathbb P^n\) to some partial compactification \(\mathcal A^*_{d_1, \ldots, d_n}\) of a moduli space of polarised abelian varieties. Since \(\pi\) has maximal variation, the rational map \(\bar \phi\) is generically finite onto its image and the author shows that the degree of \(\bar \phi^* H\) for some fixed ample line bundle \(H\) is bounded. Thus \(\bar \phi\) belongs to finitely many families of rational maps \(\mathbb P^n \dashrightarrow \mathcal A^*_{d_1, \ldots, d_n}\). The main part of the proof is to show that among these rational maps, the maps arising as a classifying map \(\bar \phi\) form a constructible algebraic set.