Sections of Lagrangian fibrations on holomorphic symplectic manifolds

TL;DR

The paper addresses constructing meromorphic sections for Lagrangian fibrations on compact hyperkähler manifolds by employing degenerate twistor (Tate–Shafarevich) deformations parameterized by $H^{1,1}(X)$. The authors develop a framework combining C-symplectic geometry, Néron models, Thom’s realizability, and Dolbeault-current techniques to produce a deformation $M'$ over which the Lagrangian fibration $\pi:(M,I_{t_0})\to X$ admits a meromorphic section. Key steps include representing fiber-homology classes by submanifolds, passing to smooth loci with abelian torsor structures, constructing sections over curves (with or without Néron models), and proving the Dolbeault class of these sections vanishes to extend deformations meromorphically to $X=\mathbb{C}P^n$. The holography principle for ample rational curves and Bishop–Barlet compactness underpin extending local holomorphic data to global meromorphic maps, culminating in the main existence result. Altogether, the work links deformation theory, algebraic geometry (relative Albanese, Shafarevich–Tate twists), and analytic tools (Dolbeault currents) to produce holomorphic sections in a meromorphic sense, with implications for the geometry of Lagrangian fibrations on hyperkähler manifolds.

Abstract

Let $M$ be a holomorphically symplectic manifold, equipped with a Lagrangian fibration $π:\; M \to X$. A degenerate twistor deformation (sometimes also called ``a Tate-Shafarevich twist'') is a family of holomorphically symplectic structures on $M$ parametrized by $H^{1,1}(X)$. All members of this family are equipped with a holomorphic Lagrangian projection to $X$, and their fibers are isomorphic to the fibers of $π$. Assume that $M$ is a compact hyperkahler manifold of maximal holonomy, and the general fiber of the Lagrangian projection $π$ is primitive (that is, not divisible) in integer homology. We also assume that $π$ has reduced fibers in codimension 1. Then $M$ has a degenerate twistor deformation $M'$ such that the Lagrangian projection $π:\; M' \to X$ admits a meromorphic section.