The dual Lagrangian fibration of known hyper-K\"ahler manifolds

Given a Lagrangian fibration $\pi :X\to {\mathbb{P}}^n$ of a compact hyper-K"ahler manifold of ${\text{K3}}^{[n]}$, ${\text{Kum}}_n$, $\text{OG10}$ or $\text{OG6}$-type, we construct a natural compactification of its dual torus fibration. Specifically, this compactification is given by a quotient of $X$ by certain automorphisms acting trivially on the second cohomology and respecting the Lagrangian fibration. It is a compact hyper-K"ahler orbifold with identical period mapping behavior as $X$.