The dual Lagrangian fibration of known hyper-Kähler manifolds

Abstract

Given a Lagrangian fibration $\pi :X\to {\mathbb{P}}^n$ of a compact hyper-Kähler 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-Kähler orbifold with identical period mapping behavior as $X$.

arXiv comment41 pages; errors corrected; the main statement is slightly modified

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