Isotrivial Lagrangian fibrations of compact hyper-Kähler manifolds

This article initiates the study of isotrivial Lagrangian fibrations of compact hyper-Kähler manifolds. We present four foundational results that extend well-known facts about isotrivial elliptic fibrations of K3 surfaces. First, we prove that smooth fibers of an isotrivial Lagrangian fibration are isogenous to a power of an elliptic curve. Second, we exhibit a dichotomy between two types of isotrivial Lagrangian fibrations, which we call A and B. Third, we give a classification result for type A isotrivial Lagrangian fibrations. Namely, if a type A isotrivial Lagrangian fibration admits a rational section, then it is birational to one of two straightforward examples of isotrivial fibrations of hyper-Kähler manifolds of K3^[n]-type and Kum_n-type. Finally, we prove that a genericity assumption on the smooth fiber of an isotrivial Lagrangian fibration without multiple fibers ensures that the fibration is of type A.