The Mumford--Tate group of a very general Lagrangian fiber

Abstract

For a primitive symplectic variety $X$ admitting a holomorphic Lagrangian fibration $f: X \to B$, we classify the possible Mumford--Tate groups of the very general fiber. We extract as a consequence that the discriminant of $f$ has codimension one in $B$. In the special case when $X$ is a hyper-Kähler manifold which is general among those admitting a Lagrangian fibration, and $b_2(X) \geq 7$, we further show that the very general fiber of $f$ is Hodge-generic. Finally, as an application of our classification, we confirm an expectation of Li--Tosatti that when $f$ is not isotrivial, the special Kähler metric on $B$ has positive holomorphic sectional curvature on the regular values of $f$.