Meromorphic Group Actions and the Support Theorem for Lagrangian Fibrations

We prove several new results about Lagrangian fibrations on holomorphic symplectic complex spaces, under the assumption that the total space is K\"ahler (but possibly non-compact or singular) and that the base is a complex manifold. First, we construct a meromorphic action by a family of meromorphic groups. Second, we use this structure, together with Hodge-theoretic methods, to prove a version of Ng\^o's support theorem for Lagrangian fibrations. Along the way, we prove a freeness theorem for the cohomology of compact K\"ahler spaces equipped with a meromorphic group action.