Hodge theory and Lagrangian fibrations on holomorphic symplectic manifolds

Abstract

The purpose of this paper is to establish several new results about the Hodge theory of Lagrangian fibrations on (not necessarily compact) holomorphic symplectic manifolds. Let $M$ be a holomorphic symplectic manifold of dimension $2n$ that is Kähler but not necessarily compact, and let $\pi :M\to B$ be a Lagrangian fibration. We establish a relationship between the bundle of holomorphic $(n+i)$-forms on $M$ and the $i$-th perverse sheaf $P_i$ in the decomposition theorem for $\pi$. This is formulated using Saito's theory of Hodge modules and the BGG correspondence (between graded modules over the symmetric and exterior algebra). Along the way, we prove a relative Hard Lefschetz theorem for the action by the symplectic form; we prove two recent conjectures by Maulik, Shen, and Yin; we give a short proof for Matsushita's theorem (about higher direct images of the structure sheaf); and we show, without using hyperkähler metrics, that every Lagrangian fibration gives rise to an action by the Lie algebra ${\mathfrak{sl}}_3(\mathbb{C})$ (in the noncompact case) or ${\mathfrak{sl}}_4(\mathbb{C})$ (in the compact case). [See the comment below.]

v2: The proof in §8 is incomplete. (It does not show that the two complexes $G$ and $G_v$ are isomorphic in the derived category of graded ${\Omega }_B$-modules, and so one cannot apply the BGG correspondence.) I know how to prove all the claimed results by a different method (that relies on the fact that Lagrangian fibrations are weak abelian); I will revise the paper at a later date