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Perverse-Hodge complexes are objects in the derived category of coherentsheaves obtained from Hodge modules associated with Saito's decompositiontheorem. We study perverse-Hodge complexes for Lagrangian fibrations andpropose a symmetry between them. This conjectural symmetry categorifies the"Perverse = Hodge" identity of the authors and specializes to Matsushita'stheorem on the higher direct images of the structure sheaf. We verify ourconjecture in several cases by making connections with variations of Hodgestructures, Hilbert schemes, and Looijenga-Lunts-Verbitsky Lie algebras.

Perverse-Hodge complexes for Lagrangian fibrations