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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. 

We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus 2 2 curves and arbitrary rank. In higher genus and arbitrary rank, we prove that P=W holds for the subalgebra of cohomology generated by even tautological classes. Furthermore, we show that all tautological generators lie in the correct pieces of the perverse filtration as predicted by the P=W conjecture. In combination with recent work of Mellit, this reduces the full conjecture to the multiplicativity of the perverse filtration. Our main technique is to study the Hitchin fibration as a degeneration of the Hilbert–Chow morphism associated with the moduli space of certain torsion sheaves on an abelian surface, where the symmetries induced by Markman’s monodromy operators play a crucial role. 

Abstract We construct natural operators connecting the cohomology of the moduli spaces of stable Higgs bundles with different ranks and genera which, after numerical specialisation, recover the topological mirror symmetry conjecture of Hausel and Thaddeus concerning $\mathrm {SL}_n$ - and $\mathrm {PGL}_n$ -Higgs bundles. This provides a complete description of the cohomology of the moduli space of stable $\mathrm {SL}_n$ -Higgs bundles in terms of the tautological classes, and gives a new proof of the Hausel–Thaddeus conjecture, which was also proven recently by Gröchenig, Wyss and Ziegler via p -adic integration. Our method is to relate the decomposition theorem for the Hitchin fibration, using vanishing cycle functors, to the decomposition theorem for the twisted Hitchin fibration, whose supports are simpler.