We prove several results concerning the intersection cohomology and the perverse filtration associated with a Lagrangian fibration of an irreducible symplectic variety. We first show that the perverse numbers only depend on the deformation equivalenc
e class of the ambient variety. Then we compute the border of the perverse diamond, which further yields a complete description of the intersection cohomology of the Lagrangian base and the invariant cohomology classes of the fibers. Lastly, we identify the perverse and Hodge numbers of intersection cohomology when the irreducible symplectic variety admits a symplectic resolution. These results generalize some earlier work by the second and third authors in the nonsingular case.
For any two degrees coprime to the rank, we construct a family of ring isomorphisms parameterized by GSp(2g) between the cohomology of the moduli spaces of stable Higgs bundles which preserve the perverse filtrations. As consequences, we prove two st
ructural results concerning the cohomology of Higgs moduli which are predicted by the P=W conjecture in non-abelian Hodge theory: (1) Galois conjugation for character varieties preserves the perverse filtrations for the corresponding Higgs moduli spaces. (2) The restriction of the Hodge-Tate decomposition for a character variety to each piece of the perverse filtration for the corresponding Higgs moduli space gives also a decomposition. Our proof uses reduction to positive characteristic and relies on the non-abelian Hodge correspondence in characteristic p between Dolbeault and de Rham moduli spaces.
We explore the cohomological structure for the (possibly singular) moduli of $mathrm{SL}_n$-Higgs bundles for arbitrary degree on a genus g curve with respect to an effective divisor of degree >2g-2. We prove a support theorem for the $mathrm{SL}_n$-
Hitchin fibration extending de Cataldos support theorem in the nonsingular case, and a version of the Hausel-Thaddeus topological mirror symmetry conjecture for intersection cohomology. This implies a generalization of the Harder-Narasimhan theorem concerning semistable vector bundles for any degree. Our main tool is an Ng^{o}-type support inequality established recently which works for possibly singular ambient spaces and intersection cohomology complexes.
Let $p$ be a prime number. We prove that the $P=W$ conjecture for $mathrm{SL}_p$ is equivalent to the $P=W$ conjecture for $mathrm{GL}_p$. As a consequence, we verify the $P=W$ conjecture for genus 2 and $mathrm{SL}_p$. For the proof, we compute the
perverse filtration and the weight filtration for the variant cohomology associated with the $mathrm{SL}_p$-Hitchin moduli space and the $mathrm{SL}_p$-twisted character variety, relying on Grochenig-Wyss-Zieglers recent proof of the topological mirror conjecture by Hausel-Thaddeus. Finally we discuss obstructions of studying the cohomology of the $mathrm{SL}_n$-Hitchin moduli space via compact hyper-Kahler manifolds.
We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus $2$ 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 Markmans monodromy operators play a crucial role.
