We establish P=W and PI=WI conjectures for character varieties with structural group GLn and SLn which admit a symplectic resolution, i.e. for genus 1 and arbitrary rank, and genus 2 and rank 2. We formulate the P=W conjecture for resolution, and prove it for symplectic resolutions. We exploit the topology of birational and quasi-étale modifications of Dolbeault moduli spaces of Higgs bundles. To this end, we prove auxiliary results of independent interest, like the construction of a relative compactification of the Hodge moduli space for reductive algebraic groups, or the intersection theory of some singular Lagrangian cycles. In particular, we study in detail a Dolbeault moduli space which is specialization of the singular irreducible holomorphic symplectic variety of type O'Grady 6. PreliminariesIn this section we introduces preliminary notions and results which will be useful throughout the paper. For further details we refer to [3,29,30,18]. Perverse sheavesAn algebraic variety X is an irreducible separated scheme of finite type over C. Denote by D b c (X) the bounded derived category of Q-constructible complexes on X. Let D : D b c (X) → D b c (X) be the Verdier duality functor. The full subcategoriesX) of the t-structure is the abelian category of perverse sheaves. The truncation functors are denoted p τ ≤k : D b c (X) → p D b ≤k (X), p τ ≥k : D b c (X) → p D b ≥k (X), and the perverse cohomology functors are p H k := p τ ≤k p τ ≥k : D b c (X) → Perv(X).

Let C be a smooth projective curve of genus $2$ . Following a method by O’Grady, we construct a semismall desingularisation $\tilde {\mathcal {M}}_{Dol}^G$ of the moduli space $\mathcal {M}_{Dol}^G$ of semistable G-Higgs bundles of degree 0 for $G=\mathrm {GL}(2,\mathbb {C}), \mathrm {SL}(2,\mathbb {C})$ . By the decomposition theorem of Beilinson, Bernstein and Deligne, one can write the cohomology of $\tilde {\mathcal {M}}_{Dol}^G$ as a direct sum of the intersection cohomology of $\mathcal {M}_{Dol}^G$ plus other summands supported on the singular locus. We use this splitting to compute the intersection cohomology of $\mathcal {M}_{Dol}^G$ and prove that the mixed Hodge structure on it is pure, in analogy with what happens to ordinary cohomology in the smooth case of coprime rank and degree.

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We give a lattice-theoretic characterization for a manifold of OG10 type to be birational to some moduli space of (twisted) sheaves on a K3 surface. We apply it to the Li-Pertusi-Zhao variety of OG10 type associated to any smooth cubic fourfold. Moreover we determine when a birational transformation is induced by an automorphism of the K3 surface and we use this to classify all induced birational symplectic involutions.