P=W conjectures for character varieties with symplectic resolution

We establish P=W and PI=WI conjectures for character varieties with structural group GL n SL which admit a symplectic resolution, i.e., genus 1 arbitrary rank, 2 rank 2. formulate the conjecture prove it resolutions. exploit topology of birational quasi-étale modifications Dolbeault moduli spaces Higgs bundles. To this end, we auxiliary results independent interest, like construction relative compactification Hodge space reductive algebraic groups, projectivity de Rham space. In particular, study in detail is specialization singular irreducible holomorphic variety type O’Grady 6.