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We prove de Cataldo-Hausel-Migliorini's P=W conjecture in arbitrary rank forparabolic Higgs bundles labeled by the affine Dynkin diagrams $\tilde{A}_0$,$\tilde{D}_4$, $\tilde{E}_6$, $\tilde{E}_7$, and $\tilde{E}_8$. Our proofrelies on the study of the tautological classes on the Hilbert scheme of pointson an elliptic surface with respect to the perverse filtration.

Perverse filtrations, Hilbert schemes, and the P=W conjecture for parabolic Higgs bundles