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We initiate and develop a framework to handle the specialization morphism asa filtered morphism for the perverse, and for the perverse Leray filtration, onthe cohomology with constructible coefficients of varieties and morphismsparameterized by a curve. As an application, we use this framework to carry outa detailed study of filtered specialization for the Hitchin morphismsassociated with the compactification of Dolbeault moduli spaces in [de-2018].

Perverse Leray filtration and specialization with applications to the Hitchin morphism