We explore the cohomological structure for the (possibly singular) moduli of 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 SL 𝑛 -Hitchin fibration extending de Cataldo's 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 Ngô-type support inequality established recently which works for possibly singular ambient spaces and intersection cohomology complexes.