  We explore the cohomological structure for the (possibly singular) moduli of
$\mathrm{SL}_n$-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 $\mathrm{SL}_n$-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 Ng\^{o}-type support inequality established recently
which works for possibly singular ambient spaces and intersection cohomology
complexes.
