On the intersection cohomology of the moduli of SLn$\mathrm{SL}_n$‐Higgs bundles on a curve

Abstract

We explore the cohomological structure for the (possibly singular) moduli of SL n $\mathrm{SL}_n$ -Higgs bundles for arbitrary degree on a genus g $g$ curve with respect to an effective divisor of degree > 2 g − 2 $>2g-2$ . We prove a support theorem for the SL n $\mathrm{SL}_n$ -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 Ngô–type support inequality established recently which works for possibly singular ambient spaces and intersection cohomology complexes.