  A monolayer of jacutingaite (Pt$_2$HgSe$_3$) has recently been identified as
a novel quantum spin Hall insulator. By first-principles calculations, we study
its Fermiology in the doped regime and unveil a type-I and type-II van Hove
singularity for hole and electron doping, respectively. We find that the common
link between the propensity for a topological band gap at pristine filling and
unconventional superconductivity at finite doping roots in the longer ranged
hybridization integrals on the honeycomb lattice. In a combined effort of
random phase approximation and functional renormalization group, we find chiral
$d$-wave order for the type-I and odd-parity $f$-wave order for the type-II
regime.

Superconductivity