Let S be a symplectic surface (for instance, a $K3$ surface, or $T^*C$ for $C$ a smooth projective curve). Let $\alpha$ be a curve class and for any $\chi \in \mathbb{Z}$, let $M_{\alpha,\chi}$ denote the moduli stack of semistable one-dimensional sheaves on $S$ with support in $\alpha$ and Euler characteristic $\chi$. It is not true in general that the cohomology of $M_{\alpha,\chi}$ is independent of $\chi$, but this becomes true if one replaces the cohomology by the so-called BPS cohomology. Such an isomorphism can in fact be chosen to preserve the perverse filtration induced by the map to the moduli space, and admits a refined version relative to the Chow variety of cycles on $S$. We will explain the statement and sketch its proof which involves the cohomological Hall algebra of sheaves on $S$ and cohomological Hecke operators.

This is joint work with Davison, Hennecart, Kinjo and Vasserot.