We prove a formula which relates Euler characteristic of moduli spaces of stable pairs on local $K3$ surfaces to counting invariants of semistable sheaves on them. Our formula generalizes Kawai- Yoshioka’s formula for stable pairs with irreducible curve classes to arbitrary curve classes. We also propose a conjectural multiple cover formula of sheaf counting invariants which, combined with our main result, leads to an Euler characteristic version of Katz- Klemm-Vafa conjecture for stable pairs.
"Stable pairs on local $K3$ surfaces." J. Differential Geom. 92 (2) 285 - 370, October 2012. https://doi.org/10.4310/jdg/1352297809