Stable pairs on local K3 surfaces

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 multi-covering 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.