A note on the boundary Dehn twist of $K3$ surfaces

Abstract: By the work of Baraglia-Konno and Kronheimer-Mrowka, the boundary Dehn twist on punctured $K3$ surfaces is nontrivial in the smooth mapping class group relative to boundary. In this short note, we prove that it becomes trivial after abelianization. The proof is based on an obstruction for $\mathrm{Spin}^\mathbb{C}$ families due to Baraglia-Konno and the global Torelli theorem of $K3$ surfaces.