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

TL;DR

This paper investigates the boundary Dehn twist on punctured K3 surfaces, showing it becomes trivial after abelianization, using obstructions from Spin^C families and the Torelli theorem.

Contribution

It proves that the boundary Dehn twist on punctured K3 surfaces is trivial after abelianization, extending previous work on its nontriviality in the smooth mapping class group.

Findings

Boundary Dehn twist is nontrivial in the smooth mapping class group.

The twist becomes trivial after abelianization.

Obstruction from Spin^C families and Torelli theorem are key to the proof.

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.