Mathematical Research Letters

Volume 30 (2023)

Number 3

Extendability of automorphisms of K3 surfaces

Pages: 821 – 863



Yuya Matsumoto (Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba, Japan)


A K3 surface $X$ over a $p$-adic field $K$ is said to have good reduction if it admits a proper smooth model over the ring of integers of $K$. Assuming this, we say that a subgroup $G$ of $\operatorname{Aut}(X)$ is extendable if $X$ admits a proper smooth model equipped with $G$-action (compatible with the action on $X$). We show that $G$ is extendable if it is of finite order prime to $p$ and acts symplectically (that is, preserves the global $2$-form on $X$). The proof relies on birational geometry of models of K3 surfaces, and equivariant simultaneous resolutions of certain singularities. We also give some examples of non-extendable actions.

This work was supported by JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers, and by JSPS KAKENHI Grant Numbers 15H05738, 16K17560, and 20K14296.

Received 6 January 2021

Received revised 20 August 2021

Accepted 30 August 2021

Published 15 December 2023