Nonsmoothable group actions on elliptic surfaces

Let G be a cyclic group of order 3, 5 or 7, and X = E ( n ) be the relatively minimal elliptic surface with rational base. In this paper, we prove that under certain conditions on n, there exists a locally linear G-action on X which is nonsmoothable with respect to infinitely many smooth structures on X. This extends the main result of [X. Liu, N. Nakamura, Pseudofree Z / 3 -actions on K3 surfaces, Proc. Amer. Math. Soc. 135 (3) (2007) 903–910].