Weights of finite cyclic actions on definite $4$-manifolds

Let $X$ be a closed, smooth, orientable, positive definite $4$-manifold with $H_1(X;\mathbb{Z})=0$. Let $p$ be a prime and suppose that $G={\mathbb{Z}}_p$ acts smoothly on $X$ (if $p=2$ we also require an assumption on how $G$ acts on $H^2(X;\mathbb{Z})$). Using equivariant Yang–Mills theory, Hambleton–Lee and Hambleton–Tanase proved (in the simply-connected case) that the fixed point set and tangential isotropy representations coincide with that of an equivariant connected sum of copies of ${\mathbb{C}\mathbb{P}}^2$ on which $G$ acts linearly. We give a new proof of this result using equivariant Seiberg–Witten theory. Furthermore we also determine the weights of all equivariant line bundles on $X$, showing that these likewise coincide with that of an equivariant connected sum of linear actions on ${\mathbb{C}\mathbb{P}}^2$.