The Nielsen realization problem for high degree del Pezzo surfaces

Abstract

Let $M$ be a smooth $4$-manifold underlying some del Pezzo surface of degree $d \geq 6$. We consider the smooth Nielsen realization problem for $M$: which finite subgroups of $\text{Mod}(M) = \pi_0(\text{Homeo}^+(M))$ have lifts to $\text{Diff}^+(M) \leq \text{Homeo}^+(M)$ under the quotient map $\pi: \text{Homeo}^+(M) \to \text{Mod}(M)$? We give a complete classification of such finite subgroups of $\text{Mod}(M)$ for $d \geq 7$ and a partial answer for $d = 6$. For the cases $d \geq 8$, the quotient map $\pi$ admits a section with image contained in $\text{Diff}^+(M)$. For the case $d = 7$, we show that all finite order elements of $\text{Mod}(M)$ have lifts to $\text{Diff}^+(M)$, but there are finite subgroups of $\text{Mod}(M)$ that do not lift to $\text{Diff}^+(M)$. We prove that the condition of whether a finite subgroup $G \leq \text{Mod}(M)$ lifts to $\text{Diff}^+(M)$ is equivalent to the existence of a certain equivariant connected sum realizing $G$. For the case $d = 6$, we show this equivalence for all maximal finite subgroups $G \leq \text{Mod}(M)$.