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 $d\ge 6$. We consider the smooth Nielsen realization problem for $M$: which finite subgroups of $\text{Mod}(M)={\pi }_0({\text{Homeo}}^{+}(M))$ have lifts to ${\text{Diff}}^{+}(M)\le {\text{Homeo}}^{+}(M)$ 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 $d\ge 7$ and a partial answer for $d=6$. For the cases $d\ge 8$, 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 $G\le \text{Mod}(M)$ 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 $G\le \text{Mod}(M)$.