The Nielsen realization problem for high degree del Pezzo surfaces

Let M be a smooth 4-manifold underlying some del Pezzo surface of degree \(d \ge 6\) $d\ge 6$ . We consider the smooth Nielsen realization problem for M: which finite subgroups of \({{\,\textrm{Mod}\,}}(M) = \pi _0({{\,\textrm{Homeo}\,}}^+(M))\) $\text{Mod}(M)={\pi }_0({\text{Homeo}}^{+}(M))$ have lifts to \({{\,\textrm{Diff}\,}}^+(M) \le {{\,\textrm{Homeo}\,}}^+(M)\) ${\text{Diff}}^{+}(M)\le {\text{Homeo}}^{+}(M)$ under the quotient map \(\pi : {{\,\textrm{Homeo}\,}}^+(M) \rightarrow {{\,\textrm{Mod}\,}}(M)\) $\pi :{\text{Homeo}}^{+}(M)\to \text{Mod}(M)$ ? We give a complete classification of such finite subgroups of \({{\,\textrm{Mod}\,}}(M)\) $\text{Mod}(M)$ for \(d \ge 7\) $d\ge 7$ and a partial answer for \(d = 6\) $d=6$ . For the cases \(d \ge 8\) $d\ge 8$ , the quotient map \(\pi \) $\pi$ admits a section with image contained in \({{\,\textrm{Diff}\,}}^+(M)\) ${\text{Diff}}^{+}(M)$ . For the case \(d = 7\) $d=7$ , we show that all finite order elements of \({{\,\textrm{Mod}\,}}(M)\) $\text{Mod}(M)$ have lifts to \({{\,\textrm{Diff}\,}}^+(M)\) ${\text{Diff}}^{+}(M)$ , but there are finite subgroups of \({{\,\textrm{Mod}\,}}(M)\) $\text{Mod}(M)$ that do not lift to \({{\,\textrm{Diff}\,}}^+(M)\) ${\text{Diff}}^{+}(M)$ . We prove that the condition of whether a finite subgroup \(G \le {{\,\textrm{Mod}\,}}(M)\) $G\le \text{Mod}(M)$ lifts to \({{\,\textrm{Diff}\,}}^+(M)\) ${\text{Diff}}^{+}(M)$ is equivalent to the existence of a certain equivariant connected sum realizing G. For the case \(d = 6\) $d=6$ , we show this equivalence for all maximal finite subgroups \(G \le {{\,\textrm{Mod}\,}}(M)\) $G\le \text{Mod}(M)$ .