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 \({{\,\textrm{Mod}\,}}(M) = \pi _0({{\,\textrm{Homeo}\,}}^+(M))\) have lifts to \({{\,\textrm{Diff}\,}}^+(M) \le {{\,\textrm{Homeo}\,}}^+(M)\) under the quotient map \(\pi : {{\,\textrm{Homeo}\,}}^+(M) \rightarrow {{\,\textrm{Mod}\,}}(M)\)? We give a complete classification of such finite subgroups of \({{\,\textrm{Mod}\,}}(M)\) 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 \({{\,\textrm{Diff}\,}}^+(M)\). For the case \(d = 7\), we show that all finite order elements of \({{\,\textrm{Mod}\,}}(M)\) have lifts to \({{\,\textrm{Diff}\,}}^+(M)\), but there are finite subgroups of \({{\,\textrm{Mod}\,}}(M)\) that do not lift to \({{\,\textrm{Diff}\,}}^+(M)\). We prove that the condition of whether a finite subgroup \(G \le {{\,\textrm{Mod}\,}}(M)\) lifts to \({{\,\textrm{Diff}\,}}^+(M)\) 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 \(G \le {{\,\textrm{Mod}\,}}(M)\).