Fixed-point-free real structures on complex surfaces of general type

Determine whether any of the known simply connected complex surfaces of general type with $K^2=2$ or $K^2=4$, in the homeomorphism classes of $\#7$ and $\#5$ respectively, admits a fixed-point-free real structure, equivalently an anti-holomorphic involution with empty real locus.

Background

The paper studies real symplectic structures, namely symplectic four-manifolds equipped with anti-symplectic involutions. Its symplectic example has a free anti-symplectic involution and therefore an empty real locus. This motivates a broader real-symplectic geography problem concerning which simply connected smooth four-manifolds admit real symplectic structures with empty real locus.

The authors compare this question with the complex-algebraic setting, where simply connected minimal complex surfaces of general type with $p_g=0$ and $K^2=1,2,3,4$ are known in the homeomorphism classes of $\#8,\#7,\#6,\#5$, respectively. A parity obstruction shows that a fixed-point-free real structure can occur only when the corresponding homeomorphism class has odd index $n$; thus the cases $n=7$ and $n=5$, corresponding to $K^2=2$ and $K^2=4$, remain as the relevant unresolved cases. The standard del Pezzo representatives in these two homeomorphism classes do admit fixed-point-free anti-holomorphic involutions, but the status for the known complex surfaces of general type is left unresolved.