Additional connected components of del Pezzo Einstein moduli spaces

Determine whether the moduli spaces of Einstein metrics on the ten del Pezzo 4-manifolds listed in Theorems 8.1 and 8.2 contain connected components beyond the component formed by the known conformally Kähler Einstein metrics.

Background

Theorems 8.1 and 8.2 classify the diffeomorphism types of compact complex or symplectic 4-manifolds that admit an Einstein metric with positive Einstein constant. The known metrics on the ten resulting del Pezzo manifolds are conformally Kähler, and the paper explains that these metrics form exactly one connected component of the corresponding Einstein moduli space.

Unlike the additional manifolds admitting Ricci-flat Kähler metrics, whose Einstein moduli spaces are connected, the global structure of the moduli spaces for the positive-Einstein-constant del Pezzo manifolds is not known. In particular, it is unresolved whether any further components exist.