Baraglia, DavidBuchdahl, NicholasTchorbadjiev, Radee Stefanov2025-07-222025-07-222025https://hdl.handle.net/2440/146223In 1958, André Weil formulated a number of famous conjectures concerning the structure of K3 surfaces: all such surfaces are deformation equivalent; they all admit K¨ahler structures; every point of the period domain arises as the period point of a K3 surface; and the biholomorphism class of a K3 surface is determined by its period point. By appealing to the fact that every Enriques surface is double-covered by a K3 surface, one can in a natural way extend the considerations of the period map of K3 surfaces to the case of Enriques surfaces. In this case, one obtains analogous results for Enriques surfaces, and it is with their proof that the present work is concerned. By making use primarily of the classification results of K3 surfaces, as well as their moduli space of Einstein metrics, we provide short, independent proofs of the results concerning Enriques surfaces which taken together fully classify this important class of compact complex surface of Kodaira dimension 0. We conclude by extending the above considerations to the case of quotients of Enriques surfaces by free anti-holomorphic involutions where we prove that these resulting compact Einstein four-manifolds are diffeomorphic, and in fact of the same deformation type.enDifferential GeometryComplex GeometryAlgebraic GeometryThesis