Generalized Shioda-Inose structures

Shioda-Inose structures establish a correspondence between the only two classes of projective compact complex surfaces with trivial canonical bundle, namely K3 surfaces and Abelian surfaces, allowing the study of certain K3 surfaces to be reduced to the study of their associated Abelian surface. In this thesis, we extend this classical theory by introducing and analyzing generalized Shioda-Inose structures. In particular, we present Morrison's work on Shioda-Inose structures of order two, the generalization by Garbagnati and Prieto-Montañez to Generalized Shioda-Inose structures of order three and the work of Piroddi on Generalized Shioda-Inose structures of order four. Finally, we summarize the state of the art on Generalized Shioda-Inose structures of order six, and discuss a possible generalization of the very definition of this geometric construction.