In this talk I will review, in an introductory manner, the latest developments in the theory of scattering amplitudes and their connection to the geometrical properties of special classes of complex varieties. I will also discuss explicit examples of correlators and scattering amplitudes which can be related to elliptic geometries and beyond. In these cases, I will highlight how some of the expected physical properties of the object computed, reflect in the specific differential forms associated to the corresponding geometries.