Quiver varieties and moduli spaces of sheaves on singular surfaces

We investigate several types of Nakajima quiver varieties and their connection to the Kleinian singularities 2/Γ. Such quiver varieties have many good properties: they are for instance irreducible and have symplectic singularities, in particular they are normal. The first part of this thesis introduces Nakajima quiver varieties, together with the necessary background on projective Deligne-Mumford stacks and framed sheaves. Following that, we prove that the punctual Hilbert schemes (when taken with their reduced scheme structures) Hilbn(2/Γ) are examples of Nakajima quiver varieties. We then show that there is a type of 'orbifold Quot scheme' generalising the Hilbert scheme, which can also be identified with a Nakajima quiver variety. Following this, we investigate another generalisation of the Hilbert scheme: that of a moduli space of framed sheaves on a certain stack compactifying 2/Γ. We show that this moduli space exists as a quasiprojective scheme. We are unable to show that it is isomorphic to a Nakajima quiver variety, but we show that it carries a canonical morphism to a Nakajima quiver variety, and this morphism is a bijection of closed points. We end by sketching some potential further directions of investigation.