Moduli spaces arise in classification problem in algebraic geometry; typically when we try to classify geometric objects we find that they have discrete invariants but these are not sufficient to classify them. Thus we would like to be able to construct moduli spaces whose points correspond to equivalence classes of the objects to be classified (with fixed discrete invariants) and whose geometry reflects the way they can vary in families. This thesis studies moduli problems for homomorphisms of sheaves over a fixed projective variety X indexed by a quiver; that is, we are looking at representations of a quiver in the category of coherent sheaves over X where the sheaves indexed by the vertices of the quiver are fixed and it is only the homomorphisms between them indexed by the arrows of the quiver which vary. More precisely, we define a the moduli functor for homomorphisms of sheaves over a fixed projective variety X and show that the construction of a moduli space for homomorphisms of sheaves over X indexed by a quiver Q can be reduced to the construction of quotients for actions of the product of the automorphism groups of the sheaves over X labelled by the vertices of Q on affine varieties. Additionally we show that the automorphism groups of the sheaves over X are linear algebraic groups. In the case that these are reductive groups, such quotients can be constructed and studied using Mumford’s classical geometric invariant theory (GIT). However in general these automorphism groups are not reductive, so a significant part of this thesis studies ways in which Mumford’s GIT can be extended to actions of non-reductive linear algebraic groups on an affine variety, before applying them to representations of quivers in homomorphisms of semisimple sheaves over X.
