Analytic Moduli Spaces of Simple (Co)Framed Sheaves

Let X be a complex space and F a coherent O x -module, A F-(co)framed sheaf on X is a pair (ε, ϕ) with a coherent O x -module ε and a morphism of coherent sheaves ϕ: F F ε (resp. ϕ: ε → F). Two such pairs (ε, ϕ) and (ε′,ϕ′) are said to be isomorphic if there exists an isomorphism of sheaves α: ε →ε′ with α° ϕ = ϕ′ (resp. ϕ′° α = ϕ). A pair (α, ϕ) is called simple if its only automorphism is the identity on ε. In this note we prove a representability theorem in a relative framework, which implies in particular that there is a moduli space of simple F- (co) framed sheaves on a given compact complex space X.