Moduli of equivariant sheaves and Kronecker–McKay modules

Abstract

We give functorial moduli construction of pure parabolic sheaves, in the sense ofÁlvarez-Cónsul and A. King, using the moduli of filtered Kronecker modules which we introduced in our earlier work. We also use a version of S. G. Langton's result due to K. Yokogawa to deduce the projectivity of moduli of parabolic sheaves. As an application of functorial moduli construction, we can get the morphisms at the level of moduli stacks.

On Faltings parabolic theta functions

“…Let M ss Y (τ ) denote the coarse moduli scheme which co-represents the moduli functor of Γ-semistable bundles on Y of local type τ (cf. [2,8]). …”

“…These technical conditions are required in order to construct the moduli space of semistable Γ-equivariant bundles of fixed local type τ using representations of Kronecker-McKay quiver (for more details on this, see [2]). …”

“…From this, it follows that λ U ∼ = D(F * ) and θ δ = θ F * . From the definition of U , we have λ U ∼ = λ N0 U and θ N0 δ = θ F * (see [2,Remark 6.3]). …”

Quiver representations and their applications

“…In [2], the authors have extended a functorial construction of [1] to the moduli of equivariant sheaves on projective Γ-schemes, for a finite group Γ, by introducing the Kronecker-McKay quiver.…”

“…In [5], a GIT-free construction of the moduli space of semistable parabolic bundles over a smooth projective curve is constructed using the analogous Faltings parabolic theta functions. In [3], it is proved that Faltings parabolic theta functions can be used to give a closed scheme-theoretic embedding of moduli of semistable parabolic bundles, using the results of [2].…”