Moduli spaces of semistable modules over Lie algebroids

TL;DR

This paper establishes foundational results on the existence, separatedness, and properness of moduli spaces of semistable modules over Lie algebroids, extending classical theories to new contexts including positive characteristic.

Contribution

It proves the existence of moduli spaces under relaxed conditions, demonstrates S-completeness, and generalizes properness of Hitchin's morphism for modules over Lie algebroids.

Findings

Moduli spaces exist for relative projective morphisms of noetherian schemes.

S-completeness of certain moduli stacks of semistable modules is established.

Properness of the Hodge-Hitchin morphism in positive characteristic is proven.

Abstract

We show a few basic results about moduli spaces of semistable modules over Lie algebroids. The first result shows that such moduli spaces exist for relative projective morphisms of noetherian schemes, removing some earlier constraints. The second result proves general separatedness Langton type theorem for such moduli spaces. More precisely, we prove S-completness of some moduli stacks of semistable modules. In some special cases this result identifies closed points of the moduli space of Gieseker semistable sheaves on a projective scheme and of the Donaldson--Uhlenbeck compactification of the moduli space of slope stable locally free sheaves on a smooth projective surface. The last result generalizes properness of Hitchin's morphism and it shows properness of so called Hodge-Hitchin morphism defined in positive characteristic on the moduli space of Gieseker semistable integrable t-connections in terms of the p-curvature morphism. This last result was proven in the curve case by de Cataldo and Zhang using completely different methods.