Moduli of Decorated Swamps on a Smooth Projective Curve

TL;DR

This paper introduces a unified framework for constructing moduli spaces of decorated vector bundles on smooth projective curves, generalizing existing concepts like parabolic and level structures.

Contribution

It defines a new stability notion for vector bundles with both global and local decorations and constructs their coarse moduli space as a GIT-quotient, unifying previous moduli space constructions.

Findings

Unified construction of moduli spaces for decorated vector bundles

Generalizes parabolic and level structure moduli spaces

Provides a GIT-quotient description of the moduli space

Abstract

In order to unify the construction of the moduli space of vector bundles with different types of global decorations, such as Higgs bundles, framed vector bundles and conic bundles, A. Schmitt introduced the concept of a swamp. In this work, we consider vector bundles with both a global and a local decoration over a fixed point of the base. This generalizes the notion of parabolic vector bundles, vector bundles with a level structure and parabolic Higgs bundles. We introduce a notion of stability and construct the coarse moduli space for these objects as the GIT-quotient of a parameter space. In the case of parabolic vector bundles and vector bundles with a level structure our stability concept reproduces the known ones. Thus, our work unifies the construction of their moduli spaces.