An introduction to moduli stacks, with a view towards Higgs bundles on algebraic curves

TL;DR

This paper introduces algebraic stacks and explores their application to constructing the moduli stack of Higgs bundles on algebraic curves, connecting classical GIT methods with modern deformation theory.

Contribution

It provides multiple constructions of the Higgs bundle moduli stack, bridging GIT, bootstrap methods, and deformation theory approaches.

Findings

Relates Nitsure's GIT moduli space to algebraic stacks

Constructs the Higgs bundle moduli stack via bootstrap and deformation methods

Clarifies the relationship between different moduli space constructions

Abstract

This article is based in part on lecture notes prepared for the summer school "The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles" at the Institute for Mathematical Sciences at the National University of Singapore in July of 2014. The aim is to provide a brief introduction to algebraic stacks, and then to give several constructions of the moduli stack of Higgs bundles on algebraic curves. The first construction is via a "bootstrap" method from the algebraic stack of vector bundles on an algebraic curve. This construction is motivated in part by Nitsure's GIT construction of a projective moduli space of semi-stable Higgs bundles, and we describe the relationship between Nitsure's moduli space and the algebraic stacks constructed here. The third approach is via deformation theory, where we directly construct the stack of Higgs bundles using Artin's criterion.