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

摘要

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 theInstitute for Mathematical Sciences at the National University of Singapore inJuly 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 onalgebraic curves. The first construction is via a "bootstrap" method from thealgebraic stack of vector bundles on an algebraic curve. This construction ismotivated in part by Nitsure's GIT construction of a projective moduli space ofsemi-stable Higgs bundles, and we describe the relationship between Nitsure'smoduli space and the algebraic stacks constructed here. The third approach isvia deformation theory, where we directly construct the stack of Higgs bundlesusing Artin's criterion.