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An Introduction to Moduli Stacks, with a View Towards Higgs Bundles On AN INTRODUCTION TO MODULI STACKS, WITH A VIEW TOWARDS HIGGS BUNDLES ON ALGEBRAIC CURVES SEBASTIAN CASALAINA-MARTIN AND JONATHAN WISE Abstract. This article is based in part on lecture notes prepared for the sum- mer 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. Contents Introduction 3 Notation 4 1. Moduli problems as functors 5 1.1. Grassmannians 5 1.2. Moduli of Riemann surfaces 8 1.3. The tangent space of a moduli functor 10 2. Moduli functors as categories fibered in groupoids 13 2.1. (Lax 2-)Functors to groupoids 13 2.2. Categories fibered in groupoids 15 3. Stacks 18 3.1. Sheaves and pretopologies 19 3.2. The isomorphism presheaf 22 arXiv:1708.08124v1 [math.AG] 27 Aug 2017 3.3. Descent for categories fibered in groupoids 24 3.4. The long-awaited definition of a stack 26 4. Fibered products of stacks 27 4.1. A working definition of 2-fibered products 28 4.2. The diagonal 29 4.3. Fibered products and the stack condition 30 5. Stacks adapted to a presite 32 5.1. Definiton of stacks adapted to a presite 32 5.2. Bootstrapping stacks adapted to a presite 33 Date: August 29, 2017. The first author was partially supported by NSF grant DMS-1101333, a Simons Foundation Collaboration Grant for Mathematicians (317572), and NSA Grant H98230-16-1-005. The second author was partially supported by an NSA Young Investigator’s Grant H98230-14-1-0107. 1 2 CASALAINA-MARTINANDWISE 6. Algebraic stacks 34 6.1. Covers 35 6.2. Representable morphisms 36 6.3. Locality to the source 37 6.4. Adapted stacks 38 6.5. Iterated adaptation 39 6.6. Algebraic stacks 42 6.7. Fiber products and bootstrapping 42 7. Moduli stacks of Higgs bundles 43 7.1. The moduli space of curves 43 7.2. The Quot stack and the Quot scheme 44 7.3. Stacks of quasicoherent sheaves 45 7.4. The stack of Higgs bundles over a smooth projective curve 46 7.5. Meromorphic Higgs bundles, and the stack of sheaves and endomorphisms 48 7.6. The stack of Higgs bundles over the moduli of stable curves 50 7.7. Semi-stable Higgs bundles and the quotient stack 51 7.8. The stack of principal G-Higgs bundles 53 8. The Hitchin fibration 54 8.1. Characteristic polynomials and the Hitchin morphism 54 8.2. Spectral covers and fibers of the Hitchin morphism 57 9. Morphisms of stacks in algebraic geometry 61 9.1. Injections, isomorphisms, and substacks 61 9.2. The underlying topological space 62 9.3. Quasicompact and quasiseparated morphisms 62 9.4. Separation and properness 62 9.5. Formal infinitesimal properties 63 9.6. Local finite presentation 64 9.7. Smooth, ´etale, and unramified morphisms 66 10. Infinitesimal deformation theory 67 10.1. Homogeneous categories fibered in groupoids 68 10.2. Deformation theory 75 10.3. Obstruction theory 95 11. Artin’s criterion for algebraicity 100 11.1. The Schlessinger–Rim criterion 101 11.2. Local finite presentation 103 11.3. Integration of formal objects 104 11.4. Artin’s theorems on algebraization and approximation 106 11.5. Algebraicity of the stack of Higgs bundles 108 11.6. Outline of the proof of Artin’s criterion 108 A. Sheaves, topologies, and descent 110 A.1. Torsors and twists 110 A.2. More on descent 114 A.3. Grothendieck topologies 117 A.4. An example of ineffective descent 119 B. The many meanings of algebraicity 122 B.1. Stacks with flat covers by schemes and stacks over the fppf site 123 B.2. Other definitions of algebraicity 126 STACKS AND HIGGS BUNDLES 3 B.3. Remarks on representability 128 B.4. Overview of the relationships among the definitions of algebraicity 129 B.5. Relationships among definitions of algebraic spaces 131 B.6. Relationships among the definitions of algebraic stacks 133 B.7. Relationships among the definitions of Deligne–Mumford stacks 133 B.8. Stacks with unramified diagonal 134 B.9. The adapted perspective 134 B.10. Conditions on the relative diagonal of a stack and bootstrapping 135 C. Groupoids and stacks 137 C.1. Torsors and group quotients 137 C.2. Groupoid objects and groupoid quotients 138 C.3. Adapted groupoid objects and adapted stacks 141 Acknowledgments 143 References 143 Introduction Stacks have been used widely by algebraic geometers since the 1960s for studying parameter spaces for algebraic and geometric objects [DM69]. Their popularity is growing in other areas as well (e.g., [MLM94]). Nevertheless, despite their utility, and their having been around for many years, stacks still do not seem to be as popular as might be expected. Of course, echoing the introduction of [Fan01], this may have something to do with the technical nature of the topic, which may dissuade the uninitiated, particularly when there may be less technical methods available that can be used instead. To capture a common sentiment, it is hard to improve on the following excerpt from Harris–Morrison [HM98, 3.D], which serves as an introduction to their section on stacks: ‘Of course, here I’m working with the moduli stack rather than with the moduli space. For those of you who aren’t familiar with stacks, don’t worry: basically, all it means is that I’m allowed to pretend that the moduli space is smooth and there’s a universal family over it.’ Who hasn’t heard these words, or their equivalent spoken at a talk? And who hasn’t fantasized about grabbing the speaker by the lapels and shaking him until he says what – exactly – he means by them? At the same time, the reason stacks are used as widely as they are is that they really are a natural language for talking about parameterization. While there is no doubt a great deal of technical background that goes into the set-up, the pay-off is that once this foundation has been laid, stacks provide a clean, unified language for discussing what otherwise may require many caveats and special cases. Our goal here is to give a brief introduction to algebraic stacks, with a view towards defining the moduli stack of Higgs bundles. We have tried to provide enough motivation for stacks that the reader is inclined to proceed to the definitions, and a sufficiently streamlined presentation of the definitions that the reader does not immediately stop at that point! In the end, we hope the reader has a good 4 CASALAINA-MARTINANDWISE sense of what the moduli stack of Higgs bundles is, why it is an algebraic stack, and how the stack relates to the quasi-projective variety of Higgs bundles constructed by Nitsure [Nit91]. Due to the authors’ backgrounds, for precise statements, the topic will be treated in the language algebraic geometry, i.e., schemes. However, the aim is to have a presentation that is accessible to those in other fields, as well, particularly complex geometers, and most of the presentation can be made replacing the word “scheme” with the words “complex analytic space” (or even “manifold”) and “´etale cover” with “open cover”. This is certainly the case up though, and including, the def- inition of a stack in §3. The one possible exception to this rule is the topic of algebraic stacks §6, for which definitions in the literature are really geared towards the category of schemes. In order to make our presentation as accessible as pos- sible, in §5 and §6 we provide definitions of algebraic stacks that make sense for any presite; in particular, the definition gives a notion of an “algebraic” stack in the category of complex analytic spaces. There are other notions of analytic stacks in the literature, but for concision, we do not pursue the connection between the definitions. The final sections of this survey (§10 and §11) study algebraic stacks infinitesi- mally, with the double purpose of giving modular meaning to infinitesimal motion in an algebraic stack and introducing Artin’s criterion as a means to prove stacks are algebraic. Higgs bundles are treated as an extended example, and along they way we give cohomological interpretations of the tangent bundles of the moduli stacks of curves, vector bundles, and morphisms between them. In the end we ob- tain a direct proof of the algebraicity of the stack of Higgs bundles, complementing the one based on established general algebraicity results given earlier. While these notes are meant to be somewhat self contained, in the sense that essentially all relevant definitions are included, and all results are referenced to the literature, these notes are by no means comprehensive. There are now a number of introductions to the topic of algebraic stacks that have appeared recently (and even more are in preparation), and these notes are inspired by various parts of those treatments. While the following list is not exhaustive, it provides a brief summary of some of the other resources available.
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