On Symplectic Resolutions in the Nonabelian Hodge Correspondence

On symplectic resolutions in the Nonabelian Hodge Correspondence

athesispresentedforthedegreeof Doctor of Philosophy of Imperial College London and the Diploma of Imperial College by Andrea Tirelli

Department of Mathematics Imperial College 180 Queen’s Gate, London SW7 2AZ

November 2018 Icertifythatthisthesis,andtheresearchtowhichitrefers,aretheproductof my own work, and that any ideas or quotations from the work of other people, published or otherwise, are fully acknowledged in accordance with the standard referencing practices of the discipline.

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ii Copyright

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iii Thesis advisor: Dr. Travis J. Schedler Andrea Tirelli

On symplectic resolutions in the Nonabelian Hodge Correspondence

Abstract

This thesis is devoted to the study of the symplectic algebraic geometry of the moduli spaces of the nonabelian Hodge correspondence: namely, we consider character varieties of (open) Riemann surfaces and the moduli spaces of semistable Higgs bundles on a closed Riemann surface and aim at giving an answer to the following two questions: are those moduli spaces symplectic singularities? Do they admit symplectic resolutions? In the case of Higgs bundles on closed Riemann surfaces, we are able to completely solve the problem, using the work of Bellamy and Schedler, [BS16b], in combination with Simpson’s Isosingularity theorem, [Sim94b, §10]. For what concerns the Betti moduli spaces, we take a di↵erent perspective and realise them as (singular open subsets of) multiplicative quiver varieties,[CS06], and we study the aforementioned problem for the latter varieties. The present work represents a ﬁrst step of a series of investigations on the symplectic algebraic geometry involved in the nonabelian Hodge correspondence.

iv To my family.

v Acknowledgments

Firstly, I would like to express my sincere gratitude to my advisor Dr. Travis Schedler for the continuous support of my PhD study and related research, for his patience, motivation, and immense knowledge. His guidance helped me in all the time of research and writing of this thesis. I could not have imagined having a better advisor and mentor for my PhD study.

Besides my advisor, I would like to thank the rest of my thesis committee: Dr. Ben Davison and Dr. Johannes Nicaise for their insightful comments and encouragement.

I would like to thank all the members of the London School of Geometry and Number Theory for making my PhD an unforgettable experience.

Last but most importantly, I would like to thank all the members of my family, to whom this thesis is dedicated: my parents and to my brother and sister for supporting me spiritually throughout writing this thesis and my my life in general.

This work was supported by the Engineering and Physical Sciences Research Coun- cil [EP/L015234/1]. The EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London and Imperial College London.

vi Contents

Introduction 1

1Preliminaryresults 7 1.1 The nonabelian Hodge correspondence ...... 7 1.1.1 Character varieties ...... 9 1.1.2 Higgs bundles ...... 10 1.2 Symplectic singularities and their resolutions ...... 13 1.3 Symplectic resolutions of quiver and character varieties ...... 15

2Symplecticresolutionsof (X, n) 20 MH 2.1 Local structure of (X, n) ...... 21 MH 2.1.1 Inﬁnitesimal deformations of Higgs bundles ...... 22 2.1.2 The Isosingularity theorem ...... 23 2.2 Proof of the main theorem ...... 25 2.3 Extensions and future directions ...... 29

3Multiplicativequivervarieties 32 3.0.1 Summary of results on character varieties ...... 32 3.0.2 Multiplicative quiver varieties with special dimension vectors .36 3.0.3 Character varieties as (open subsets of) multiplicative quiver varieties ...... 39 3.0.4 General dimension vectors ...... 40 3.0.5 Outline of the chapter ...... 43 3.1 Multiplicative quiver varieties ...... 45 3.1.1 Multiplicative preprojective algebras ...... 45 3.1.2 Reﬂection functors for ⇤q(Q) ...... 48 3.1.3 Moduli of representations of ⇤q(Q) ...... 49 3.1.4 Reﬂection isomorphisms ...... 51

vii 3.1.5 Poisson structure on (Q, ↵) ...... 51 Mq,✓ 3.1.6 Stratiﬁcation by representation type ...... 52 3.2 Punctured character varieties as multiplicative quiver varieties .... 56 3.3 Singularities of multiplicative quiver varieties ...... 63 3.3.1 Singular locus of (Q, ↵)for↵ ⌃ ...... 64 Mq,✓ 2 q,✓ 3.3.2 The q-indivisible case ...... 66 3.3.3 The q-divisible case ...... 67 3.3.4 Proof of Corollary 3.0.8 ...... 71 3.3.5 The anisotropic imaginary (p(↵),n)=(2, 2) case ...... 71 3.4 Combinatorics of multiplicative quiver varieties ...... 72 3.4.1 (2,2) cases for crab-shaped quivers ...... 73 3.5 General dimension vectors and decomposition ...... 80 3.5.1 Flat roots ...... 82

3.5.2 Fundamental and flat roots not in ⌃q,✓ ...... 85 3.5.3 Canonical decompositions ...... 87 3.5.4 Symplectic resolutions for q-indivisible flat roots ...... 90 3.5.5 Symplectic resolutions for general ↵ ...... 93 3.5.6 Classifications of symplectic resolutions of punctured character varieties ...... 94 3.5.7 Proof of Theorems 3.0.1 and 3.0.4 ...... 96

4Conclusionandfuturedirections 98 4.1 Non-emptiness of multiplicative quiver varieties ...... 99 4.2 Reﬁned decompositions for multiplicative quiver varieties ...... 100 4.3 Symplectic resolutions and singularities ...... 102 4.4 Moduli of parabolic Higgs bundles and the Isosingularity Theorem ..103 4.5 Moduli of representations of 2-Calabi–Yau algebras ...... 107 4.6 Character varieties and Higgs bundles for arbitrary groups ...... 110 4.7 Moduli spaces of objects in 2-Calabi–Yau categories ...... 111

viii Introduction

This thesis is devoted to the study of certain moduli spaces appearing in the context of the nonabelian Hodge correspondence from the perspective of symplectic algebraic geometry. More precisely, we consider character varieties of Riemann surfaces (with punctures) and the moduli spaces of semistable Higgs bundles on a smooth projective curve and tackle the problem of understanding whether they are symplectic singularities (´ala Beauville, [Bea00]) and whether they admit symplectic resolutions.

Despite the simple formulation of the problem, it turns out that its solution involves the use of many di↵erent techniques, coming from di↵erential geometry, algebra and representation theory: indeed, the nonabelian Hodge correspondence is a topic at the interface of such areas of mathematics. As such, the aim of this thesis is to initiate a series of investigations in what seems to be a very promising topic, given its multidisciplinary nature.

Higgs bundles and symplectic resolutions have become ubiquitous throughout algebraic and di↵erential geometry, representation theory and mathematical physics. For instance, Higgs bundles, which first emerged about thirty years ago in Nigel Hitchin’s study of the self-duality equations on a Riemann surface [Hit87]andinCarlosSimp- son’s work on nonabelian Hodge theory, [Sim94a; Sim94b], play a role in many di↵erent areas of mathematics, including gauge theory, Kähler and hyperkähler geometry, surface group representations, integrable systems, the Deligne–Simpson problem on products of matrices, and (more recently) mirror symmetry and Langlands duality. On the other hand, the theory of (conical) symplectic resolutions has been widely studied not only in mathematics, but also in physics, and has applications and connections to representation theory, symplectic geometry, quantum cohomology, mirror symmetry, and equivariant cohomology. For a survey on some of these connections, see, e.g.,[BPW16]and[Bra+16].

Multiplicative preprojective algebras were ﬁrst deﬁned by Crawley-Boevey and Shaw

1 in [CS06], with the aim of better understanding Katz’s middle convolution opera- tion for rigid local systems, [Kat96]. Another important application contained in the seminal paper [CS06]isthesolutionofthemultiplicativeDeligne-Simpsonprob- lem in terms of the root data of a certain star-shaped quiver. Moduli spaces of representations of these algebras, in the sense of King, [Kin94], give rise to the so called multiplicative quiver varieties,whichcanbethoughtofasamultiplicative version of Nakajima’s quiver varieties, [Nak94]. A number of authors have studied multiplicative quiver varieties since their definition: among others, Jordan, in [Jor14], considered quantizations of such varieties from a representation theoretic point of view by constructing flat q-deformations of the algebra of di↵erential operators on certain ane spaces; a more geometric approach was used by Yamakawa in [Yam08], where a symplectic structure on these moduli spaces was defined and studied – some of these results will be recalled in the next chapters. More recently, (derived) multiplicative preprojective algebras appeared in the study of wrapped Fukaya categories of certain Weinstein 4-manifolds, see [EL17]. In the recent work of Chalykh and Fairon [CF17], multiplicative quiver varieties are used to construct a new integrable system generalising the Ruijsenaars–Schneider system, which plays a central role in supersymmetric gauge theory and cyclotomic DAHAs. Moreover, we have been informed of the on-going work of Gammage and McBreen [Gam18], related to [BK16, §7], on mirror symmetry for multiplicative quiver varieties, which, at least in the case of ALE spaces (i.e. Dynkin quivers), are believed to be self- mirrors. Given their appearance in so many di↵erent contexts, it seems natural to perform a careful analysis on multiplicative quiver varieties from the point of view of symplectic algebraic geometry.

The subject of symplectic resolutions and symplectic singularities has recently gained importance in many areas of mathematics and physics. Their quantisations subsume many of the important examples of algebras appearing in representation theory (Cherednik and symplectic reﬂection algebras, D-modules on ﬂag varieties and representations of Lie algebras, quantised hypertoric and quiver algebras, etc.). More- over, there is a growing theory of symplectic duality, or three-dimensional physical mirror symmetry ([Nak16; CHZ14; Bra+16]andmanyothers),betweenpairsof these varieties. Pioneering work of Braverman, Maulik and Okounkov shows that their quantum cohomology is also deeply tied to connections arising in representation theory, which is expected to relate to derived autoequivalences of these varieties.

2 Since, as mentioned before, quiver varieties play such an important role here, it is expected that multiplicative quiver varieties will as well. Moreover, the varieties in question are instances of moduli spaces parametrizing geometric objects. The study of such spaces and their singularities is, in general, important in algebraic geometry.

In this thesis, we shall see that, for the moduli spaces we consider, the existence of symplectic resolutions is intrinsically linked to certain numerical and combinatorial data associated to such varieties. As an instance of such a connection, in Chapter 3 we study a combinatorial problem related to multiplicative quiver varieties. This aspect represents yet another reason for considering the topic very interesting.

We now outline the structure of the thesis and point out the main results contained in this work: we shall only state the most important theorems as all other ancillary results will be discussed in depth in the relevant chapters. Chapter 1 is devoted to recalling some preliminary results which will be extensively used throughout this thesis as well as to deﬁning and giving the main properties of the varieties appearing in the nonabelian Hodge correspondence, which play a key role in this work. As such, we outline the main statements of the nonabelian Hodge correspondence and then we focus on the Betti and Dolbeault moduli spaces. More- over, in order to make this thesis as self-contained as possible, we recall the deﬁnition of symplectic singularities and resolutions and present some important properties that will be extensively used in Chapters 2 and 3. Note that, for sake of brevity, we limited ourselves to treating such topics only from the perspective of symplectic algebraic geometry: as such, we leave out from our analysis some other important aspects, more closely related to mathematical physics and di↵erential geometry – such as the gauge-theoretic point of view of the nonabelian Hodge correspondence: nonetheless, for convenience of the reader, we carefully point out the relevant references in the existing literature. Finally, in the last part of the chapter, we summarise the main results of the work of Bellamy and Schedler, [BS16b], which inspired the theorems and conjectures outlined in this thesis. Indeed, in [BS16b], the authors carry out a detailed study of the symplectic algebraic geometry of Nakajima’s quiver varieties and character varieties of compact orientable surfaces and, in order to do so, they establish a general procedure which can be adapted to other contexts and, in particular, to the cases treated in this thesis.

In Chapter 2 we deal with the ﬁrst of the two main topics treated in this thesis:

3 Higgs bundles on smooth projective curves. More precisely, we consider the moduli space (X, n) of semistable Higgs bundles of rank n and degree 0 on a compact MH Riemann surface X of positive genus g and give an answer to the two following questions: is (X, n) a symplectic singularity? Does (X, n)admitasym- MH MH plectic resolution? As we have already pointed out, we approach these problems essentially using the techniques developed in [BS16b]and,inparticular,theworkof the authors on symplectic resolutions of character varieties. Another fundamental ingredient in the proof of the main results of Chapter 2 is the Isosingularity theorem, proved by Simpson in [Sim94b, §10], which establishes an ´etale-local isomorphism at corresponding points between the Betti, the de Rham and the Dolbeault moduli spaces. Combining these results, we manage to prove the following

Theorem. (Theorems 2.2.3, 2.2.4, 2.2.5) The following holds:

1. The moduli space (X, n) is a symplectic singularity; MH 2. (X, n) admits a symplectic resolution if and only if g(X), the genus of X, MH is 1 or (n, g(X)) = (2, 2) or n =1.

As we shall see in Chapter 2,inthecasewhen(g(X),n) =(2, 2) and g(X),n>1, we 6 show that (X, n)isfactorialandhasterminalsingularities,whichimpliesthat MH symplectic resolutions can not exist. We limit ourselves to consider Higgs bundles, instead of the more general G-Higgs bundles: nonetheless, it seems that some of the ideas used for (X, n)couldbeextendedtothemodulispacesofsemistable MH G-Higgs bundles and we give an account of this aspect at the end of the chapter.

In Chapter 3, we focus on the Betti side of the nonabelian Hodge correspondence in the noncompact case: namely, we consider character varieties of Riemann surfaces with punctures. In order to study such varieties, we consider moduli spaces of representations of multiplicative preprojective algebras – a class of quiver algebras deﬁned by Crawley-Boevey and Shaw in [CS06]. Considering such algebras and their representations enables us to study the symplectic algebraic geometry of character varieties of open surfaces and is of interest in its own sake. We begin the chapter by introducing, following [CS06], multiplicative preprojective algebras,denoted

q Q0 by ⇤ (Q), where Q is a quiver and q (C⇤) aparameter;wethenrecalltheir 2 most important properties, from the perspective of both algebraic geometry and representation theory. We outline the construction of the varieties parametrising

4 their (semistable) representations and we study the singularities of such varieties. We detect the singular locus and, in certain cases, prove that the singularities are symplectic. Note that one of the most critical aspects in the study of multiplicative quiver varieties is related to the fact that it is still not known when they are nonempty. In Chapter 3,wediscussinmoredepththisissueandthecasesinwhich one is able to give a positive answer. Nonetheless, all the theorems contained in the chapter have to take into account this subtlety. This is shown, e.g. in one of the main results, which is reported below. In the following result, we will assume that the dimension vector deﬁning the multiplicative variety lies in a special region,

Q0 denoted by ⌃q,✓,ofthesetofalldimensionvectors,N ,whereQ0 denotes the set of vertices fo the quiver Q.Thedeﬁnitionofsuchasubsetanditspropertiesare given in full details in Chapter 3.

Theorem. (Theorem 3.0.7) Let ↵ ⌃ and assume that ↵ =2 for N and 2 q,✓ 6 2 q,✓ p()=2. Then, assuming it is non-empty, (Q, ↵) satisﬁes the following: Mq,✓

• its normalisation is a symplectic singularity;

• if ↵ is q-indivisible, then for suitable generic ✓0, it admits a symplectic resolution of the form (Q, ↵) (Q, ↵); Mq,✓0 !Mq,✓

• if ↵ = m for ⌃q,✓ and m 2, and q,✓(Q, ) = , then q,✓(Q, ↵) 2 M 6 ; M does not admit a symplectic resolution. Moreover, for suitable generic ✓0, (Q, ↵) is a singular factorial terminalisation. In fact, (Q, ↵) itself Mq,✓0 Mq,✓ contains a singular, factorial, terminal open subset.

In order to apply the above result to the nonabelian Hodge correspondence, we also construct a connection between multiplicative quiver varieties and punctured character varieties by proving that, with a clever choice of the quiver and the dimension vector, one can embed any punctured character variety – by which we mean character varieties of Riemann surfaces with punctures – inside an appropriate multiplicative quiver variety as a singular open subset, see [CS06], which is where the inspiration for this result came from.

We then proceed to proving some generalisations to the above result, to multiplicative quiver varieties whose dimension vector does not lie in ⌃q,✓ and, to achieve this goal, we consider a decomposition of multiplicative quiver varieties which is analogous to Crawley-Boevey’s decomposition of (ordinary) quiver varieties, see [Cra02;

5 BS16b]. Moreover, we perform some combinatorial computations to classify all the possible (2, 2) cases, for which we conjecture that there exists a symplectic resolution. Analogously to Chapter 2,thetechniquesweusetostudymultiplicativequiver varieties are taken from the analysis carried out in [BS16b] for Nakajima’s quiver varieties.

The last chapter contains a number of conjectures that arise naturally from the work carried out in Chapters 2 and 3.Moreprecisely,fortheDolbeaultmodulispaces, we conjecture that the results proved in Chapter 2 could be extended to the case of parabolic (G-)Higgs bundles; on the other hand, motivated by [CS06, §3], [EE07] and the work of Bocklandt, Galluzzi and Vaccarino of moduli spaces of representations of 2-Calabi–Yau algebras, [BGV16], we conjecture that, in the non-Dynkin case, multiplicative preprojective algebras are indeed 2-Calabi–Yau: we give some arguments in favour of such a conjecture and outline some possible applications. Finally, in the last part of the chapter, we discuss some more general ideas for how to give a uniﬁed proof of the results contained in Chapters 2 and 3.Inorderto do that, we consider the setting of 2-Calabi–Yau categories and make conjectures on the geometric properties of the associated moduli spaces of (polystable) objects. If such conjectures were true, they would represent general theorems, of which the results on the Betti and Dolbeault moduli spaces are particular cases. We hope to come back to these very interesting topics in future research.

6 The diculty lies not so much in developing new ideas as in escaping from old ones. John Maynard Keynes 1 Preliminary results

As outlined in the Introduction, in this chapter we recollect the main results and ﬁx the notation which will be used throughout the following chapters. More precisely, the ﬁrst section is devoted to an introduction to the nonabelian Hodge correspondence and the associated moduli spaces. Then, we provide the reader with the necessary notions of birational geometry needed to understand the content of the following chapters and lastly, we give a brief account on the main results of [BS16b], whose techniques are used extensively throughout the thesis.

1.1 The nonabelian Hodge correspondence

As suggested by the name, the nonabelian Hodge correspondence can be thought of as a nonabelian version of the well known Hodge Theorem, which gives an isomor- n phism between the de Rham cohomology HdR(X, C) and the Dolbeault cohomology Hq(X, ⌦p)ofacompactK¨ahlermanifoldX.Puttingthisresulttogether p+q=n with the classical de Rham Theorem, one obtains isomorphisms:

n n q p HB(X, C) ⇠= HdR(X, C) ⇠= H (X, ⌦ ), p+q=n M

7 n where HB(X, C)isthesingularcohomologyofthemanifoldX with coecients in the ﬁeld C. In the nonabelian Hodge correspondence, C is substituted by a complex algebraic group G and the above cohomology spaces are replaced by the so-called Betti, de Rham and Higgs (or Dolbeault) moduli spaces respectively, which all have a much richer geometric structure than their abelian counterparts. More explicitly, in the case of G = GL(n, C), one considers the following spaces.

• The Betti moduli space B(X, n): the space of representations ⇡1(X) M ! GL(n, C)modulotheconjugationactionofGL(n, C), which is also known as the character variety X(g, n), recalled below, in Section 1.1;

• The de Rham moduli space dR(X, n): the moduli space of rank n holomor- M phic vector bundles on X equipped with a ﬂat connection;

• The Higgs (or Dolbeault) moduli space H (X, n): the moduli space of semistable M Higgs bundles of degree 0 and rank n,whichwerecallbelow,inSection1.1.

From the work of Hitchin [Hit87], Donaldson [Don87], Corlette [Cor88]andSimpson [Sim92], we know that one can construct natural bijections

(X, n) = (X, n) = (X, n). (?) MB ⇠ MdR ⇠ MH

Moreover, the nonabelian Hodge correspondence states that much more is true: indeed, as a consequence of results in [Sim94b], we have the following:

Theorem 1.1.1. Denote by : (X, n) (X, n) and : (X, n) MB !MdR MdR ! (X, n) the natural bijections mentioned in (?). Then, one has that MH

(1) in an isomorphism of the associated complex analytic spaces;

(2) is a homeomorphism of topological spaces.

The fact that Theorem 1.1.1.(1)isnotjustaset-theoreticbijection,butaniso- morphism of complex analytic spaces, is a key ingredient in the proof of the main result of Chapter 2,asitenablesustotransfertheformalisomorphismgivenby the Isosingularity theorem to another formal isomorphism between di↵erent spaces. This will become clear in Section 2.1 of the next chapter.

8 In general, the moduli spaces above are analytic varieties: let sm(X, n)denote M⇤ their smooth locus, for = B,dR and H.Thesearecomplexmanifoldswiththe ⇤ same real manifold structure. Indeed, we have the following

Theorem 1.1.2. [Sim94b] The complex manifolds sm(X, n), sm(X, n) and MB MdR sm(X, n) are isomorphic as real di↵erentiable manifolds. MH Remark 1.1.3. Even though they have the same real di↵erentiable structure, the above manifolds are not isomorphic as complex manifolds.

Remark 1.1.4. Even though we stated the above theorem in the special case when G = GL(n, C), it is worth pointing out that, by the work of Simpson, [Sim94b], the result holds for an arbitrary algebraic group G.Weshallcomebacktothispoint and explain the implications of such a general version of the above theorem at the end of the next chapter, in Section 2.3.

In the next two subsections, we study in more depth the properties of the Betti and Dolbeault moduli spaces.

1.1.1 Character varieties

Let X be a smooth complex projective curve of genus g,andletG be a reductive algebraic group over C.ConsiderthespaceYG = Hom(⇡1(X),G)ofhomomorphisms from the fundamental group of X to the group G. Using the presentation by generators and relations of ⇡1(X), we can give YG the structure of ane variety: indeed, we know that ⇡ (X) = a ,...,a ,b ,...,b /R, 1 ⇠ h 1 g 1 gi g where R is the relation R = i=1[ai,bi]and[a, b]denotesthecommutator[a, b]:= 1 1 2g aba b .Then,wecanembedQ YG into G as follows:

Y G2g,⇢ (⇢(a ),...,⇢(b )), G ! 7! 1 g

2g which is equivalent to considering YG as the subvariety of G cut out by the equation

[Ai,Bi]=1, i=1 Y

9 for A ,B G, i =1,...,g.ThegroupG acts by conjugation on G and one may i i 2 deﬁne the G-character variety of X as the categorical quotient (see [MFK02])

X(g, G)=YG G.

Algebraically, this is just

G X(g, G)=Spec(C[YG] ),

the spectrum of the ring of G-invariant functions on YG. Remark 1.1.5. In the notation X(g, G)weomittedX since the character variety depends only on the topology of X,i.e.onitsgenus,andnotonthecomplex structure.

Despite their simple deﬁnition, character varieties have a very rich geometry and have been the subject of a large body of literature. For the purposes of this thesis, we will be interested in the case where G = GL(n, C)andwewillusethenotation X(g, n) for the character variety X(g, GL(n, C)), where X is a compact Riemann surface of genus g.

1.1.2 Higgs bundles

In what follows we shall give a brief overview of Higgs bundles and their moduli spaces. Our main references are [GR15] and the seminal papers of Hitchin [Hit87] and Simpson [Sim92; Sim94a; Sim94b].

Deﬁnition 1.1.6. A Higgs bundle on X is a pair (E,), where E is a holomorphic vector bundle on X and , the Higgs ﬁeld,isanEnd(E)-valued 1-form on X,i.e. H0(End(E) ⌦1 ). 2 ⌦ X

In order to have a moduli space with a meaningful geometric structure, one has to consider bundles of ﬁxed rank and degree which satisfy the stability condition deﬁned below. Without such a stability condition, the moduli problem would not admit a coarse moduli space in the category of schemes and one would have to consider (X, n) as an algebraic stack, which we want to avoid. Nonetheless, the MH interested reader may consult [CW18].

10 Even though we will limit ourselves to studying Higgs bundles on Riemann surfaces, it is important to point out that the deﬁnition we give below extends to higher dimensional projective manifolds, see [Sim94b]. For the sake of completeness, we report it here.

Definition 1.1.7. Let M be an d-dimensional complex Kähler manifold. A Higgs bundle on E is given by a pair (E,), where E is a holomorphic vector bundle on M and is a section of the sheaf End(E) ⌦1 satisfying the following integrability ⌦ M condition: =0. ^ Remark 1.1.8. Clearly, the definition of Higgs bundle on a curve is a particular case of the above definition, since, when d =1,theintegrabilityconditionaboveis automatically satisfied.

Deﬁnition 1.1.9. A Higgs bundle (E,) on X is semistable if for any nontrivial subbundle F of E such that (F ) F ⌦1 ,onehas ⇢ ⌦ X

µ(F ) µ(E),  where µ(E):=deg(E)/rank(E) is the slope of a vector bundle. The Higgs bundle (E,)issaidtobestable if the strict inequality holds for all strinct subbundles. Moreover, (E,)issaidtobepolystable if it is either stable or a direct sum of stable Higgs bundles with the same slope.

Remark 1.1.10. Explaining the origin of the notion of Higgs bundles and the use of such a terminology, ﬁrst introduced by Hitchin, is beyond the scope of this thesis, and thus we would like only to highlight that Higgs bundles were deﬁned in the context of the study of certain self-duality equations on a Riemann surface. The interested reader may wish to refer to the original paper [Hit87]andtothereferences mentioned in the introduction for further details.

As mentioned before, imposing the (semi)stability condition makes it possible to have a moduli space that has suciently nice properties. The construction of such a moduli space was firstly carried out by Hitchin, in the rank 2 case [Hit87], and generalized to arbitrary rank by Nitsure [Nit91], via the use of Geometric Invariant Theory. We summarise this in the following theorem, which describes the structure of the moduli space in the case when the g(X) 2. Before stating this result, we 11 need to give a definition of S-equivalence between Higgs bundles. This definition goes as follows: given a semistable Higgs bundle (E,), it is possible to show that it admits a Jordan-Hölder filtration

E E, 0 ⇢···⇢ of -invariant subbundles such that Ei/Ei 1 is a stable Higgs bundle of the same slope for every i.Theisomorphismclassoftheassociatedgradedbundle iEi/Ei 1 is unique, and semistable Higgs bundles are said to be S-equivalent if their associated graded bundles are isomorphic.

We now deﬁne the moduli functor for the problem of parametrizing S-equivalence classes of semistable Higgs bundles and report the fundamental result that states that such a moduli functor has a coarse moduli space which is a quasi-projective variety.

Definition 1.1.11. Let X be a smooth projective curve of positive genus g, S a C-scheme and fix d, n two integers with n 1. A family of semistable Higgs bundles on X of degree d and rank n parametrized by S is an S-flat family of vector bundles of rank n and degree d on S X, X S,togheterwithamorphism ⇥ E! ⇥

: ⇡⇤K , E!E⌦ X where ⇡ : S X X is the projection on the second factor and K the canonical ⇥ ! X bundle on X, such that, for every closed point s S,therestriction( , ) X is 2 Es s ! a semistable Higgs bundle on X.Giventwofamilies( , ) and ( 0, 0)ofsemistable E E Higgs bundles of rank n and degre d on X parametized by S,wesaythattheyare

S-equivalent if, for each closed point s S, Higgs bundles ( , )and( 0 , 0 )are 2 Es s Es s S-equivalent. We denote by MH (X, n, d)themodulifunctor

M (X, n, d):Sch Sets, H C ! associating to any C-scheme S the set of S-equivalence classes of families of semistable Higgs bundles of degree d and rank n on X.

Remark 1.1.12. Analogous deﬁnitions are given for the case of stable Higgs bundles on X.

12 Theorem 1.1.13. [Hit87; Nit91] Let g 2. The moduli functor M (X, n, d) H admists a coarse moduli space, denoted by (X, n, d). (X, n, d) is a quasi- MH MH projective variety, which contains the moduli space of stable Higgs bundles s (X, n, d) MH as an open smooth subvariety.

In what follows we will use the notation (X, n)forthemodulispace (X, n, 0). MH MH (X, n)ispreciselytheobjectofstudyofChapter2,andinSection2.2 we will MH characterize the singularities of such a quasi-projective variety and determine when it admits a symplectic resolution. For this, we shall ﬁrst recall some basic facts about symplectic singularities and resolutions.

1.2 Symplectic singularities and their resolutions

The theory of symplectic singularities and symplectic resolutions was first defined by Beauville in [Bea00] and, since then, it has seen a tremendous development, see e.g. [Fu06]. In this section, we aim at recollecting the basic facts about this theory and state a number of results which shall be used in the next chapters. For the sake of completeness, we shall also report the proofs of such results. Apart from giving the definition of a symplectic singularity and symplectic resolution, we shall recall some standard definitions coming from birational geometry, e.g. the definitions of terminal singularity and factorial variety. For such topics, our reference is [Ish14].

Deﬁnition 1.2.1. Let Y be a normal algebraic variety over C.Then,wesaythat Y is a symplectic singularity if the smooth locus of Y , U = Y Y sing,carriesa \ holomorphic symplectic 2-form !U such that, for every resolution of singularities

⇢ : Y˜ Y the pull-back ⇢⇤! extends to a holomorphic 2-form on Y˜ . ! YU Remark 1.2.2. One could alternatively define symplectic singularities by requiring the existence of a resolution of singularities that satisfies the condition of the above definition. It turns out that this is equivalent to requiring the pull-back of the symplectic form to extend for every resolution of singularities.

Deﬁnition 1.2.3. Given a symplectic singularity (X, !U ), we say that a resolution

⇢ : X˜ X is symplectic if the extension of ⇢⇤! is a holomorphic symplectic 2-form. ! U Remark 1.2.4. The reader should refer to [Fu06, §2], for a list of examples of symplectic singularities, symplectic resolutions and an account on symplectic singularities

13 which do not admit symplectic resolutions. In this context, our study provides further examples of symplectic singularities which do not admit a symplectic resolution. For related examples, see, e.g.,[Bel09; BS13; BS]onsymplecticsingularitiesarising from linear quotients. Remark 1.2.5. Note that, following Beauville [Bea00], we define symplectic singularities using holomorphic 2-forms. One can also define them requiring the symplectic form to be algebraic, and it appears that the symplectic structures defined on the moduli spaces under consideration are indeed algebraic.

We now define factorial varieties and terminal singularities. In order to give such definitions, we shall fix some hypotheses and notation: let X be a normal variety such that the canonical divisor KX is Q-Cartier and let f : Y X aresolutionof ! singularities. Then, one has that

K f ⇤K + a E , Y ⇠ X i i i X where i aiEi is the (possibly empty) formal sum of divisors contained in the ex- ceptionalP locus of the resolution f and ai are rational numbers, called discrepancies. Deﬁnition 1.2.6. In the above setting, X is said to have terminal singularities if ai > 0foreveryi.

Deﬁnition 1.2.7. An algebraic variety X is said to be factorial if every Weil divisor is a Cartier divisor.

Finally, we shall deﬁne crepant resolutions.

Deﬁnition 1.2.8. In the above setting, a resolution ⇡ : Y X is crepant if the ! following holds:

⇡⇤K K . X ⇠ Y Remark 1.2.9. Symplectic resolutions are particular examples of crepant resolutions: indeed, if f : Y (X, !)isasymplecticresolution,thenthefactthatf ⇤! extends ! to a symplectic form on Y implies that its top wedge power is non-degenerate and, as such, it generates the canonical sheaf ⌦ .Inparticular,thisimpliesthatf ⇤K Y X ⇠ KY .

We now recall a standard result on the existence of crepant resolutions which we shall use multiple times in this thesis.

14 Proposition 1.2.10. Let X be a factorial variety with terminal singularities. Then, X does not admit crepant resolutions.

Proof. Let f : Y X be a resolution of singularities. Since X is factorial, by van ! der Waerden purity, see [Deb01,Section1.40],onehasthattheexceptionallocusof f is a divisor, and, therefore, the summation on the right hand side of the equation

K f ⇤(K )+ a E , Y ⇠ X i i i X is nonempty. Note also that ai > 0 because of the terminality assumption. There- fore, f cannot be crepant and the proposition is proved.

To end this section, we recall an important result of Namikawa, which will be used later in this thesis to prove that (X, n), is a symplectic singularity. It is a crite- MH rion that enables one to prove that an algebraic variety is a symplectic singularity by checking that the variety has rational Gorenstein singularities and that the smooth locus admits a symplectic form.

Proposition 1.2.11. [Nam01b, Theorem 6] Let Y be a complex algebraic variety. Then Y is a symplectic singularity if and only if Y has rational Gorenstein singularities and the regular locus U of Y admits an everywhere non-degenerate holomorphic closed 2-form.

1.3 Symplectic resolutions of quiver and character varieties

Before outlining the main results of the work [BS16b], from which we take a great deal of inspiration, we recall the basic deﬁnitions and ﬁx the notations from the theory of quiver representations.

For a ﬁnite quiver Q,letQ0 and Q1 denote the set of vertices and the set of arrows of Q,respectively.Moreover,foranarrowa Q ,leth(a)andt(a)denotethehead 2 1 and the tail of a,respectively.Foradimensionvector↵ NQ0 , we will denote by 2 Rep(Q, ↵)thespaceofrepresentationsofQ of dimension ↵,whichisnaturallyacted upon by the group GL(↵):= GL(↵ ). i Q0 i 2

Q Q0 The coordinate vector at vertex i is denoted ei.ThesetN of dimension vectors is partially ordered by ↵ if ↵ for all i and we say that ↵>if ↵ with i i 15 Q0 ↵ = .Thevector↵ is called sincere if ↵i > 0foralli.TheRingelformonZ is 6 deﬁned by ↵, = ↵ ↵ . h i i i t(a) h(a) i Q0 a Q1 X2 X2 Let (↵, )= ↵, + ,↵ denote the corresponding Euler form and set p(↵)= h i h i 1 ↵, ↵ .Thefundamentalregion (Q)isthesetof0= ↵ NQ0 with connected h i F 6 2 support and with (↵, e ) 0foralli. i 

Q0 Q0 If i is a loopfree vertex, so p(ei)=0,thereisareflectionsi : Z Z defined ! by s (↵)=↵ (↵, e )e .Therealroots(respectivelyimaginaryroots)arethe i i i elements of ZQ0 which can be obtained from the coordinate vector at a loopfree vertex (respectively an element of the fundamental region) by applying some ± sequence of reflections at loopfree vertices. Let R+ denote the set of positive roots. Recall that a root is isotropic imaginary if p()=1andanisotropic imaginary if p() > 1. We say that a dimension vector ↵ is indivisible if the greatest common divisor of the ↵i is one.

The work of Bellamy and Schedler [BS16b]onsymplecticresolutionsofquiverand character varieties is crucial in this thesis for, at least, two reasons: ﬁrst, the techniques used in the next chapters on Higgs moduli spaces and multiplicative quiver varieties are largely inspired by [BS16b]; secondly, such a paper represents the starting point of the mathematical investigations which form the core of the present work. Indeed, the interest in understanding the singularities of the moduli spaces appearing in the nonabelian Hodge correspondence was ﬁrstly prompted by a remark in the Introduction of [BS16b].

The main results of [BS16b] in the context of Nakajima’s quiver varieties concern the study of the singularities of such varieties: ﬁrst, the authors are able to detect the singular points by identifying a necessary and sucient condition for a point to be smooth, called ✓-canonical stability.Secondly,usingCrawley-Boevey’scanonical decomposition of dimension vectors, they prove that it is sucient to reduce the study to quiver varieties whose dimension vector lies in a special region, denoted with ⌃,✓,ofthefundamentaldomainofthequiverunderlyingthevariety–thiswill be the approach of Chapter 3 in the context of multiplicative quiver varieties. (Or- dinary) quiver varieties are moduli spaces of semistable representations of deformed preprojective algebras. Let us give some deﬁnitions.

16 Let Q be a quiver and let Q be the doubled quiver so that there is a natural identifi- cation T ⇤Rep(Q, ↵)=Rep(Q, ↵). The group G(↵) acts symplectically on Rep(Q, ↵) and the corresponding moment map is µ :Rep(Q, ↵) G(↵), where we have iden- ! tified G(↵) with its dual using the trace form. An element CQ0 is identified with 2 the tuple of scalar matrices (iIdV )i Q0 G(↵). The ane quotient, denoted here i 2 2 1 as M,0(Q, ↵), µ ()// G (↵)parameterizessemi-simplerepresentationsofthede- formed preprojective algebra ⇧(Q):= Q/( (aa a a) p ), where C a Q1 ⇤ ⇤ i Q0 i i 2 2 Q0 pi is the length-zero path at the vertex i.Moreingeneral,given✓ Z ,one P P 2 can consider ✓-semistable (in the sense of King, [Kin94]) representations of ⇧(Q), 1 ✓ ss 1 which form a subset µ () of the set µ ()ofallrepresentationsof⇧(Q)of dimension ↵. The coarse moduli sapce of families of ✓-semistable representations of 1 ✓ ss ⇧ (Q)istheGITquotientµ () // G (↵), which we denote by M,✓(Q, ↵). In the following statements, we assume the dimension vectors to lie in a region denoted by ⌃,✓,whichistheadditiveanalogueoftheset⌃q,✓ mentioned in Theorem 3.0.7, which we shall define later on in the chapter.

In this setting, the main results of [BS16b]areasfollows.

Theorem 1.3.1. [BS16b, Theorem 1.5] Let ↵ ⌃ . Then M (Q, ↵) admits 2 ,✓ ,✓ a projective symplectic resolution if and only if ↵ is indivisible or one has that 1 (gcd(↵),p(gcd(↵) ↵)) = (2, 2).

The above theorem is a consequence of following two results.

Theorem 1.3.2. [BS16b, Theorem 1.6] Let ↵ ⌃ , and suppose that one has 2 ,✓ 1 (gcd(↵),p(gcd(↵) ↵)) = (2, 2). Let ✓0 be a generic stability parameter such that

✓0 ✓.IfM (Q, ↵) is the blowup of M (Q, ↵) along the reduced singular locus, ✓0 ,✓ then the canonical morphism ⇡ : M✓ (Q, ↵) M,✓(Q, ↵) is a projective symplectic f 0 ! resolution of singularities. f Theorem 1.3.3. [BS16b, Theorem 1.7] If ↵ ⌃ is ⌃-divisible, and one has that 2 ,✓ 1 (gcd(↵),p(gcd(↵) ↵)) =(2, 2), then M (Q, ↵) does not admit a proper symplectic 6 ,✓ resolution.

In this thesis, we shall prove analogues of the above results in di↵erent settings, i.e. in the context of moduli spaces of Higgs bundles and multiplicative quiver varieties: more precisely, for the Dolbeault moduli spaces we are able to completely classify the

17 cases in which symplectic resolutions do exist, while the treatment of the problem for multiplicative quiver varieties is much more subtle; for such varieties we are able to prove an analogue of [BS16b,Theorem1.7],while[BS16b,Theorem1.6]remains aconjecture.ThiswillbemadepreciseinChapter3 and Chapter 4.

Another important part of the work [BS16b]concernsthestudyofthesymplectic algebraic geometry of the Betti moduli spaces of the nonabelian Hodge correspondence. Since such theorems will be crucial in Chapter 2,weshallrecallthem.

Remark 1.3.4. In what follows, we shall use the notation from the nonabelian Hodge correspondence, to denote the character variety of a compact Riemann surface.

Theorem 1.3.5. [BS16b, Proposition 8.5] The Poisson variety (X, n) is a sym- MB plectic singularity.

Theorem 1.3.6. [BS16b, Corollary 8.16] Suppose g>1 and (g, n) =(2, 2). Then, 6 the symplectic singularity (X, n) does not admit a symplectic resolution. MB

Although we will not prove the above results, one should note that two di↵erent strategies are proposed in [BS16b]fortheirproofs:

(A) Since a symplectic resolution is a crepant resolution, to prove that the former can not exist, it suces to prove that the latter does not exist. To this end, as proved in Proposition 1.2.10,ifY is a normal variety which is factorial and has terminal singularities, then Y does not admit a crepant resolution, as we have shown earlier in this chapter. In [BS16b,Theorem8.15]itisshownthat (X, n) has these properties under the assumptions of Theorem 1.3.6; MB (B) one may also prove Theorem 1.3.6 by noting that if a symplectic resolution exists, then the same is true for the formal (or ´etale) neighbourhood at every point. This gives an alternative proof of the above result because in [BS16b, Remark 1.21] it is pointed out that the formal neighbourhood of (0,...,0) in the SL(n, C)-character variety is isomorphic to the formal neighbourhood of a certain quiver variety, which in turn, from the proof of [KLS06,TheoremB], does not admit a symplectic resolution when g>1and(n, g) =(2, 2). 6

To prove the main result of Chapter 2,wewilladoptstrategy(B)viatheuseofthe Isosingularity theorem (Theorem 2.1.7), and then apply strategy (A) ´etale locally:

18 in Chapter 2 we will carry out in details the aforementioned steps. Moreover, note that, in principle a similar analysis could be carried out for multiplicative quiver varieties, to prove e.g. an analogue of [BS16b,Theorem1.7]:aswehavealready pointed out, such a result still is a conjecture and the main reason for this is that, at the moment, there is no local ´etale model for multiplicative quiver varieties. In the last chapter of this thesis, we shall outline a possible approach to reach this goal and, therefore, prove that, as well as for ordinary quiver varieties, multiplicative quiver varieties admit a symplectic resolution when the combinatorial condition of [BS16b,Theorem1.7]issatisﬁed.

19 You get in life what you have the courage to ask for. Oprah Winfrey 2 Symplectic resolutions of (X, n) MH

The aim of the chapter is to show how the results of Bellamy and Schedler [BS16b] on symplectic resolutions of quiver and character varieties can be used to derive information on symplectic resolutions for the moduli space (X, n)ofsemistable MH Higgs bundles of degree 0 and rank n on a compact Riemann surface X of genus g. As outlined in the Introduction of this thesis, we prove that for g>1and (g, n) =(2, 2) the aforementioned moduli space does not admit such a resolution. 6 On the other hand, we show that, in the cases when (g, n)=(2, 2) or X is an elliptic curve, (X, n) does admit a symplectic resolution (note that in the case MH of SL(2)-Higgs bundles such a resolution was constructed for g =2in[KY08]). For the proof of these results, a central tool is the so-called Isosingularity Theorem, proved by Simpson in the seminal paper [Sim94b], which we give an account of in the next section. We point out that the results contained in this chapter have been published in [Tir18].

The main results of this chapter can be summarised as follows.

Theorem (Theorems 2.2.3, 2.2.4, 2.2.5 and 2.2.7 below). The following holds true:

(A) The moduli spaces (X, n) are symplectic singularities. MH 20 (B) They admit projective symplectic resolutions exactly in the cases g =1and (g, n)=(2, 2).

Part (A) of the above result is proved by using Namikawa’s criterion [Nam01b, Theorem 6], Simpson’s Isosingularity Theorem (Theorem 2.1.3)andthehyperk¨ahler structure on the moduli space of stable Higgs bundles. Furthermore, part (B) follows from a combination of the aforementioned Isosingularity Theorem and (a formal analogue of) one of the main results in the [BS16b], (Theorem 1.3.6). Finally, to consider the elliptic curve case, a result of Franco, Garcia-Prada and Newstead [FGN13]givesacleargeometricdescriptionofthemodulispace (X, n), when MH X is an elliptic curve, which allows us to see that (X, n)doesadmitasymplectic MH resolution (Theorem 2.2.7).

Symplectic resolutions for Higgs bundles have been considered by Kiem and Yoo in [KY08]: in their work they prove that the moduli space of SL(2)-Higgs bundles admits a symplectic desingularization if and only if g =2.Theirmethodsaredif- ferent from the one exploited here and involve computations of the so-called stringy E-function of an algebraic variety: the authors prove that the stringy E-function of the SL(2)-Higgs moduli space is fractional when g =2,whichimpliesthenon- 6 existence of a symplectic resolution – indeed, it can be proved that, if a variety admits a crepant resolution, then its stringy E-function is a polynomial. In the genus 2 case, [KY08]followsKirwan’sdesingularizationmethod,[Kir85].

2.1 Local structure of (X, n) MH

The aim of this section is twofold: firstly, we describe the local structure of the moduli space (X, n)byunderstandingthefirstorderdeformationcomplexata MH point of such a moduli space and, thus, computing its Zariski tangent space; such calculations were first carried out by Biswas and Ramanan in [BR94]andlateronby Göthen and King in [GK05]inthemoregeneralset-upofmodulispacesoftwisted quiver sheaves; secondly, we recall a fundamental result of Simpson, the so-called Isosingularity Theorem, which is crucial in the proof of the main theorem of this chapter; essentially, our result follows by applying the Isosingularity Theorem to the work [BS16b]. We shall closely follow [BR94], but specialise to the case of Higgs bundles, instead of considering the more general context of G-Higgs bundles.

21 2.1.1 Infinitesimal deformations of Higgs bundles

When studying the geometry of a moduli space parametrizing certain geometric objects, it is very useful to understand the (inﬁnitesimal) deformations of such objects: indeed, they carry important information about the local structure of the moduli space around a point.

Let X be such a compact Riemann surface of positive genus and let (E,)bea Higgs bundle on X.Thereisanaturalcomplexofvectorbundlesassociatedto (E,): namely, consider the map of vector bundles

ad() : End(E) End(E) K , ! ⌦ X given by the endomorphism Lie bracket with ,

ad()(f)=[, f], where [ , ]isintendedtoactonlyontheendomorphismpartof.Onecanconsider the complex C•(E,), given by

0 End(E) End(E) K 0, ! ! ⌦ X ! where the second map is ad(). To state the main result of [BR94], we have to define the deformation functor associated to the pair (E,). Going into the details of the general theory of deformation functors and spaces is beyond the scopes of this thesis, so we limit ourselves to give the definitions in our setting – for a general treatment of the topic, see [Har10, §3.15]. Given a Higgs bundle (E,)onX,we define its formal deformation functor as

: Art Set, F(E,) C ! from the category of local Artinian C-algebras to the category of sets as follows: given A Ob(Art ), (A) is the set of isomorphism classes of families of Higgs 2 C F(E,) bundles ( , ) on Spec(A) X such that there exists an isomorphism between the E ⇥ restriction ( , ) m X and (E,), where mA is the closed point of Spec(A). Then, E | A⇥

22 the deformation space of (E,)isdeﬁnedas

Def(E,):= (E,)(C["]), F where C["]istheringofdualnumbers.

We can now state the main result of [BR94].

Theorem 2.1.1. [BR94, §2.3] The deformation space of the Higgs bundle (E,) is canonically isomorphic to the ﬁrst hypercohomology group of the deformation com- 1 plex, H (C•(E,)), 1 Def(E,) ⇠= H (C•(E,)). Remark 2.1.2. One can check that, if (E,)isstable,thentheinﬁnitesimaldefor- mation space Def(E,)coincideswiththeZariskitangentspaceT (X, n), (E,)MH where, with a slight abuse of notation, we used the same notation for the Higgs bundle (E,)anditsS-equivalence class in (X, n). This result, in combination MH with the above theorem, can be used to give an alternative proof of the computation of the dimension of (X, n). MH Understanding the deformation and Zariski tangent spaces of the variety (X, n) MH is important in view of the next section, where we discuss the Isosingularity theorem, which gives a more precise description of the local structure of the moduli spaces (X, n), (X, n)and (X, n)aroundapoint. MH MB MdR

2.1.2 The Isosingularity theorem

In this section we present the Isosingularity Theorem, proved by Simpson in [Sim94b, §10], which is the key tool to prove the main statement of this chapter. The beauty of Simpson’s result relies on the fact that it establishes the existence of a formally local isomorphism between the moduli spaces (X, n)and (X, n): recall from MB MH Chapter 1, §1.1, that such spaces are isomorphic as topological spaces and, on the smooth locus, as real di↵erentiable manifolds, but not as complex manifolds: it is thus somehow surprising that at the level of formal local neighbourhoods around corresponding points, (X, n)and (X, n)areisomorphic.Itfollowsfrom MB MH Artin’s approximation theorem, [Art69], that such formally local isomorphisms induce étale-local isomorphisms, which is the key ingredient in our discussion on the existence of symplectic resolutions for (X, n). MH 23 The Isosingularity theorem is proved using Goldman-Millson deformation theory, [GM88], which was developed to study the local structure of the moduli space of representations of the fundamental group of a compact Kähler manifold – and, hence, of the Betti space (X, n): in [Sim94b], such arguments are extended to (X, n) MB MdR and (X, n). Note that Simpson’s extension of the work [GM88], applies in the MH general context of the nonabelian Hodge correspondence for an arbitrary algebraic group G,whereoneconsidersprincipalG-Higgs bundles, flat principal G-bundles and representations with values in G for the Dolbeault, de Rham and Betti spaces respectively.

We follow the notation of the Introduction and denote with (X, G), for = M⇤ ⇤ B,dR,H the Betti, de Rham and Dolbeault moduli spaces for the algebraic group G, respectively. Moreover, note that the result stated below is actually a consequence of the original formulation, [Sim94b,Proposition10.5],oftheIsosingularityTheorem. Nevertheless, we shall use the same name for such a result.

Theorem 2.1.3 (Isosingularity). Let X be a compact Riemann surface and G a reductive algebraic group. Moreover, suppose x (X, G) and x0 (X, G) 2MdR 2MH are two corresponding points via the nonabelian Hodge theorem (Theorem 1.1.1). Then, the following isomorphism of formal completions holds:

\(X, G) = \(X, G) . MdR x ⇠ MH x0

Remark 2.1.4. One can deduce an important corollary from the above theorem using a result of Artin ([Art69,Corollary2.6]):indeed,usingthisresultonecanprovethat Theorem 2.1.3 implies that (X, G)and (X, G)are´etale-locallyisomorphic MH MB at corresponding points. We will make use of this consequence later in the chapter when studying the singularities of H (X, G), for G = GL(n, C). Some conjectural M results concerning the general case will be mentioned at the end of the chapter.

Corollary 2.1.5. In the hypotheses and notation of Theorem 2.1.3, there exist ´etale neighbourhoods U and V of x (X, G) and x0 (X, G) respectively, which 2MdR 2MH are isomorphic.

From now on and until the end of Section 2.2,wespecializetothecaseofinterest, namely when G = GL(n, C), even though the results of this section hold for arbitrary G and can be proved using the same methodology.

24 As a consequence of the above result, one can relate formal completions of the spaces (X, n)and (X, n)atcorrespondingpoints.Tothisend,oneneeds MB MH the following result, which relates formal and analytic completions of a variety at a point. The reader should refer to [Tay02,Chapter13]foraproofofthisresultanda detailed treatment of the relations between the complex algebraic and the analytic points of view.

Proposition 2.1.6. Let V be an algebraic variety and let V an denote the space V considered as a complex analytic space. For x a point in V , there is an isomorphism of locally ringed spaces an Vx ⇠= Vx .

b d With the above proposition at hand, one can prove the following theorem, which is the crucial tool to transfer the results about symplectic resolutions from the context of character varieties to that of Dolbeault moduli spaces.

Corollary 2.1.7. There is an isomorphism between the formal completions of the spaces (X, n) and (X, n) at corresponding points. MB MH

Proof. The result is a consequence of Theorem 2.1.3,Proposition2.1.6 and part (1) of Theorem 1.1.1.Indeed,forx apointin (X, n)onehasthefollowingchain MB of isomorphisms

an an B\(X, n)x = B\(X, n) = dR\(X, n) = dR\(X, n)x = H\(X, n)x , M ⇠ M x ⇠ M x0 ⇠ M 0 ⇠ M 00 where x0 = (x)andx00 = (x0)andtheﬁrstandthethirdisomorphismscomefrom Proposition 2.1.6,thesecondisaconsequenceofTheorem1.1.1,andthefourthis precisely the Isosingularity Theorem.

2.2 Proof of the main theorem

This section is devoted to the proof of the main result of this chapter, which follows from Simpson’s Isosingularity theorem applied to the work of Bellamy and Schedler, [BS16b].

First, we recall an important property of the moduli space (X, n), proved by MH Simpson in [Sim94b].

25 Proposition 2.2.1. [Sim94b, Theorem 11.1] The moduli space (X, n) is irre- MH ducible and of dimension 2n2(g 1) + 2.

From the above, one can prove a key fact about the singularities of the moduli space (X, n). In order to do this, we will need the following result, which gives an MH estimate on the codimension of the strictly semistable locus of the moduli space (X, n), i.e. the locus given by Higgs bundles which are semistable but not MH stable, stps(X, n)= (X, n) s (X, n). stps(X, n)coincideswiththelocus MH MH \MH MH of strictly polystable Higgs bundles, hence the notation stps(X, n). MH Lemma 2.2.2. The following holds true:

codim( stps(X, n)) 2. MH

Proof. In this proof we will use the shortened notations and stps for (X, n) Mn Mn MH and stps(X, n), respectively. The ﬁrst step is to give an explicit description of the MH locus stps:fromDeﬁnition1.1.2, a polystable Higgs bundle which is not stable is M the direct sum stable Higgs bundles. Therefore, for any partition n =(n1,...,nk) of n there is a set-theoretic map

s s ⌫n : n := n, ((E1,1),...,(Ek,k)) ( Ei, i). M Mn1 ⇥···⇥Mnk !M 7!

Note that this map ⌫n is algebraic. Up to the action of a permutation group on the image, the map ⌫n is injective so that

dim( )=dimIm(⌫ ). Mn n

Moreover, it is clear that stps = Im(⌫ ), Mn n n (n) 2[P here (n) denotes the set of partitions of n.Therefore,thefollowingholdstrue: P

stps dim n =maxn (n) dim n . M 2P { M }

Let n =(n ,...,n ) (n)beapartitionsuchthatthemaximumaboveisattained 1 k 2P and let k = l(n)beitslength:notethat1 k n.Then,theaboveequalitycan  

26 be written as k dim stps = (2n2(g 1) + 2). Mn i i=1 X The desired estimate can be written as

dim stps dim 2. Mn  Mn

On the other hand, note that, in our setting, the following inequality holds true:

k k k 2 (2n2(g 1) + 2) = 2(g 1) n2 +2k 2(g 1) n . i i  i i=1 i=1 ! i=1 ! X X X From the computation above, one sees that the left hand side of the inequality is precisely dim stps, while the right hand side is dim 2, which follows from Mn Mn k the fact that i=1 ni = n.Therefore,theproofisconcluded. P We are now ready to prove part (A) of the main result of this chapter.

Theorem 2.2.3. Assume that g(X) 2. Then, the moduli space (X, n) is a MH symplectic singularity.

Proof. In order to prove the theorem one needs to verify that the hypotheses of Proposition 1.2.11 are satisfied. To this end, one needs to prove that (X, n)is MH normal, its singularities are rational Gorenstein, and the smooth locus of (X, n) MH admits a holomorphic symplectic form. The first two properties are proved by noting that being normal and rational Gorenstein are étale-local properties and, thus, by Corollary 2.1.5, (X, n)isnormalandhasrationalGorensteinsingularitiesif MH and only if the same holds for (X, n): but this is true from Theorem 1.3.5. MB For the third part, it is a well known result ([Hit87; Sim94a], see also [Wel07]) that s (X, n)admitsahyperkählerstructureand,thus,aholomorphicsymplectic MH structure (this is true, more generally, for Higgs bundles of arbitrary degree d). Such a holomorphic symplectic structure can be extended to the smooth locus: indeed, by Lemma 2.2.2 we know that the codimension of the complement of the stable locus inside the smooth locus is at least 2, and the stable locus is dense in the smooth locus since the latter is irreducible (by Proposition 2.2.1). Therefore, all the hypotheses of Proposition 1.2.11 are satisfied and the theorem is proved.

27 Part (B) of the main result is proved in the following

Theorem 2.2.4. When g>1 and (g, n) =(2, 2), the moduli space (X, n) does 6 MH not admit a symplectic resolution.

Proof. Suppose by contradiction that when g>1and(g, n) =(2, 2) such a resolu- 6 tion ⇢ : (X, n) exists and let x be the point in (X, n) that corresponds M!MH MH to the trivial representation, call it Id, in the Betti moduli space B(X, n)viathe f M homeomorphism given by Theorem 1.1.1.(2).Then,bytheIsosingularityTheorem, there is an étale neighbourhood V of x in (X, n)isomorphictoanétaleneigh- MH bourhood U of Id in (X, n). Furthermore, by assumption, via this étale-local MB isomorphism, from the resolution ⇢,onecanconstructasymplecticresolution˜⇢ of U. But U is factorial and terminal: factoriality of U follows from the proof of [BS16b, Theorem 8.15]; on the other hand, the fact that U has terminal singularities follows from [Nam01a,Corollary1].Sincethisisacontradiction,thetheoremfollows.

The existence of a symplectic resolution in the case when (g, n)=(2, 2) is shown in the following result.

Theorem 2.2.5. When g(X)=2the blow-up : (X, 2) (X, 2) along MH !MH the radical ideal deﬁning the singular locus of H (X, 2) is a symplectic resolution M f of (X, 2). MH

Proof. Note that the corresponding statement for the Betti moduli space (X, 2) MB is proved in [BS16b,Theorem8.11,Corollary8.12].Wecantransferthisresultto the Higgs moduli space via the Isosingularity theorem – more precisely, via Corollary 2.1.5:indeed,thehomeomorphism := of Theorem 1.1.1 preserves the singular locus and, moreover, the property of being a symplectic resolution is formally local on the target. Thus, by the formal local isomorphism given by Theorem 2.1.7,one has that the variety (X, 2) is smooth and the map satisﬁes the property of MH Deﬁnition 1.2.3 and, hence, the theorem follows. f

Given the above theorems, the only case left to consider is that of elliptic curves. Recall that, from [BS16b], it is possible to show that, for X an elliptic curve, the character variety X(g, n)doesadmitasymplecticresolution,(see[BS16b,Propo- sition 8.13]). Moreover, an analogous result can be obtained for the moduli space (X, n)usingaresultshownin[FGN13], which relies on techniques from [Hit87]. MH 28 Theorem 2.2.6. [FGN13, Theorem 3.15] Consider the moduli space (X, n, d), MH where X is an elliptic curve, and let h =gcd(n, d). Then, there exists an isomorphism h ↵ :Sym T ⇤X (X, n, d). n,d !MH

With such an explicit geometric description of (X, n, d), one can prove the MH following result.

Theorem 2.2.7. Let X be an elliptic curve. Then, the moduli space (X, n) is MH a symplectic singularity and it admits a symplectic resolution

n Hilb T ⇤X (X, n). ! M H

Proof. We know from the Theorem 2.2.6 that the moduli space (X, n)isisomor- MH phic to the n-th symmetric power of the cotangent bundle to the elliptic curve X. Via this isomorphism, we can induce a (generic) symplectic structure on (X, n) MH so that, by deﬁnition, the isomorphism ↵n,0 is, generically, a symplectomorphism. Moreover, it is a well-known fact ([Fu06,Example2.4])that,givenasmoothsym- plectic surface S,foranyn 1, the variety SymnS is a symplectic singularity and there exists a symplectic resolution

HilbnS SymnS. !

Therefore, setting S = T ⇤X the theorem follows.

Remark 2.2.8. E. Franco has pointed out to us that the map ↵n,d is actually a symplectomorphism over the stable locus.

2.3 Extensions and future directions

In this last section we present some research questions connected to the problems studied in the chapter. While in Chapter 4 we point a general conjectural strategy to study the singularities of moduli spaces of (semistable) objects in 2-Calabi–Yau categories, of which (X, n)isanexample,(see[FSS17]), here we investigate MH possible generalizations of the main theorem of this chapter to the moduli spaces (X, G), for = B,dR,H and G an arbitrary algebraic group. M⇤ ⇤

29 Another interesting problem, directly related to Theorem 2.2.5,wouldbetoinves- tigate whether the symplectic resolution in the (2, 2) case has a moduli theoretic interpretation: such a question is not new; in fact, it was treated previously by Seshadri, in [Ses77], in the case of ordinary vector bundles, where the resolution is interpreted as a moduli space of semistable parabolic bundles. Moreover, in the case of Higgs bundles on elliptic curves, an answer was given by Groechenig in [Gro14], h where the Hilbert scheme Hilb T ⇤X is seen to be isomorphic to a moduli space of parabolic Higgs bundles.

So far, this question, in the case of rank 2 Higgs bundles on a genus 2 surface, remains open; on the other hand, some progress has been made in understanding more about the geometry of the resolution (X, 2) of Theorem 2.2.5:indeed,in MH ajointworkwithEmilioFranco,[FT], we investigate conjectural relations between f the variety (X, 2) and the famous example of irreducible holomorphic symplectic MH manifold, found by O’Grady in the celebrated paper [O’G99]. This is done via the f work of Ein, Donagi and Lazarsfeld, [DEL97], on degenerations of the integrable system structure deﬁned on the moduli space of Higgs bundles, constructed through the Hitchin ﬁbration. In [O’G99], O’Grady considered the singular moduli space of rank 2 torsion-free semistable sheaves on a K3 surface S with trivial determinant and second Chern class equal to 4 – in such a case, the moduli space contains strictly semistable singular points – and proved that, by performing a blow-up of the singular locus of such a variety, one can construct a symplectic resolution which is a irreducible holomorphic symplectic (IHS, for short) manifold of dimension 10: the main outcome of the work [O’G99]wasthatsuchavarietyrepresentedanew example of IHS manifold, since it was proved not to be equivalent, up to deformation of birational transformations, to any other IHS manifold known at that time.

To end this section, we outline a possible approach to solve the problem of understanding the symplectic geometry of the moduli spaces (X, G), for an arbitrary MH algebraic group G. In order to do that, we need to study more in depth the Isosingu- larity theorem, which proved to be crucial already for the case when G = GL(n, C). In particular, what is most important for us is how the formal isomorphism of Theorem 2.1.3 is constructed. This turns out to be an indirect construction, as what is defined in the proof of such a result is not a formal isomorphism between \(X, G) and \(X, G) ,butamoregeneralresult:indeed,inhisproof, MdR x MH x0 Simpson shows that both \(X, G) and \(X, G) are formally isomorphic to MdR x MH x0 30 athirdformalscheme,whichisdefinedstartingfromthedataoftheproblem.We shall not give the precise definition of such a formal scheme, which can be found in [Sim94b, §10, p. 67]: essentially it is the formal completion at 0 of the good quotient W/H,ofaquadraticcone,W , modulo the action of a reductive subgroup H of G. As explained above, such a formal local isomorphism actually descends from an étale-local isomorphism. Therefore, understanding the symplectic geometry of the good quotient W/H,wouldbeenoughtosolvethecorrespondingproblemfor the moduli spaces (X, G), for = B,dR and H.Itisshownin[BS16b, §8] that, M⇤ ⇤ in the case when G = GL(n, C), such a quotient is a quiver variety, which is what the authors use to prove the theorems of symplectic resolutions of the moduli spaces X(g, n). It would be interesting to understand whether this conclusion is true in the general case of a reductive group G.

31 Life isn’t about ﬁnding yourself. Life is about creating yourself. George Bernard Shaw 3 Multiplicative quiver varieties

As mentioned in the Introduction, this third chapter is devoted to the study of the symplectic algebraic geometry of moduli spaces of representations of multiplicative preprojective algebras.Inparticular,ouraimistocarryoutasimilaranalysistothe one in the previous chapter; namely we shall: understand whether multiplicative quiver varieties are symplectic singularities and classify, modulo a conjecture which we pose at the end of the chapter, all the possible cases in which they admit symplectic resolutions. We apply the results on multiplicative quiver varieties to the study of the symplectic algebraic geometry of character varieties of surfaces with punctures. Since they are the easiest to state and perhaps of the broadest interest, we ﬁrst summarise the relevant results for the latter varieties and then expand the focus and give an account of the theorems proved for multiplicative quiver varieties.

3.0.1 Summary of results on character varieties

Fix a connected compact Riemann surface X of genus g 0, let S = p ,...,p { 1 k}⇢ X be a subset of k 0points,andﬁxatuple =( ,..., )ofconjugacyclasses C C1 Ck i GLn(C),i =1,...,k.LetX := X p1,...,pk be the corresponding punc- C ⇢ \{ } tured surface, and let i be the homotopy class in ⇡1(X)ofsomechoiceofloop around the puncture pi (having the same free homotopy class as a small counter-

32 clockwise loop around pi). We deﬁne the character variety of X with monodromies in as follows: Ci

X(g, k, ):= : ⇡ (X) GL ( ) . (3.1) C { 1 ! n | i 2 Ci}

As recalled in Section 3.2 below, this has the structure of an ane algebraic variety.

Note that X (or X)doesnotappearinthenotationontheleft-handside,sincethe result does not depend on the choice of X up to isomorphism (only the identiﬁcation of ⇡1(X)isrelevant).

Observe that, in order for this character variety to be nonempty, we must have k det( )=1,whereweletdet( )bedeﬁnedasthedeterminantofanyelement i=1 Ci Ci m m of .Letusassumethisfromnowon.Givenm 1weletm =( ,..., ). Q Ci ·C C1 Ck k We call q-divisible if = m 0 for m 2and det( 0)=1.Callitq-indivisible C C ·C i=1 Ci if it is not q-divisible. Below, q-indivisibility willQ be the most important criterion for the existence of symplectic resolutions for X(g, k, ). C For each ,lettheminimalpolynomialofanyA be (x ⇠ ) (x ⇠ ), Ci 2Ci i,1 ··· i,wi ordered so that the sequence ↵ := rank(A ⇠ ) (A ⇠ )hastheproperty i,j i,1 ··· i,j that ↵ ↵ is non-increasing in j (for 0 j w 1, setting ↵ = n). This i,j i,j+1   i i,0 is possible since the non-increasing property obviously holds when all the ⇠i,j are equal. The following quantities will be important:

k w 1 k w i i ` := ↵ ,p(↵):=1+n2(g 1) + n` + ↵ ↵ ↵2 . (3.2) i,1 i,j i,j+1 i,j i i=1 j=1 i=1 j=1 X X X X X The quantity 2p(↵)isthe“expecteddimension”ofthecharactervariety,whichis its actual dimension in many cases, as explained below.

The sequences ↵i,j,togetherwiththerankn,cannaturallycanbearrangedona star-shaped undirected graph, call it ,withnodelabeledbytherank,n,andthe C vertices of the i-th branch labeled ↵i,1,...,↵i,w 1,inordergoingawayfromthe i node – the crab-shaped quiver associated to the variety, which will be important later on, is obtained by orienting the edges of and then adding g loops at the C node.

Our main results on character varieties can be summarised as follows. We divide separately into the genus 0 and the positive genus cases.

33 Theorem 3.0.1. Let g =0and ﬁx n and conjugacy classes 1,..., k GLn(C) C C ✓ as above.

• If `<2n, then one of the following exclusive possibilities occur, and can be computed by an explicit algorithm:

– X(0,k, ) is empty; C – X(0,k, ) is a point; C

– There is a canonical datum (n0,k0, 0 ,..., 0 ,◆) of n0

(a) the graph and labeling ↵ are twice one of those listed in Theorem 3.4.1 below (note in this case k 3, 4, 5 and n 24); 2{ }  (b) the graph is ane Dynkin and the labeling ↵ = m is an imaginary CG root for this graph (here k 3, 4 ); 2{ }

(c) The pair ( ,↵) is obtained from an ane Dynkin one, ( 0 ,m), ap- CG CG pearing in the previous case, by attaching single vertex, at which ↵ has a

prime value p dividing the collection 0, which is attached to an extending C vertex of 0 (i.e., one with valued at one). Again, here k 3, 4 . CG 2{ } Moreover, in all cases except case (b) above, dim (0,k, )=2p(↵). X C

As mentioned in the theorem, the technique used to show non-existence of symplectic resolutions is by identifying an open singular factorial terminal subset.

34 Remark 3.0.2. Regarding the excluded cases of the theorem, we have some expec- tations: In case (a), conjecturally a symplectic resolution can be obtained by ﬁrst applying geometric invariant theory and then blowing up the singular locus; this would follow from Conjecture 4.5.1. No conjectures are made for cases (b) and (c), but in the analogous ordinary (Nakajima) quiver varieties, case (b) does admit a symplectic resolution (via framing the quiver and then applying geometric invariant theory), whereas case (c) does not admit one (by [BS16b]).

In case (c) above, it actually follows from the proof of the theorem that the normalisation of (0,k, )isasymplecticsingularity,sinceapplyinggeometricinvariant X C theory results in a partial crepant resolution by a singular factorial terminal variety. (Note that this partial crepant resolution itself cannot admit a symplectic resolution, but we cannot deduce the same property for (0,k, )itself,assuchanimplication X C is only known for conical varieties, by [BS16a,Theorem2.2]).

Remark 3.0.3. Note that, when k 2, the character variety is always a point (or  empty).

Theorem 3.0.4. Suppose that g 1. Then the following holds:

• If is q-indivisible, then (g, k, ) admits a projective symplectic resolution C X C (via geometric invariant theory). Therefore, its normalisation is a symplectic singularity. Moreover, it has dimension 2p(↵).

• If is q-divisible, then unless one of the following conditions is satisﬁed, then C the normalisation of (g, k, ) is a symplectic singularity which does not admit X C a symplectic resolution (in fact, it contains a singular terminal factorial open subset):

(a) g =2,k =0, and n =2;

(b) g =1,k =0;

Moreover, in all cases except case (b), dim (g, k, C)=2p(↵). X Remark 3.0.5. The cases (a), (b), and (c) are analogous to those in Theorem 3.0.1. However, here, in cases (a) and (b), a symplectic resolution is known to be obtained by the procedures described in Remark 3.0.2;see,e.g.,[BS16b,Section8].Incase

35 (c), on the other hand, by analogy with the case of Nakajima quiver varieties (see [BS16b]), we expect a symplectic resolution not to exist, but we leave the case open. As in Remark 3.0.2,weknowthatthenormalisationofthevarietyincase(c)isa symplectic singularity (it follows from the proof of the theorem).

The proof of these theorems is given in Section 3.5.7;theyareconsequencesofthe main results of the chapter on multiplicative quiver varieties (particularly Corollary 3.5.27).

Remark 3.0.6. Actually, the results above (slightly modiﬁed) should apply not merely to character varieties, but to twisted character varieties, where we replace

⇡1(X)byaﬁnitecentralextension,correspondingtosettingtherelation

g k

[Ai,Bi] Mj i=1 j=1 Y Y to be a root of unity times the identity matrix. To prove such a statement would require a straightforward generalisation of [Cra13]andofSection3.2 below. With this in hand, these results would follow from Corollary 3.5.27 just as before. For some more details, see Section 3.0.4,wherewedescriberoughlyhowtotranslate this corollary into the setting of twisted character varieties.

3.0.2 Multiplicative quiver varieties with special dimension vectors

Q0 Given a quiver Q together with a tuple of non-zero complex numbers q (C⇥) ,one 2 can deﬁne the multiplicative preprojective algebra ⇤q(Q), over the semisimple ring CQ0 (see Section 3.1.1 below). To a representation, we associate a dimension vector in NQ0 .Givenfurthermoreastabilityparameter✓ ZQ0 ,onecandeﬁneavariety, 2 denoted (Q, ↵), which is a coarse moduli space of ✓-semistable representations Mq,✓ of ⇤q(Q)ofdimensionvector↵.Itisnaturaltoaskwhatthedimensionvectors of ✓-stable representations are. Towards this end, one considers a combinatorially-

Q0 deﬁned subset ⌃q,✓ N of the set of all possible dimension vectors (deﬁned in ✓ Section 3.1.1 below). It has the property that, for ↵ ⌃ ,the✓-stable locus is 2 q,✓ dense in (Q, ↵) (and it is always open). However, it is unknown in general Mq,✓ if (Q, ↵) is non-empty. It is expected, but not known, that these conversely Mq,✓

36 describe all dimension vectors of stable representations, i.e.:

q Q0 If there is a ✓-stable representation of ⇤ (Q)ofdimension↵ N ,then↵ ⌃q,✓. 2 2 (*) In the case ✓ = 0, Crawley-Boevey pointed out a work in progress with Hubery towards a proof of (*). We prove a weakened version of (*) below (Corollary 3.5.18), replacing ⌃ by a larger set. Note that, if (*) holds and furthermore (Q, ↵) = q,✓ Mq,✓ 6 ; for all ↵ ⌃ ,thenputtogetherwewouldobtainacharacterisationoftheset⌃ : 2 q,✓ q,✓ in this case, ↵ ⌃ if and only if there exists a ✓-stable representation of dimension 2 q,✓ ↵. However, this is, again, unknown.

To deﬁne (Q, ↵), we require ↵ ✓ =0,andforittobenon-empty,werequire Mq,✓ · that q↵ := q↵i =1.LetN := ↵ Q0 q↵ =1,↵ ✓ =0.Wecall i Q0 i q,✓ N 2 { 2 | · } avector↵ N q-indivisible if 1 ↵/N for any m 2. Equivalently, writing 2Q q,✓ m 2 q,✓ ↵ = m for m =gcd(↵i), we have that q is a primitive m-th root of unity. Note that, if ↵ N is indivisible, it is clearly q-indivisible, although the converse does 2 q,✓ not hold in general. Note that, unlike in the case of character varieties, here the q in “q-(in)divisible” refers to an actual parameter; see Remark 3.0.12 for an explanation how the two notions nonetheless coincide.

We denote by p the following function:

1 p : NQ0 Z,p(↵)=1 (↵, ↵) 0, ! 2 where ( , )denotestheCartan–TitsformassociatedtothequiverQ (see Sec- tion 1.3 for more details). Geometrically, 2p(↵)givesthe“expecteddimension”of (Q, ↵)(whichistheactualdimensionif↵ ⌃ and (Q, ↵) = :see Mq,✓ 2 q,✓ Mq,✓ 6 ; Remark 3.1.23 below).

One of the main results of this chapter, proved in Section 3.3,isthefollowing:

Theorem 3.0.7. Let ↵ ⌃ and assume that ↵ =2 for N and p()=2. 2 q,✓ 6 2 q,✓ Then, assuming it is non-empty, (Q, ↵) satisﬁes the following: Mq,✓

• its normalisation is a symplectic singularity;

• if ↵ is q-indivisible, then for suitable generic ✓0, it admits a symplectic resolution of the form (Q, ↵) (Q, ↵); Mq,✓0 !Mq,✓

37 • if ↵ = m for ⌃q,✓ and m 2, and q,✓(Q, ) = , then q,✓(Q, ↵) 2 M 6 ; M does not admit a symplectic resolution. Moreover, for suitable generic ✓0, (Q, ↵) is a singular factorial terminalisation. In fact, (Q, ↵) itself Mq,✓0 Mq,✓ contains a singular, factorial, terminal open subset.

Implicit in Theorem 3.0.7 is the fact (see Lemma 3.1.16 and Corollary 3.1.24 below) that, for all ↵ ⌃ , (Q, ↵) (Q, ↵)isaprojectivebirationalPoisson 2 q,✓ Mq,✓0 !Mq,✓ morphism for suitable ✓0. This implies, by deﬁnition, that it is a symplectic resolution if the source is smooth symplectic. In the last part of the theorem, by singular factorial terminalisation, we mean a projective birational Poisson morphism with source a singular factorial terminal variety.

In the case of generic ✓,thetheoremcanbesimpliﬁedasfollows,avoidingtheneed to check if a vector is in ⌃q,✓. First, note that ⌃q,✓,bydeﬁnition,isasubsetofthe set of roots for the quiver (which equals the set of roots of the associated Kac-Moody Lie algebra in suitable cases).

Corollary 3.0.8. Fix an imaginary root ↵ for Q. Let q be such that q↵ =1. Let ✓ be generic (inside the hyperplane ✓ ↵ =0 ). Then: { · }

(i) ↵ ⌃ if and only if ↵ is q-indivisible or anisotropic. If ↵ is q-indivisible, 2 q,✓ (Q, ↵) is smooth symplectic. Mq,✓ (ii) Assume ↵ is q-divisible and anisotropic. Moreover, assume that ↵ =2 for 6 p()=2and q =1. Then (Q, ↵) is a (normal) symplectic singularity. Mq,✓ (iii) Under the assumptions of (ii), we have the following:

q 1 – If there exists a ✓-stable representation of ⇤ (Q) of dimension m ↵, for some m 2, then (Q, ↵) is singular, factorial, and terminal, and Mq,✓ hence does not admit a symplectic resolution.

– If, on the other hand, there are no ✓-stable representations of ⇤q(Q) of dimension r↵ for all rational r<1, then (Q, ↵) is smooth. Mq,✓

Note that, in the general case with ↵ ⌃ ,theabovecorollaryalwaysdescribes 2 q,✓ the source of projective birational Poisson morphisms obtained by suitably varying ✓.

38 Remark 3.0.9. Note that every r↵, r Q<1 appearing in the theorem is also in ⌃q,✓, 2 by part (i). Thus, if there exists a ✓-stable representation of every dimension in

⌃q,✓,theninpart(iii)wearenecessarilyintheﬁrstcase.Thisconditionholdsin the additive case (with RQ0 ), by [BS16b], but there is no further evidence that 2 this holds here. Also, note that the two cases are not exhaustive, so it could happen 1 that there are some stable representations of dimension r↵ but not when r = m . In this (unexpected) situation, it would require more detailed analysis to determine whether a symplectic resolution exists.

Remark 3.0.10. In the case left out of the theorem, where ↵ =2 for some N 2 q,✓ satisfying p()=2–wecallthisthe“(2, 2)-case” –, we conjecture, as in the special case of character varieties of rank two local systems on genus two surfaces handled in [BS16b], that (Q, ↵)hasasymplecticresolutionobtainedbyblowingupthe Mq,✓ singular locus of (Q, ↵), for suitably generic ✓0. However, in order to prove Mq,✓0 this, it is necessary to understand the ´etale local structure of (Q, ↵), while at Mq,✓ the moment all of our techniques are global in nature. In Section 4.5,wediscuss an approach to understand the local structure of the multiplicative quiver varieties, based on the conjectural 2-Calabi–Yau property of the multiplicative preprojective algebra for non-Dynkin quivers.

Remark 3.0.11. Note that, as part of Corollary 3.0.8,when✓ is generic (and ↵ 2 ⌃ ), we prove normality of the variety (Q, ↵), see Proposition 3.3.10.More- q,✓ Mq,✓ over, we conjecture normality for all ✓ (as well as for the (2, 2)-case). Such a result requires a local understanding of the varieties (Q, ↵), which would again follow Mq,✓ from the conjectural 2-Calabi–Yau property for ⇤q(Q)whenQ is not Dynkin: see Section 4.5.

3.0.3 Character varieties as (open subsets of) multiplicative quiver varieties

In Section 3.2,extendingresultsof[CS06] and [Yam08], we explain how character varieties identify as natural open subsets of the multiplicative quiver varieties for crab-shaped quivers (Theorem 3.2.6). Namely, the character variety identiﬁes as an open subset of a multiplicative quiver variety for the crab-shaped quiver described

Q0 in Section 3.0.1,withappropriateparameterq (C⇥) .Theopensubsetisdeﬁned 2 by requiring the loops to act invertibly (which means that, in the genus zero case,

39 the character variety equals the multiplicative character variety).

Remark 3.0.12. By the above correspondence, a collection of conjugacy classes C✓ GLn(C⇥)isq-divisible in the sense of Section 3.0.1 (where q is not yet a parameter) if and only if, for the associated quiver Q,dimensionvector↵ NQ0 ,andparameter 2 Q0 q (C⇥) ,thevector↵ is q-divisible in the sense of Section 3.0.2.Wehopethat 2 this abuse of notation aids understanding.

It is then an interesting question which character varieties exhibit the di↵erent properties discussed above and, in particular, which ones are the “(2,2)”-cases where an “O’Grady” type resolution is expected? (We already mentioned these cases in Theorems 3.0.1 and 3.0.4,under(a)).TheclassificationisachievedinTheorems 3.4.1 and 3.4.3:alloftheseareinthegenuszerocase,withthreetofivepunctures and particular monodromy conditions, as classified in Theorem 3.4.1,exceptfortwo cases in Theorem 3.4.3.Thelattercasescorrespondtoonce-puncturedtoriandto closed genus two surfaces (with even rank and rank two local systems, respectively, the former having particular monodromy about the puncture).

3.0.4 General dimension vectors

Although it is dicult to study directly quiver varieties of dimensions ↵/⌃ ,in 2 q,✓ the additive setting this issue is alleviated by Crawley-Boevey’s canonical decomposition, expressing an arbitrary variety as a product of varieties for dimension vectors in ⌃ [Cra02,Theorem1.1](extendedto✓ =0in[BS16b,Proposition2.1]).In q,✓ 6 Theorem 3.5.17 below, we provide a version of this decomposition in the multiplicative setting using reﬂection functors, following the proof of [Cra02], which is weaker in the sense that the dimension vectors of the factors need not be in ⌃q,✓,andhence the factors could further decompose – although it is not known in general if they do. One of the reasons why we must give the weaker statement is the unavailability of (*); see Section 4.2 for more details. Along with this, we prove a more general suf- ﬁcient criterion for varying ✓ to produce a symplectic resolution (Theorem 3.5.22), that does not require dimension vectors to be in ⌃q,✓. Using these results, in Theo- rem 3.5.26,weareabletoextendTheorem3.0.7 to general dimension vectors. The content of Theorems 3.5.17 and 3.5.26 can be summarised in the following. Here,

⌃q,✓ is a larger set than ⌃q,✓,consistingofrootsforwhichacertainmultiplicative iso moment map is ﬂat; ⌃ ⌃q,✓ is the subset of isotropic roots. See Sections 3.1 e q,✓ ✓ 40 and 3.5 for details on these deﬁnitions.

Theorem 3.0.13. Assume that (Q, ↵) is non-empty. Mq,✓ (1) (k) (i) iso (1) There is a unique decomposition ↵ = + + with ⌃q,✓ N 2 ⌃q,✓, ··· 2 [ · satisfying the properties: e

(a) When p((i))=0then (i) ⌃ ; 2 q,✓ (b) Every other element (i) has the property ((i),) 0 for every real root  N ; 2 q,✓ iso (c) Any other decomposition into ⌃q,✓ N 2 ⌃q,✓ satisfying (a) and (b) is a re- [ · ﬁnement of this one. e

(2) The direct sum map produces an isomorphism (of reduced varieties), using the decomposition in (1),

k (i) (Q, ) ⇠ (Q, ↵). Mq,✓ !Mq,✓ i=1 Y

(i) iso (3) Assume that this decomposition has neither elements N 2 ⌃q,✓(Q, ↵) nor 2 · (i) =2↵ for ↵ N and p(↵)=2. Then: 2 q,✓

• The normalisation of q,✓(Q, ↵) is a symplectic singularity; M (i) (i) • Each factor q,✓(Q, ) with / ⌃q,✓ admits a symplectic resolution; M 2 • If for any factor (i) there exists a ✓-stable representation of dimension (i) = 1 (i) with m 2, then (Q, ↵) does not admit a symplectic resolution. In m Mq,✓ fact, it has an open, singular, terminal, factorial subset.

Putting everything together, in Corollary 3.5.27,weareabletogiveaclassification of crab-shaped settings whose multiplicative quiver varieties admit symplectic resolutions. By Theorem 3.2.6, we also deduce the corresponding statement for character varieties (Theorems 3.0.1 and 3.0.4), which are open subsets of these varieties, for ✓ =0andforcertainvaluesoftheparameterq.Tostatetheresult,firstrecall that the Jordan quiver is the quiver with one vertex and one arrow (a loop). The fundamental region (Q)consistsofthosenonzerovectors↵ NQ0 with connected F 2 41 support and with (↵, e ) 0foralli. As we explain below, by applying certain i  reflection functors, we can reduce to this case. We give a simplified version of the statement of Corollary 3.5.27 below; see the full statement for precise details.

Corollary 3.0.14. Let Q be a crab-shaped quiver and ↵ N a vector in the 2 q,✓ fundamental region with ↵ > 0 for all i Q . Further assume that (Q, ↵) is not i 2 0 one of the following cases:

1 (a) := 2 ↵ is integral, q =1, and (Q, ) is one of the quivers in Theorem 3.4.1 and Theorem 3.4.3;

(b) Q is ane Dynkin of type A˜0 (i.e., the Jordan quiver with one vertex and

one arrow), D˜ 4 or E˜6, E˜7, E˜8) and ↵ is a q-divisible multiple of the indivisible imaginary root of Q.

Then:

• The normalisation of q,✓(Q, ↵) is a symplectic singularity; M

• If ↵ is q-indivisible, q,✓(Q, ↵) admits a symplectic resolution; M • If ↵ is q-divisible, and ↵ is not: (c) a prime multiple of one of the quivers listed in Theorem 3.5.16.(b2) below (a framed ane Dynkin quiver with dimension vector (1,m) with m q-divisible) with ✓ =0, then (Q, ↵) does not · Mq,✓ admit a symplectic resolution (it contains an open singular factorial terminal subset).

Thus, after reducing to the fundamental region, a symplectic resolution exists if and only if the dimension vector is q-indivisible, unless we are in one of the following three open cases:

(a) twice one of the dimension vectors appearing in Theorems 3.4.1 and 3.4.3 below, which correspond to one of certain (twisted) character varieties of a sphere with 3 to 5 punctures (with rank at most 24), of a once-punctured torus (of rank 4) or a closed genus two surface (of rank 2);

(b) a q-divisible imaginary root on an extended Dynkin quiver (of type A˜0, D˜ 4, E˜6, E˜7,

or E˜8), which corresponds to either a (twisted) character variety of a closed

42 torus (type A), or a (twisted) character variety of a sphere with 3 or 4 punctures (type E or D,respectively)withparticularrankandmonodromyconditions;

(c) a prime multiple of the vector (1,`)onaframedaneDynkinquiver(again of type A˜ , D˜ , E˜ , E˜ ,orE˜ )with✓ =0,whichcorrespondsagaintoa 0 4 6 7 8 · certain character variety of a once-punctured torus or a sphere with 3 or 4 punctures.

Here when we say “correspond”, we mean precisely that, for ✓ =0,themultiplicative quiver varieties equal the given (twisted) character varieties, whereas for the genus 1case,thelatteristheopensubsetoftheformerwherethetransformations corresponding to loops are invertible. Setting ✓ =0givesapartialresolution(which 6 may be an actual one, as in case of ✓ generic and ↵q-indivisible). Remark 3.0.15. For the ordinary (untwisted) character varieties of closed genus one or two surfaces appearing in the lists for cases (b) and (a) above, a symplectic resolution exists, as we already noted in Section 3.0.1. The proof makes use of Poincar´e–Verdier duality for closed surfaces; perhaps a suitable generalisation of this for orbifolds would allow us to extend those results to the orbifold case. If so, we could remove A˜0 from case (b) and the quiver with one vertex and two loops from (a).

3.0.5 Outline of the chapter

The outline of the chapter is as follows: in Section 3.1 we recall the deﬁnition of multiplicative preprojective algebras and outline some of their algebraic properties. These are needed in the construction, via Geometric Invariant Theory (GIT), of their moduli spaces of semistable representations, following [Kin94]. In Deﬁnition

3.1.5,weintroducethesubset⌃q,✓ appearing in the theorems above. There are a number of properties of multiplicative quiver varieties with dimension vector in ⌃q,✓ that were originally formulated and proved in [CS06]inthecaseofatrivialstability condition, i.e., ✓ = 0, and that have been extended here in the more general case of ✓-semistable representations, i.e., for ✓ =0. 6 In Section 3.2 we prove the correspondence between multiplicative quiver varieties and chatacter varieties of punctured surfaces outlined above. In order to build such a correspondence we exploit [CS06, Lemma 8.2 and Theorem 1.1]. Note that

43 an instance of the correspondence between multiplicative quiver varieties and local systems on punctured surfaces already appeared in [Yam08], where a proof is given for the case of the punctured projective line. Our result applies to all genera. Thanks to this correspondence, and to the results proved in Section 3.3,weare able to extend the work of Bellamy and Schedler from closed Riemann surfaces to open ones. Another interesting aspect of this correspondence is that it could be conjecturally combined with the Non-abelian Hodge Theorem to extend the main results of the previous chapter, in the context of moduli spaces of parabolic Higgs bundles. More details on this topic are provided in Section 4.5,wherepossiblefuture research directions of the present work are discussed.

Section 3.3 contains the proof of Theorem 3.0.7.Acarefulstudyofthesingularities of multiplicative quiver varieties is carried out: ﬁrst, we detect which points in these moduli spaces are indeed singular, proving the following

Proposition 3.0.16 (Proposition 3.3.3 below). Let ↵ ⌃ . Then, the singular 2 q,✓ locus of (Q, ↵) is given by the strictly ✓-semistable points. Mq,✓

The remaining part of the section is devoted to the study of the nature of the singular locus. To this end, we use techniques from the work [BS16b]ofBellamyandSchedler to prove that, under suitable hypotheses, the singularities are symplectic. We also prove that, under certain conditions, the moduli space (Q, ↵)containsanopen Mq,✓ subset which is singular, factorial, and terminal. As a consequence, (Q, ↵)does Mq,✓ not admit a symplectic resolution. Moreover, for generic ✓,weseethattheopen subset is the entire variety. The only case left out by Theorem 3.0.7,when↵ =2 for N and p()=2,ismoresubtlethantheothers.Thecorrespondingresult 2 q,✓ in the context of ordinary quiver varieties, treated in [BS16b], is based on the study of the local structure of such varieties. In our case, such a tool is still not available, but will hopefully be the object of future research.

In Section 3.4,namelyinTheorems3.4.1 and 3.4.3, we combinatorially classify all the the pairs (Q, ↵), formed by a crab-shaped quiver and a corresponding dimension 1 vector in the fundamental region such that (p(gcd(↵) ↵), gcd(↵)) = (2, 2). This is relevant as these are the cases expected, for generic ✓,toadmit“O’Grady”-type resolutions (i.e., by blowing up the singular locus). This is also important since, by Theorem 3.0.13, it allows us to recognise whether a dimension vector in the fundamental region is expected to admit a symplectic resolution or not.

44 In Section 3.5 we face the problem of existence of symplectic resolutions of multiplicative quiver varieties for general dimension vectors. In order to do so, we follow the approach of Bellamy and Schedler. In particular, we prove that a multiplicative quiver variety has a canonical decomposition into natural factors, see Theorem 3.5.17.ThiscanbeviewedasamultiplicativeanalogueofCrawley-Boevey’sdecom- position [Cra02], and we follow his proof, obtaining some more factors due to the unavailability of (*) and some local structure results. Our result makes it possible to solve the problem by understanding it only at the level of such indecomposable factors, which are multiplicative quiver varieties with particular dimension vectors. We then extend the GIT construction of symplectic resolutions by varying ✓ to dimension vectors not in ⌃ (Theorem 3.5.22); this includes multiplicative analogues of framed quiver varieties such as Hilbert schemes of C2 and of hyperk¨ahleral- most locally Euclidean spaces. In Theorem 3.5.26 we make use of our canonical decomposition and, modulo some cases for which the question remains still open, we classify all multiplicative quiver varieties with arbitrary dimension vector that admit a symplectic resolution. As an application of this result, by restricting to crab-shaped quivers, we give an explicit classiﬁcation of the character varieties of punctured surfaces admitting symplectic resolutions (Corollary 3.5.27), combining Theorem 3.5.26 and the results of Section 3.2.

3.1 Multiplicative quiver varieties

In this section we give the deﬁnition of multiplicative quiver varieties following [CS06]andrecallsomebasicpropertiesofsuchmodulispaceswhichwillbeuseful in the arguments of the proof of our main theorems. In addition to these known results, we prove a new one, concerning the normality of the aforementioned varieties, provided that certain codimension estimates hold.

3.1.1 Multiplicative preprojective algebras

We now deﬁne the quiver algebras whose moduli of representations are the varieties of interest for this chapter. To this purpose, let Q be a quiver, ﬁxed once and for all in this section. First, recall that, for a vector CQ0 ,thedeformed preprojective 2 algebra ⇧(Q)isthequotientofthepathalgebraCQ of the doubled quiver Q by

45 the relation

[x, x⇤]= iei,

x Q1 i Q0 X2 X2 where x⇤ denotes the dual loop to x in Q1; it is well-known that Nakajima quiver varieties can be interpreted as moduli spaces of (✓-semistable) representations of such algebras. As one might expect, the deﬁning relation for multiplicative preprojective

Q0 algebras is a multiplicative analogue of the above equation: choose q (C⇥) and 2 define A(Q)tobetheuniversallocalisationofthepathalgebraCQ such that 1+xx⇤ and 1 + x⇤x are invertible, for x Q .Then,following[CS06, Definition 1.2], the 2 1 multiplicative preprojective algebra ⇤q(Q)isdefinedasthequotientofA(Q)bythe relation < "(x) (1 + xx⇤) = qiei, x Q i Q0 Y2 1 X2 where "(x)equals1ifx Q and 1otherwiseandtheproductisorderedbyan 2 1 arbitrary choice of ordering “<”onQ1.Itisknown,by[CS06,Theorem1.4]that, up to isomorphism, ⇤q(Q)doesnotdependontheorientationofthequiverorthe chosen ordering on Q1. Remark 3.1.1. When the quiver Q is clear from the context, we will use the shortened notation ⇤q in place of ⇤q(Q).

Analogously to the additive case mentioned above, representations of ⇤q(Q)are representations of the underlying quiver Q, (Vi)i Q , (j)j Q ,satisfyingtheaddi- { 2 0 2 1 } tional relations:

Id + ⇤ is an invertible endomorphism of V for all j Q Vh(j) j j h(j) 2 1

"(j) (Id + ⇤) = q Id for all i Q , Vh(j) j j i Vi 2 0 j Q ,h(j)=i 2 Y1 where, for an edge j Q , ⇤ denotes the linear map , j⇤ being the dual edge of 2 1 j j⇤ j in the doubled quiver.

For a positive vector ↵ NQ0 ,wedenotebyRep(⇤q,↵)thesetofrepresentationsof 2 q ↵i ⇤ with Vi = C for all i.Thiscanbegivenanobviousaneschemestructurevia the subset of matrices satisfying the obvious polynomial equations. We will work below with the reduced subvariety of this ane scheme. Remark 3.1.2. By taking determinants of the deﬁning relation for the multiplicative

46 preprojective algebra, one can easily see that if ⇤q has a representation of dimension vector ↵,thenq↵ = q↵i =1,which,thus,isanecessaryconditiontobe i Q0 i 2 satisﬁed in order to haveQ non-empty moduli spaces.

The following results, which will be used in the next sections, are proved in [CS06]. It is worth pointing out that, even though we work over C,thesestatementshold true over an arbitrary ﬁeld K.

Proposition 3.1.3. If X and Y are ﬁnite-dimensional representations of ⇤q, then

1 dim Ext q (X, Y ) = dim Hom q (X, Y ) + Hom q (Y,X) (dimX, dimY ) ⇤ ⇤ ⇤

The following result concerns the geometry of the space Rep(⇤q,↵)ofrepresentations

q Q0 of the algebra ⇤ , when a dimension vector ↵ N is ﬁxed. Deﬁne g↵ as g↵ := 2 1+ ↵2. i Q0 i 2 P Proposition 3.1.4. Rep(⇤q,↵) is an ane variety, and every irreducible component has dimension at least g +2p(↵). The subset T Rep(⇤q,↵) of representations X ↵ ⇢ with trivial endomorphism algebra, End(X)=C, is open and, if non-empty, smooth of dimension g↵ +2p(↵).

In order to state the next result, we introduce a subset of the set of all possible dimension vectors for representations of ⇤q,whichisthemultiplicativeanalogueof the set ⌃ introduced by Crawley-Boevey in [Cra02]andextensivelyusedin[BS16b]

(generalised to ⌃,✓)toprovetheexistenceofacanonical decomposition for quiver varieties with general stability conditions. In the next deﬁnition, one should keep in mind that ✓ ZQ0 will play the role of a stability condition for a GIT quotient. 2

Q0 Q0 Q0 ↵ Deﬁnition 3.1.5. Fix q (C⇥) and ✓ Z and set Nq,✓ := ↵ N q = 2 2 { 2 | 1,↵ ✓ =0 .LerR+ denote the set of positive roots of Q and deﬁne R+ := R+ N . · } q,✓ \ q,✓ Then,

r + (i) ⌃q,✓ := ↵ Rq,✓ p(↵) > p for any decomposition ( 2 i=1 X ↵ = (1) + + (r) with r 2,(i) R+ . ··· 2 q,✓

When ✓ =0,weshallusetheshortenednotation⌃q in place of ⌃q,0.

47 Deﬁnition 3.1.6. A dimension vector ↵ N is said to be q-indivisible if 1 ↵/ 2 q,✓ m 2 N for all m 2. Equivalently, for ↵ = m and indivisible, then q is a primitive q,✓ m-th root of unity.

The following property is part of the motivation for the deﬁnition of ⌃q (which we generalise in Proposition 3.1.22 to the ✓ =0case): 6 Proposition 3.1.7. [CS06, Theorem 1.11] If ↵ ⌃ , then, if non-empty, Rep(⇤q,↵) 2 q is a complete intersection, equidimensional of dimension g↵ +2p(↵), and the set of simple representations is open and dense.

3.1.2 Reflection functors for ⇤q(Q)

As in the additive case, one can define reflection functors for the multiplicative preprojective algebra ⇤q(Q): let v a loopfree vertex in Q and define

Q0 Q0 (ev,ew) uv :(C⇥) (C⇥) ,uv(q)w = qv qw. !

It is easy to see that the map uv satisﬁes the following identity:

↵ sv(↵) (uv(q)) = q ,

where sv is the reflection map defined in Subsection 1.3.Themainresultconcerning such maps is analogous to the properties of reflections functors for ⇧(Q).

Proposition 3.1.8. [CS06, Theorem 1.7] If v is a loopfree vertex and q =1, then v 6 q there is an equivalence of categories Fq from the category of representations of ⇤ to the category of representations of ⇤uv(q), acting on dimension vectors through the reﬂection sv. The inverse equivalence is given by Fuv(q).

We will need also reﬂections on ✓.Deﬁne

Q0 Q0 rv : Z Z ,rv(✓)w = ✓w (ev,ew)✓v. !

Definition 3.1.9. The map (q, ✓, ↵) (u (q),r (✓),s (↵)), is called a reflection. If 7! v v v ✓ =0orq =1,itiscalledanadmissible reflection. v 6 v 6

48 We will explain below isomorphisms of multiplicative quiver varieties, due to Ya- makawa, which are closely related to the above equivalence.

3.1.3 Moduli of representations of ⇤q(Q)

We shall now outline the construction of the varieties of interest for the present chapter. As mentioned above, the general definition involves a stability condition ✓ ZQ0 ,whichwefixfortherestofthissection. 2 Given a dimension vector ↵,fromtheseminalworkofKing,[Kin94], one can introduce the following definition of ✓-(semi)stability on Rep(Q, ↵).

Deﬁnition 3.1.10. Let M be a ﬁnite-dimensional representation of ⇤q such that dimM ✓ =0. M is said to be ✓-semistable if, for any sub-module N M · ⇢

✓ dimN 0. · 

M is said to be ✓-stable if the strict inequality holds for any proper submodule. Finally, M is said to be ✓-polystable if it is a direct sum of ✓-stable representations. ✓ s ✓ ss Given a set (or scheme) X of representations, let X and X denote the ✓-stable and ✓-semistable loci, respectively.

We will also need the corresponding partial ordering:

Deﬁnition 3.1.11. We say that ✓0 ✓ if every ✓0-semistable representation is also ✓-semistable.

Remark 3.1.12. By [Kin94, Proposition 3.1], one has that the above deﬁnition of stability coincides with the usual one coming from GIT: indeed, consider the character

✓i ✓ :GL(↵) C⇥, (gi)i Q0 (det gi) . ! 2 7! i Q0 Y2 It defines a linearisation on the trivial line bundle Rep(Q, ↵) C of the action of ⇥ GL(↵)onRep(Q, ↵); thus, one can define the notion of ✓-(semi)stability àla Mum- ford, [MFK02]. The aforementioned result of King proves that M is ✓-(semi)stable if and only if it is ✓-(semi)stable.

Using the notion above one can construct the moduli space of (semistable) representations of ⇤q of dimension ↵ as follows (see [Yam08, §2], for the details): deﬁne

49 ✓ ss Rep (Q, ↵) Rep(Q, ↵)tobethesubsetof✓-semistable representations and set ⇢ Rep(Q, ↵)tobe

Rep(Q, ↵)= (j)j Q Rep(Q, ↵) det(1 + jj⇤) =0,j Q1 . { 2 1 2 | 6 2 }

One can then consider the map

:Rep(Q, ↵) GL(↵), ! deﬁned by the formula

< "(j) ((j)j Q )= (1 + jj⇤) . 2 1 j Q Y2 1

Q0 Let us identify C⇥ also with the scalar matrices in GL(↵i), and hence (C⇥) also

Q0 q with a subset of GL(↵). Fixing q (C⇥) ,onehasthatRep(⇤(Q),↵) is the 2 1 set-theoretic preimage (q). Thus, one can give the following Deﬁnition 3.1.13. The multiplicative quiver variety (Q, ↵) is the GIT quotient Mq,✓

✓ ss 1 (Q, ↵):=(Rep (Q, ↵) (q)) // GL(↵). Mq,✓ \

Remark 3.1.14. The reason for the terminology in the previous definition is apparent: indeed, the equations defining the multiplicative preprojective relation are modifications of the ones used to define the usual deformed preprojective algebras, whose moduli of (semistable) representations are indeed Nakajima quiver varieties.

It is worth recalling a fundamental result of King, which gives a moduli-theoretic interpretation—in the sense of (representable) moduli functors—to (Q, ↵). Mq,✓ Theorem 3.1.15. [Kin94, Propositions 3.1 and 3.2] Assume ✓ ZQ0 . Then, 2 (Q, ↵) is a coarse moduli space for families of ✓-semistable representations up Mq,✓ to S-equivalence.

Here two ✓-semistable representations are S-equivalent if and only if they have the same composition factors into ✓-stable representations (i.e., they have ﬁltrations whose subquotients are isomorphic ✓-stable representations). This means that every point in (Q, ↵)hasauniquerepresentativewhichis✓-polystable, up to Mq,✓ isomorphism.

50 Precisely as in [BS16b,Lemma2.4],wehavethefollowinginstanceofthewell-known principle of GIT:

Lemma 3.1.16. [BS16b, Lemma 2.4] Let ↵ N be such that (Q, ↵) = 2 q,✓ Mq,✓ 6 . Take ✓0 ✓. Then we have a projective Poisson morphism (Q, ↵) ; Mq,✓0 ! 1 ✓ ss 1 ✓ ss (Q, ↵) induced by the inclusion (q) 0 (q) . Mq,✓ ✓

We caution that this morphism need not be surjective (and indeed the source could be empty when the target is not). However, in many cases, as we will see, it produces a symplectic resolution.

3.1.4 Reflection isomorphisms

There is a multiplicative analogue of the Lusztig-Ma↵ei-Nakajima reflection isomorphisms of quiver varieties (see in particular [Maf02,Theorem26]),duetoYa- makawa, which makes use of the reflection functors Fq.Letusextendthedefinition

Q0 of q,✓(Q, ↵)to↵ Z by setting it to be empty in the case that ↵i < 0forsome M 2 i.

Theorem 3.1.17. [Yam08, Theorem 5.1] An admissible reﬂection

(q, ✓, ↵) (u (q),r (✓),s (↵)) 7! v v v induces an isomorphism of multiplicative quiver varieties,

(Q, ↵) = (Q, s (↵)) Mq,✓ ⇠ Muv(q),rv(✓) v

3.1.5 Poisson structure on (Q, ↵) Mq,✓

In order to construct a Poisson structure on (Q, ↵), we shall use the theory of Mq,✓ quasi-Hamiltonian reductions, ﬁrst developed in [AMM98]forthecaseofrealman- ifolds, and then treated by Boalch, [Boa07], and Van den Bergh [Van08a; Van08b] in the holomorphic and algebraic settings. To this end, note that the map de- ﬁned above is a group valued moment map for the quasi-Hamiltonian action of

51 GL(↵)onRep(Q, ↵). Thus, the variety (Q, ↵)canbeconsideredasthequasi- Mq,✓ Hamiltonian reduction of Rep(Q, ↵)modulotheactionofGL(↵). From the properties of such a reduction, we obtain that (Q, ↵) is a Poisson variety. Moreover, Mq,✓ deﬁning s ✓ s 1 (Q, ↵):=(Rep (Q, ↵) (q))/ GL(↵), Mq,✓ \ ✓ s ✓ ss where Rep (Q, ↵) Rep (Q, ↵)denotesthe✓-stable locus, one has the follow- ⇢ ing result, which will be crucial in proving that (Q, ↵)isasymplecticsingu- Mq,✓ larity. Note that, in the above deﬁnition the quotient is the usual orbit space, if we replace GL(↵)byPGL(↵)=GL(↵)/C⇥,asapointinthestablelocushastrivial stabiliser group under PGL(↵).

Proposition 3.1.18. [Yam08, Theorem 3.4] s (Q, ↵), if non-empty, is an equidi- Mq,✓ mensional algebraic symplectic manifold and its dimension is 2p(↵).

3.1.6 Stratification by representation type

An important result proved in [CS06, §7] concerns a natural stratiﬁcation of the ane quotient (Q, ↵) = Rep(⇤q,↵)// GL(↵), which parametrises semisimple Mq,0 representations of the algebra ⇤q.Thisstratiﬁcationanditsgeneralisation,proved below, to the case of ✓-semistable representations are important in order to understand the singular locus of (Q, ↵). Mq,✓ ✓ ss q Consider M Rep (⇤ ,↵). Replace it by the unique ✓-polystable representation 2 which is S-equivalent to it (see the discussion after Theorem 3.1.15). M is then said (1) (r) to be of representation type ⌧ =(k1, ; ...; kr, ) if it can be decomposed into the direct sum M = M k1 M kr ,whereM is a ✓-stable representation of ⇤q ⇠ 1 ··· r i of dimension vector i, i =1,...,r,andMi Mj for i = j. 6 q ⌧ Proposition 3.1.19. If ⌧ is a representation type for ⇤ , then the set Cq,✓(Q, ↵) of ✓-semistable representations of type ⌧ is a locally closed subset of (Q, ↵), which, Mq,✓ if non-empty, has dimension r 2p((i)). (Q, ↵) is the disjoint union of the i=1 Mq,✓ ⌧ strata Cq,✓(Q, ↵), where ⌧ runsP over the set of representation types that can occur for ⇤q.

Proof. First, note that the case when ✓ =0istreatedin[CS06]andprovedin Lemma 7.1 therein. For the case when ✓ =0weusethesamearguments.Indeed, 6

52 the fact that (Q, ↵)isadisjointunionofsubsetsofaﬁxedrepresentationtype Mq,✓ is immediate from the fact that the decomposition of a ✓-polystable module into ✓- stable modules is unique. This, in turn, holds because, for ✓-stable modules M and N, we have dim Hom(M,N) 1, with equality if and only if M and N are isomor-  ⌧ phic. Moreover, to prove that each Cq,✓(Q, ↵) is locally closed and of the dimension prescribed by the lemma, one can adapt the proof [Cra01,Theorem1.3]:indeed, those arguments can be repeated in this case as well, replacing Rep(Q, ↵)with ✓ ss 1 1 Rep (Q, ↵), µ↵ ()with (q), the word ‘(semi-)simple’ with ‘✓-(semi)stable’ in the proof, and noting that everything goes through in the same way because ✓ ss Rep (Q, ↵)isopeninRep(Q, ↵). The only di↵erence is that, in this case, we ✓ ss do not claim irreducibility, since Rep (Q, )isnotknowntobeirreducible,even when ⌃ . 2 q,✓

✓ ss q We will need also the following property of Rep (⇤ (Q),↵):

✓ ss q Lemma 3.1.20. Every irreducible component of Rep (⇤ (Q),↵) has dimension at least g↵ +2p(↵) and the set of ✓-stable representations form an open subset of q Rep(⇤ (Q),↵) which, if non-empty, is smooth of dimension g↵ +2p(↵).

Proof. For the ﬁrst part, Lemma 6.2 in [Cra03]provesthestatementinthecase ✓ ss q when ✓ = 0, of which the above result is a consequence since Rep (⇤ (Q),↵)is an open subset of Rep(⇤q(Q),↵): indeed, every irreducible component of the former variety is contained in only one irreducible component of the latter and, hence, the dimension estimate holds. For the second part, one just needs to note that, if X is a ✓-stable representation, then End(X)=C and, hence, by [CS06,Theorem1.10] deﬁnes a smooth point of Rep(⇤q(Q),↵), which implies that it is a smooth point of ✓ ss q Rep (⇤ (Q),↵).

For the proof of the following propositions, apply the strategy carried out in [Cra03, §6, 7] and [CS06, §7]: the only change is that, in the deﬁnition of representation of top-type,onehastoreplacetheword‘simple’withtheword‘✓-stable’ and use Proposition 3.1.19 instead of [CS06,Lemma7.1]andLemma3.1.20 instead of [CS06, Theorem 1.1].

✓ ss q Proposition 3.1.21. The inverse image in Rep (⇤ (Q),↵) of the stratum of (1) (r) representations of type ⌧ =(k1, ; ,...; kr, ) has dimension at most g↵ +p(↵)+ r (l) l=1 p( ). P 53 ✓ ss q Proposition 3.1.22. Let ↵ ⌃ . Then, if non-empty, Rep (⇤ (Q),↵) is a 2 q,✓ ✓ ss complete intersection in Rep (Q, ↵), equidimensional of dimension g↵ +2p(↵). ✓ s q ✓ ss q The locus of ✓-stable representations Rep (⇤ (Q),↵) is dense inside Rep (⇤ (Q),↵).

Remark 3.1.23. Note that a consequence of the above proposition is that, if ⇡ : ✓ ss q s Rep (⇤ (Q),↵) (Q, ↵)istheprojectionmap,theimage (Q, ↵):= !Mq,✓ Mq,✓ ✓ s q ⇡ Rep (⇤ (Q),↵)) is dense in the moduli space (Q, ↵). As a corollary of Mq,✓ this and Proposition 3.1.18 (or Proposition 3.1.19), one has that every component of (Q, ↵)hasdimension2p(↵). Mq,✓ Ausefulcorollaryofthepropositionisthefollowingcriterionforbirationalityofthe maps (Q, ) (Q, ). Together with Lemma 3.1.16,thisexplainsthat Mq,✓0 !Mq,✓ these maps will be resolutions of singularities when the source is smooth.

Corollary 3.1.24. Let ↵ ⌃ be such that (Q, ↵) = . Take ✓0 ✓. Then, 2 q,✓ Mq,✓ 6 ; the morphism (Q, ) (Q, ) is birational. Mq,✓0 !Mq,✓

✓ ss q ✓ ss q Proof. By Deﬁnition 3.1.11,Rep0 (⇤ ,↵)isasubsetofRep (⇤ ,↵), and it is open. Since the ✓-stable locus is dense in the ✓-semistable locus, this implies that the ✓-stable, ✓0-semistable locus is dense in the ✓0-semistable locus. So the ✓-stable locus is dense in (Q, ). Thus, birationality of the desired map is ensured Mq,✓0 ✓ stable by the fact that the ✓-stable locus (Q, ), which is open and dense inside Mq,✓0 (Q, ), is mapped isomorphically to the ✓-stable locus of (Q, ). Mq,✓0 Mq,✓

Using the above results, one can derive an important geometric property of the moduli space (Q, ↵). For reasons which are clear in the proof of the propo- Mq,✓ sition, we assume that a certain codimension estimate holds. As usual, let ⇡ : ✓ ss q Rep (⇤ ,↵) (Q, ↵)denotethequotientmap. !Mq,✓ Lemma 3.1.25. Assume ↵ ⌃ and let ⌧ be a stratum. The following inequality 2 q,✓ holds true:

1 ⌧ 1 ⌧ ✓ ss codimRep (⇤q,↵)(⇡ (Cq,✓(Q, ↵))) codim q,✓(Q,↵)(Cq,✓(Q, ↵)). 2 M

Proof. By Proposition 3.1.22,onehasthat

1 ⌧ 1 ⌧ codim(⇡ (C (Q, ↵))) = g +2p(↵) dim ⇡ (C (Q, ↵)). q,✓ ↵ q,✓

54 Moreover, from Proposition 3.1.21 it follows that

r r 1 ⌧ (l) (l) g +2p(↵) dim ⇡ (C (Q, ↵)) g +2p(↵) g p(↵) p( )=p(↵) p( ). ↵ q,✓ ↵ ↵ Xl=1 Xl=1 On the other hand, by Proposition 3.1.19,onehasthat

r 1 p(↵) p((l))= dim (Q, ↵) dim C⌧ (Q, ↵) , 2 Mq,✓ q,✓ l=1 X which, combined with the above inequality, leads to the desired statement.

By taking the minimum of these codimensions, we immediately conclude:

Corollary 3.1.26. Let Z denote the complement inside (Q, ↵) of the set of Mq,✓ ✓-stable representations s (Q, ↵), i.e., Z is the union of all the non-open strata Mq,✓ of (Q, ↵). Then, the following inequality holds: Mq,✓

1 1 ⌧ codim ⇡ (Z) min codim Cq,✓(Q, ↵). 2 ⌧=(1,↵) 6 Proposition 3.1.27. Consider ↵ ⌃ and assume that all strata in the non- 2 q,✓ empty multiplicative quiver variety (Q, ↵) have codimension at least 4, i.e., Mq,✓ assume that ⌧ min dim q,✓(Q, ↵) dim Cq,✓(Q, ↵)) 4. ⌧=(1,↵) M 6 Then, the variety (Q, ↵) is normal. Mq,✓

Proof. The arguments to prove the above statement are analogous to the ones used in [BS16b,Proposition8.3].Inparticular,weshalluseacriterionprovedbyCrawley- Boevey, [Cra03,Corollary7.2].Wefirstdealwiththecasewhen✓ =0andthen explain how to adapt the arguments for general ✓. When ✓ =0, (Q, ↵)is Mq,0 the categorical quotient Rep(⇤q,↵)// GL(↵)ofananevarietymoduloareductive q group. Thus, we only need to show that Rep(⇤ ,↵)satisfiesSerre’scondition(S2) and that certain codimension estimates hold true. The first condition is ensured by the fact that, by Proposition 3.1.7,Rep(⇤q,↵)isacompleteintersectionand, hence, Cohen-Macaulay, which indeed implies condition (S2). Now, denote by S the open subset S (Q, ↵)ofsimplerepresentations,whichisnon-emptybyour ⇢Mq,0 assumption. S is contained in the smooth locus and hence is normal. Moreover, let Z

55 denote its complement in (Q, ↵)anddenotewith⇡ :Rep(⇤q,↵) (Q, ↵) Mq,0 !Mq,0 the quotient map; then, by Corollary 3.1.26, one has

q 1 1 ⌧ dim Rep(⇤ ,↵) dim ⇡ (Z) min (dim q,0(Q, ↵) dim Cq,0(Q, ↵)), 2 ⌧=(1,↵) M 6 and the right hand side is greater or equal than two by assumption. Thus, all the hypotheses of [Cra03,Corollary7.2]aresatisfiedandwecanconcludethat (Q, ↵)isnormal.Forthecasewhen✓ =0,keepinginmindthatnormalityisa Mq,0 6 local property, we fix a point x (Q, ↵)andaimatprovingnormalityatx.This 2Mq,✓ is achieved by choosing an open neighbourhood V of x such that the restriction to 1 1 ⇡ (V )oftheprojectionmorphism⇡ (V ) V is an ane quotient (note that this ! can be done thanks to the properties of the GIT construction). One can now repeat the same arguments as for the ✓ =0case,notingthat,byProposition3.1.19,the estimates above hold true also in this more general setting: being Cohen-Macaulay 1 is a local statement and thus the previous part of the proof ensures that ⇡ (V ), q which is open in Rep(⇤ ,↵), satisfies such a property. Moreover, defining S✓ to be the subset of V of ✓-stable representations, then one may proceed as in the first part of the proof to obtain the desired conclusion.

Remark 3.1.28. In the next sections, we will examine some cases in which the tech- nical assumption in the previous result is satisﬁed, thus giving explicit examples of when (Q, ↵)isnormal. Mq,✓

3.2 Punctured character varieties as multiplicative quiver varieties

In this section, we explain how it is possible to realise certain character varieties as particular examples of multiplicative quiver varieties by considering quivers of special type, the so-called crab-shaped quivers. Such character varieties parametrise representations of the fundamental group of a compact Riemann surface with a ﬁnite number of punctures, where the monodromies at closed loops around such punctures are ﬁxed to lie in (the closure of) certain conjugacy classes. As mentioned in the introduction, these varieties are central in the ground-breaking work of Hausel, Lettelier and Rodriguez-Villegas, who studied ways to compute their Betti numbers using arithmetic methods, see [HLR11; HLR13]. We use the language of quiver

56 Riemann surfaces introduced by Crawley-Boevey in [Cra13]. Moreover, in what follows, we shall adopt the term punctured character variety to refer to the character variety of a Riemann surface with punctures.

Fix a connected compact Riemann surface X of genus g 0, let S = p ,...,p { 1 k}⇢ X be the set of punctures and ﬁx a tuple =( ,..., )ofconjugacyclasses C C1 Ck i GLn(C),i =1,...,k.Recallthatthefundamentalgroup⇡1(X S) of the C ⇢ \ punctured surface X S admits the following presentation: \

⇡ (X S)= a ,...,a ,b ,...,b ,c ,...,c [a ,b ] ... [a ,b ]c ... c =1, 1 \ h 1 g 1 g 1 k | 1 1 · · g g 1 · · k i

1 1 where [a, b]=aba b denotes the commutator. Note that the generators c1,...,ck represent homotopy classes of closed loops around the punctures, in the same free homotopy classes as small counter-clockwise loops around the punctures. Thus, a representation of ⇡ (X S)whosemonodromiesaboutthepuncturesareinthecon- 1 \ jugacy classes is given by a tuple of matrices (A ,...,A ,B ,...,B ,C ,...,C ) Ci 1 g 1 g 1 k 2 2g GLn(C) 1 k,satisfyingtherelation ⇥C ⇥···⇥C

g k

[Ai,Bi] Cj = I. i=1 j=1 Y Y Given the above, from the fact that isomorphic representations correspond to con- jugate matrices, one has that the character variety X(g, k, )associatedtothepair C (X, S) and monodromies lying in the conjugacy classes ﬁxed above is isomorphic to the ane quotient

2g X(g, k, ):= (A1,...,Ag,B1,...,Bg,C1,...,Ck) GLn(C) 1 k C { 2 ⇥ C ⇥···⇥C | g k [Ai,Bi] Cj = I // G L n(C). } i=1 j=1 Y Y Remark 3.2.1. Note that the closures are ane varieties and hence the quotient Ci is indeed that of an ane variety by an algebraic group.

We shall now explain how to realise the variety X(g, k, )asanopensubsetofa C multiplicative quiver variety, using an equivalence of categories proved in [Cra13]. As mentioned above, such a correspondence holds when one considers the so-called crab-shaped quivers, i.e., quivers such that there exists a vertex v satisfying the

57 following condition: the set of arrows is formed by loops at v and a ﬁnite number of legs ending at v. See Figure 3.1.Astar-shaped quiver is a crab-shaped quiver with no loops.

••

• •••

•

Figure 3.1: A crab-shaped quiver with 2 loops and 3 legs, of length 2, 3 and 1 respectively.

For the remainder of this section, the following notation will be used: g,forthe number of loops around the central vertex; k for the number of legs and li,for i =1,...,k,forthelengthofthei-th leg. As we shall see, g contains the information regarding the genus of the surface, while the integers k and li encode information about the (prescribed) conjugacy classes of the monodromies of the loops around the punctures.

Deﬁnition 3.2.2. [Cra13, §2] A Riemann surface quiver isaquiverwhoseset of vertices has the structure of a Riemann surface X with ﬁnitely many connected components. is said to be compact if X is compact. A point p X is called 2 marked if it is a head or a tail of an arrow of .

Deﬁnition 3.2.3. Given a Riemann surface quiver , the component quiver [] of , is the quiver whose set of vertices is the set of connected components of and arrows given by [a]:[p] [q]foranyarrowa : p q,wherep and q are points of ! ! X and [p]denotestheconnectedcomponentofX containing p.

Remark 3.2.4. Although, by definition, there are in general infinitely many vertices, we will consider (Riemann surface) quivers with finitely many arrows.

Following closely [Cra13, §5, §8], starting from a Riemann surface quiver , it is pos- q sible to deﬁne two categories of representations, Rep(⇡()) and Rep ⇤ ([]), whose equivalence is the key point to proving the correspondence between multiplicative quiver varieties and punctured character varieties.

Fix a quiver Riemann surface and let Xi i I the set of connected components { } 2

58 of the underlying Riemann surface X.Foreachi I let D be the set of marked 2 i D points of contained in Xi.Moreover,letD = iDi:ﬁx (C⇥) , bi Xi Di [ 2 2 \ and, for each p D ﬁx a loop l ⇡ (X D ,b)aroundp. 2 i p 2 1 i \ i i

Rep⇡() is deﬁned to be the category whose objects are given by collections

(V ,⇢,⇢ ,⇢⇤)consistingofrepresentations⇢ : ⇡ (X D ,b) GL(V ), for i I i i a a i 1 i \ i i ! i 2 and linear maps ⇢ : V V and ⇢⇤ : V V for each arrow a : p q in , where a i ! j a j ! i ! Xi =[p]andXj =[q], satisfying

1 1 p ⇢i(`p) =1Vi + ⇢a⇤⇢a and q⇢j(`q)=1Vj + ⇢a⇢a⇤ and whose morphisms are the natural ones. Consider the component quiver [] and deﬁne Q to be the quiver obtained from

[] by adjoining gi loops at each vertex i,wheregi is the genus of Xi.Moreover, I q deﬁne q (C⇥) by qi = p D p.WedeﬁneRep⇤([])0 to be the category of 2 2 i q representations of the multiplicativeQ preprojective algebra ⇤ (Q)inwhichthelinear maps representing the added loops are invertible.

Lemma 3.2.5. [Cra13, Proposition 2] There is an equivalence of categories

q Rep ⇡() Rep ⇤ ([])0. '

This induces a GL(↵)-equivariant isomorphism of ane algebraic varieties,

q Rep (⇡(),↵) ⇠ Rep(⇤ ([])0,↵), !

↵i deﬁned as the collections of representations with Vi = C for all i.

Proof. The ﬁrst statement is precisely [Cra13,Proposition2].Forthesecond,both q Rep(⇡(),↵)andRep(⇤([])0,↵)areacteduponbythegroupGL(↵)andthe above equivalence of categories implies that there is a GL(↵)-equivariant bijection q as desired. Moreover, Rep(⇡(),↵)andRep(⇤([]0,↵)areeasilyseentobeane algebraic varieties, deﬁned as tuples of matrices satisfying certain polynomial relations, with certain polynomials inverted. To see that the above map is a GL(↵)- equivariant algebra isomorphism, observe that the proof of [Cra13,Proposition2] uses explicit invertible polynomial formulae. Therefore, the map descends to an isomorphism between the ane quotients for the action of GL(↵).

59 In order to explain how the above equivalence of categories implies the correspondence between character varieties and preprojective algebras, we shall explain how it is possible to encode the datum of a number of conjugacy classes into a star-shaped quiver. We follow [CS06, §8] and [Cra04, §2]: ﬁx k conjugacy classes 1,..., k in C C GLn(C), for k 1. We can encode the datum of such conjugacy classes in a combinatorial object as follows: take A and let w 1bethedegreeofitsminimal i 2Ci i polynomial, for i =1,...,k;chooseelements⇠ij C⇥,1 i k, 1 j wi,such 2     that (A ⇠ I) (A ⇠ I)=0. i i1 ····· i iwi

By construction, one may take ⇠ij as the roots of the minimal polynomial of Ai,for i =1,...,k.Theclosureoftheconjugacyclass is then determined by the ranks Ci of the partial products

↵ =rank(A ⇠ I) (A ⇠ I), ij i i1 ····· i ij for A and 1 j w 1. In addition, if we set ↵ = n,wegetadimension i 2Ci   i 0 vector ↵ for the following quiver Qw

[1, 1] [1, 2] [1,w 1] 1 ⌦ s s qqq s ⌦ [2,1] [2,2] [2,w2 1] ⌦ ⌦ ⇡ 0 s s qqq s J] s J q q q J q q q J q q q [k, 1] [k, 2] [k, w 1] s s qqq s k

Now, for every i 1,...,k ,lett be an integer 2anddeﬁneaRiemannsurface 2{ } i quiver as follows: its underlying Riemann surface X is given by the disjoint union

1 X = X0 Pi,j, t i 1,...,k ,j 1,...,t 2{ }G2{ i} where X0 is an arbitrary closed Riemann surface of genus g (the choice does not 1 1 matter), and Pi,j is simply a copy of P for the index (i, j). For each pair of indices

60 1 (i, j)ﬁxapointpi,j Pi,j,and,fori =1,...,k, pi X0.DeﬁneD as before to be 2 2

D = p p . { i,j}[{ l}

The arrows of are listed as follows:

• ai,0 : pi,1 pi,fori =1,...,k; !

• ai,j : pi,j+1 pi,j,fori =1,...,k, j =1,...,ti 1. !

Figure 3.2: An example of a Riemann surface quiver associated with a tuple of conjugacy classes.

Unfolding the deﬁnition for the objects of the category Rep⇡(), one has that for such a Riemann surface quiver these are representations

⇢ : ⇡ (X p ,...,p ) GL(V ), 1 0 \{ 1 k} ! and linear maps ⇢ : V V, ⇢ : V V i,0 i,1 ! i,j i,j+1 ! i,j and

⇢⇤ : V V ,⇢⇤ : V V , i,0 ! i,1 i,j i,j ! i,j+1 for i =1...,k and j =1,...,t 1, such that, if l ⇡ (X p ,...,p )isthe i i 2 1 0 \{ 1 k} loop around pi, i =1,...,k,thelinearautomorphism⇢(li)satisﬁesthecondition

i⇢(li)=1V + ⇢i,0⇢i,⇤0

and the linear maps ⇢i,j and ⇢i,j⇤ satisfy the equations:

1 i,j +11Vi,j+1 =1Vi,j+1 + ⇢i,j⇤ ⇢i,j,

i,l1Vi,l =1Vi,l + ⇢i,l⇢i,l⇤ ,

61 for i =1,...,k, j =1,...,t 1andl =1,...,t,which,settingj = l 1and i i summing the equations involving operators on the same space Vi,l,canberewritten as 1 1 ⇢(li)=i 1V + i ⇢i,0⇢i,⇤0, 1 ⇢i,j⇤ 1⇢i,j 1 ⇢i,j⇢i,j⇤ =(i,j i,j)1Vi,j , 1 =1 + ⇢ ⇢⇤ , i,ti Vi,ti Vi,ti i,ti i,ti for i =1,...,k and j =1,...,t 1. Now, we specialise to the case t = w 1and i i i assume that dim Vi,j = ↵i,j and dim V = n,wherewi and ↵i,j are deﬁned as before.

Through some simple algebraic computations, it is possible to see that, given ⇠i,j as before, it is possible to ﬁnd corresponding i,j,deﬁnedas

1 ⇠i,j 0 = k ,i,j = ⇠ , i=1 ⇠i,1 i,j+1 Q such that the above sets of equations can be rewritten in terms of linear operators and ,fori =1,...,k and j = i, . . . , w 1, i,j i,j i

i1 i2 i3 i,wi 1 V Vi1 Vi2 ... Vi,w 1 i i1 i2 i3 i,wi 1 satisfying

⇢(l ) = ⇠ 1 i i1 i1 i1 V =(⇠ ⇠ )1 (1 j

i,wi 1 i,wi 1 =(⇠i,wi ⇠i,wi 1)1Vi,w 1 , i which, by [Cra04, Theorem 2.1], implies that ⇢(li) lies in the closure of the conjugacy class , i =1,...,k.Infact,thistheoremsaysthatthisisanecessaryandsucient Ci condition; thus, given a representation

⇢ : ⇡ (X p ,...,p ) GL(V ), 1 0 \{ 1 k} ! where dim V = n and ⇢(l ) ,forprescribedconjugacyclasses ,..., in i 2 Ci C1 Ck GLn(C), we can ﬁnd linear maps ⇢i,j and ⇢i,j⇤ and vector spaces Vi,j of dimension

↵i,j as above, such that the tuple (V, ⇢,Vi,j.⇢i,j,⇢i,j⇤ )isanobjectofthecategory

Rep⇡(). Then, combining this with Lemma 3.2.5, one has the following result.

Theorem 3.2.6. There is an isomorphism between the character variety X(g, k, ) C

62 q and the ane quotient ([],↵):=Rep⇤([],↵)0 // GL(↵). Mq,0 q Remark 3.2.7. From itsf deﬁnition, one can see that Rep ⇤ ([],↵)0 is an open ane GL(↵)-invariant subset of Rep(⇤q[],↵)whichisobtainedbyinvertingcer- tain GL(↵)-invariant functions (the determinants of the linear transformations corresponding to loops at the node). Since the quotient in the ane case is obtained by passing to G-invariant functions, i.e., Spec B//G =SpecBG,wededucethat the ane quotient ([],↵)canbeidentiﬁedwithanopensubsetofthemul- Mq,0 tiplicative quiver variety q,0([],↵). This is important because, as outlined in f M the following section, in order to show the non-existence of symplectic resolutions we prove that certain such varieties contain an open subset which is factorial and terminal.

Remark 3.2.8. We note that, in the star-shaped case, this result follows from [CS06, Section 8]. Moreover, in the general case, Yamakawa proves a similar result to the one obtained in this section in the language of local systems on punctured surfaces, see [Yam08,Theorem4.14]formoredetails.

3.3 Singularities of multiplicative quiver varieties

Throughout this section, which is devoted to the study of the singularities (Q, ↵) Mq,✓ and to the proof of Theorem 3.0.7,weusethenotationintroducedinSection3.1.

In order to carry out this analysis, in Section 3.3.1,wedescribethesingularlocus of the varieties in question. As one might expect, for ↵ ⌃ , this is given by 2 q,✓ the locus of strictly semistable representations. This follows because these varieties are Poisson, the stable locus is symplectic and smooth, and its complement has codimension at least two. Since a generically non-degenerate Poisson structure on asmoothvarietycanonlydegeneratealongadivisor(thevanishinglocusofthe Pfaan of the Poisson bivector), we conclude that the entire smooth locus is non- degenerate. Since the strictly semistable locus is degenerate, it must therefore be singular. Moreover, in the case where ↵ is q-indivisible, a symplectic resolution can be obtained by varying ✓,byLemma3.1.16 and Corollary 3.1.24.Thisprovesthe second statement of Theorem 3.0.7.

In Section 3.3.3,wecompletetheproofofTheorem3.0.7 by considering strata of representation type ⌫,where↵ = n and ⌫ is a partition of n.Wecompute

63 their codimension. As a consequence, taking to be q-indivisible, for suitable ✓0 ✓, (Q, ↵)hassingularitiesincodimension 4. Hence by Flenner’s theorem Mq,✓0 [Fle88], its normalisation is a symplectic singularity, which proves the ﬁrst statement of Theorem 3.0.7. Finally, we show, using Drezet’s criterion of factoriality, that the singularities along most strata ⌫ are factorial and terminal. This proves the ﬁnal statement of Theorem 3.0.7. Note that Section 3.3.3 closely follows [BS16b], where the analogous strata are considered for ordinary quiver varieties.

3.3.1 Singular locus of (Q, ↵) for ↵ ⌃ Mq,✓ 2 q,✓

Before proving the main statement, we need an auxiliary result which is valid for any variety endowed with a Poisson structure.

Lemma 3.3.1. Let X be a smooth variety and ⇡ 2TX a generically non- 2^ degenerate Poisson bivector. Let D the degenerate locus of ⇡. Then, if nonempty, D is a divisor.

Proof. By generic non-degeneracy, dim X has to be even, therefore dim X =2d. Deﬁne the top polyvector ﬁeld = d⇡.Then,D coincides with the zero locus of ^ .Ontheotherhand, is a section of a line bundle and, therefore, its zero locus is adivisor(ifnonempty).

This implies the following criterion for the singular locus of a Poisson variety:

Corollary 3.3.2. Let X be a Poisson variety which is smooth and symplectic in the complement of a closed Poisson subvariety Z X which has codimension at least ✓ two everywhere. Then Z equals the set-theoretic singular locus of X.

Proof. Suppose for a contradiction that X is smooth at a point z Z.SinceZ is a 2 closed Poisson subvariety, the Poisson structure of X is degenerate at z.Itfollows from Lemma 3.3.1 that the degeneracy locus of X has codimension 1 at z. However, this locus is contained in Z,whichhascodimensionatleasttwoatz.Thisisa contradiction.

Proposition 3.3.3. Let ↵ ⌃ . The smooth locus of (Q, ↵) is s (Q, ↵). 2 q,✓ Mq,✓ Mq,✓

64 Proof. By Proposition 3.1.18 s (Q, ↵) is smooth and symplectic. Let Z be the Mq,✓ complement. It is the union of all the non-open strata of (Q, ↵). There are Mq,✓ finitely many and these all have purely even dimension; hence Z has codimension at least two everywhere (as s is dense and it has purely even dimension, 2p(↵)). Mq,✓ Furthermore, we claim that Z is a Poisson subvariety, i.e., all Hamiltonian vector fields are tangent to it. Indeed, Hamiltonian vector fields descend from GL(↵)- invariant Hamiltonian vector fields on representation varieties. These integrate to formal automorphisms which commute with the G-action, which hence preserve the stratification by conjugacy classes of stabiliser. Therefore, the hypotheses of Corollary 3.3.2 are satisfied, and the statement follows.

Remark 3.3.4. It is reasonable to ask if a stronger statement is true, which makes sense for general ↵:aretheconnectedcomponentsoftherepresentationtypestrata the symplectic leaves? Equivalently, do the Hamiltonian vector fields span the tangent spaces to the representation type strata? If so, then (a) (Q, ↵)hasfinitely Mq,✓ many symplectic leaves, and (b) the representation type strata are all smooth. The converse statement also holds: if a stratum is smooth and it is a union of finitely many symplectic leaves, its Poisson structure must be non-degenerate outside a locus of codimension at least two. So Lemma 3.3.1 implies that it is actually non- degenerate.

Let us comment brieﬂy on conditions (a) and (b). First, if (Q, ↵)isasymplectic Mq,✓ singularity, it has ﬁnitely many symplectic leaves, by [Kal06, Theorem 2.5]. Next, (1) (r) for a representation type ⌧ =(k1, ; ...; kr, ), there is an obvious surjection ( s (Q, (1)) s (Q, (r)))dist C⌧ (Q, ↵)withsmoothsource,wherethe Mq,✓ ⇥···⇥Mq,✓ ! q,✓ dist refers to the open subset where the elements of the i-th and j-th factors are unequal for all i and j. It seems reasonable to expect this to be a covering, in which case the stratum is smooth.

In order to prove the ﬁrst statement of Theorem 3.0.7,weneedacriterionforthe normalisation of a variety to have symplectic singularities. This is an extension of [BS16b,Lemma6.12],using[Kal09,Theorem1.5]:

Proposition 3.3.5. Let X be a Poisson variety and assume that ⇡ : Y X is a ! proper birational Poisson morphism from a variety Y with symplectic singularities.

Then the normalisation X0 of X has symplectic singularities. Moreover, the induced map ⇡ : Y X0 is Poisson. ! 65 Proof. In [BS16b, Lemma 6.12], the result is proved under the assumption that X is in fact normal. To conclude the lemma from this result, we may apply [Kal09, Corollary 1.4, Theorem 1.5]. By these results (and their proofs), given a Poisson variety X,thenormalisationX0 has a unique Poisson structure such that the normalisation map ⌫ : X0 X is a Poisson morphism. The map ⇡ factors through ! ⌫,andtheinducedmap⇡0 : Y X0 must be Poisson, since the Poisson bracket ! on is the unique extension of the Poisson bracket on to a biderivation OX0 OX .ThenthefactthatX0 has symplectic singularities follows from OX0 ⇥OX0 !OY [BS16b,Lemma6.12].

Remark 3.3.6. For convenience, we will apply this result even in the case where ⇡ is a symplectic resolution. However, in this case, the statement follows from deﬁnitions, without really requiring the results of [BS16b; Kal09], as follows. The map ⇡ : Y X factors through ⇡0 : Y X0,whichinducesonX0 auniquePoisson ! ! structure such that ⇡0 is Poisson; as Y is non-degenerate and its symplectic form is pulled back from X0, X0 must also be non-degenerate on the smooth locus. By deﬁnition, X0 is then a symplectic singularity. Since ⇡ is dominant, the Poisson structure on X is uniquely determined from the one on Y ,andmustbetheone obtained from X0 via the inclusion .Thisprovesthelaststatement. OX !OX0 Remark 3.3.7. Actually, in the above proposition, the biconditional holds: X has symplectic singularities if and only if Y does. Moreover, one can generalise to the case where X is a nonreduced Poisson scheme: in this case, the map ⇡ factors through the reduced subvariety Xred, which is canonically Poisson by [Kal09,Corol- lary 1.4].

3.3.2 The q-indivisible case

We now prove the second statement of Theorem 3.0.7.Supposethat ⌃ is 2 q,✓ q-indivisible.

First suppose that is real. In this case, by [CS06,Theorem2.1],⇤q(Q)admitsa simple rigid representation X and any other representation Y of the same dimension must be isomorphic to X,whichmeansthatthevariety (Q, )isapoint.So Mq,✓ the questions about symplectic singularities and resolutions are trivially answered.

Next suppose that is imaginary. in this case one may proceed as follows: by choos-

66 ing a generic stability parameter ✓0 ✓,thereisaprojectivesymplecticresolution

⇡ : (Q, ) (Q, ); Mq,✓0 ! M q,✓

s indeed, by [Yam08,Proposition3.5],for✓0 generic, the stable locus (Q, ), Mq,✓0 which is smooth, coincides with the semistable locus and hence we can ﬁnd ✓0 ✓ such that is smooth and symplectic. Moreover, the fact that the morphism ⇡ Mq,✓0 exists and is projective and Poisson follows, in the ✓ = 0 case, from the deﬁnitions of ane and GIT quotient and, for general ✓,fromLemma3.1.16. Finally, birationality of ⇡ is ensured by Corollary 3.1.24.Thus,wecanconcludethat (Q, )admits Mq,✓ asymplecticresolution,givenbythemorphism⇡.ByProposition3.3.5 (or Remark 3.3.6), this implies that the normalisation of (Q, ) has symplectic singularities. Mq,✓

3.3.3 The q-divisible case

In this subsection, we prove the ﬁrst and third statements of Theorem 3.0.7.Wemay assume that ↵ is q-divisible: this is automatic in the third part, whereas in the ﬁrst part, the result follows from the second part (proved in the preceding subsection) in the q-indivisible case. This means that ↵ is anisotropic, by the following result:

Lemma 3.3.8. Let ↵ N be q-divisible. Then ↵ ⌃ only if ↵ is anisotropic. 2 q,✓ 2 q,✓ Conversely, if ↵ = m and ⌃ is anisotropic, then ↵ ⌃ . 2 q,✓ 2 q,✓

Proof. This is a generalisation of [Cra02,Proposition1.2](inviewofRemarks3.5.7 and 3.5.8), with the same proof. For details, see Corollary 3.5.20.(ii) below (whose proof is independent of any of the results of this section).

Recall that a weighted partition of n is a sequence ⌫ =(l1,⌫1; ...; lk,⌫k)suchthat ⌫ ⌫ and k l ⌫ = n.If⌫ is a partition of n,weshalldenoteby⌫ the 1 ··· k i=1 i i representation typeP (l1,⌫1; ...,lk,⌫k).

Lemma 3.3.9. 1. The set ⌃ contains m m 1 ; q,✓ { | } 2. dim C⌫(Q, n)=2 k +(p() 1) k ⌫2 ; q,✓ i=1 i ⇣ ⌘ P ⌫ 3. for (p(),n) =(2, 2), dim (Q, n) dim C (Q, n) 4 for all ⌫ =(1,n). 6 Mq,✓ q,✓ 6

67 4. for (p(),n) =(2, 2) and ⌫ =(1,n), one has dim (Q, n) dim C⌫(Q, n) 6 6 Mq,✓ q,✓ 8 unless one of the following holds: (i) (p(),n)=(2, 3) and ⌫ =(1, 2; 1, 1); (ii) (p(),n)=(3, 2) and ⌫ =(1, 1; 1, 1).

Proof. The arguments are completely analogous to those of [BS16b,Lemma6.1], except here that we use the dimension estimates given by Proposition 3.1.19.The ﬁrst statement is a consequence of Lemma 3.3.8.

Note that the above result has the following interesting consequence.

Proposition 3.3.10. Assume that all ✓-stable representations of dimension

Proof. This is an immediate consequence of Proposition 3.1.27 and point (3) of Lemma 3.3.9,giventhat,byassumptionon✓,allstrataexceptfortheopenone have codimension greater than 4.

Now, let ↵ ⌃ be q-divisible. Write ↵ = n for q-indivisible and n 2. 2 q,✓ For generic ✓0 ✓,theonlystrataof (↵)arethosegivenbymultipartitions Mq,✓0 of n times .If(p(),n) =(2, 2), then, taking into account Remark 3.1.23,all 6 non-open strata have codimension at least four by Lemma 3.3.9.(3). Therefore (Q, ↵) is a symplectic singularity by Flenner’s Theorem [Fle88]. Now, the map Mq,✓0 (Q, ↵) (Q, ↵)isbirational,projective,andPoissonbyCorollary3.1.24 Mq,✓0 !Mq,✓ and Lemma 3.1.16. Therefore, the normalisation of (Q, ↵)isitselfasymplectic Mq,✓ singularity by Proposition 3.3.5. This proves the ﬁrst statement of Theorem 3.0.7.

It remains to prove the ﬁnal statement of Theorem 3.0.7.Forthispurpose,assume that ↵ = n for n 2andthat N (not necessarily q-indivisible or in ⌃ ), 2 q,✓ q,✓ such that there exists a ✓-stable representation of dimension .LetU be the union of all the strata indexed by ⌫ for ⌫ aweightedpartitionsofn,

⌫ U := Cq,✓(Q, ↵). ⌫ [ As well as for the previous lemma, to prove the following result one can repeat verbatim the arguments in [BS16b,Lemma6.2].

68 Lemma 3.3.11. The subset U is open in (Q, ↵).If✓ is generic and is Mq,✓ q-indivisible, this subset is the entire variety.

In order to prove that U is factorial, we shall follow the approach of [BS16b], which was itself inspired by some results of Drezet, [Dre91], on factoriality of points in moduli spaces of semistable sheaves on rational surfaces. Assuming the notation ✓ ss q above, with ⇡ :Rep (⇤ ,↵) (Q, ↵)denotingthequotientmap,deﬁne !Mq,✓ 1 V := ⇡ (U). We aim at proving that V is a local complete intersection and that it is factorial and normal. We shall then descend the factoriality property to the subvariety U.

Proposition 3.3.12. V is a local complete intersection, factorial and normal.

The proof of this proposition follows closely the arguments used in [BS16b,Propo- sition 6.5] and uses the Serre’s normality criterion and Grothendieck’s criterion for factoriality, which we state below.

Proposition 3.3.13 (Serre’s criterion for normality). Let R be a Noetherian ring. The following are equivalent:

• R is a normal ring, and

• R has properties (R1) and (S2).

Proposition 3.3.14 (Grothendieck’s criterion for factoriality). Let B be noetherian local ring. If B is a complete intersetion and regular in codimension 3, then B is  factorial.

✓ ss q Proof of Proposition 3.3.12. Since V is open inside Rep (⇤ ,↵), Proposition 3.1.22 implies that it is a local complete intersection. To prove normality and factoriality, recall that a local complete intersection satisﬁes Serre’s S2 property, so Serre’s criterion implies that it is normal if it is smooth outside a locus of codimension at least 2. Moreover, by a result of Grothendieck ([KLS06, Theorem 3.12]), a local complete intersection which is smooth outside a locus of codimension at least 4 is factorial. Put together, to show that V is normal and factorial, it suces to show that it is smooth outside of a locus of codimension at least 4. For this, one can repeat verbatim the arguments used in [BS16b,Proposition6.5],replacingCorollary6.4and Lemma 6.1 with Lemma 3.1.25 and Lemma 3.3.9,respectively.

69 In order to descend factoriality from V to U we use Drezet’s method, which we state below. In particular, [BS16b, Theorem 6.7] holds true in this context as well with no change in the proof of the result, as we have already made sure that all of the tools used there are still applicable here, Proposition 3.3.12 being the most important one.

Theorem 3.3.15 ([Dre91], Theorem A). Let V be a factorial, normal G-variety, with categorical quotient ⇠ : V U := V//G. Assume that there exists an open ! subset U U such that s ⇢

(a) the complement to Us has codimension at least two in U,

1 (b) V := ⇠ (U ) U is a principal G-bundle; and s s ! s

Let x U and y T (x) a lift in V (so that G y is closed in V ). The following are 2 2 · equivalent:

(i) The local ring is a unique factorization domain. OU,x (ii) For every line bundle M on U , there exists an open subset U U containing 0 s 0 ⇢ both x and Us such that M0 extends to a line bundle M on U0.

(iii) For every G-equivariant line bundle L on V , the stabilizer of y acts trivially

on the ﬁber Ly.

Theorem 3.3.16. U is a factorial variety.

We omit the proof, as it is exactly the same as in [BS16b,Corollary6.9].

Using the previous theorem and the estimates on the codimension of the singular locus, one can conclude that (Q, ↵)doesnotadmitasymplecticresolution. Mq,✓ We state this formally below, where we also recall our running hypotheses for the reader’s convenience.

Theorem 3.3.17. Let ↵ = n ⌃ be anisotropic imaginary, for n 2, such that 2 q,✓ there exists a ✓-stable representation of ⇤q of dimension , and (p(),n) =(2, 2). 6 Then (Q, ↵) has an open subset which is factorial, terminal, and singular. Mq,✓ Hence it does not admit a symplectic resolution. Moreover, if ✓ is generic and ↵ is q-indivisible, then this open subset is the entire variety.

70 Proof. The subset U is singular, since it contains the non-open stratum (n, ). It is factorial by Theorem 3.3.16. Under the assumptions, the singular strata in U all have codimension at least four, hence also the singular locus. Thus, U is terminal by [Nam01a], since it has symplectic singularities and the singular locus has codimension at least four.

This completes the proof of the third and ﬁnal statement of Theorem 3.0.7.Itsproof is complete.

3.3.4 Proof of Corollary 3.0.8

Write ↵ = m for q-indivisible. Note that for ✓ generic, then the only possible decompositions of ↵ are into multiples of .If↵ is q-indivisible, it therefore follows trivially that ↵ ⌃ ;sincetheonlystratumin (Q, ↵)istheopenoneof 2 q,✓ Mq,✓ stable representations, it also follows from Proposition 3.1.18 that is smooth Mq,✓ symplectic. Suppose that ↵ is q-divisible. It then follows from Corollary 3.5.20.(ii) below that ↵ is in ⌃q,✓ if and only if it is anisotropic. This completes the proof of part (i).

Part (ii) follows from Proposition 3.3.10 and Theorem 3.0.7. The ﬁrst statement of part (iii) follows from Theorem 3.0.7. Finally, the last statement follows from 3.1.18 because, in this case, there is only one stratum in (Q, ↵), consisting of ✓-stable Mq,✓ representations.

3.3.5 The anisotropic imaginary (p(↵),n)=(2, 2) case

The only case left out in this analysis is that of 2↵ ⌃ for ↵ N satisfying 2 q,✓ 2 q,✓ p(↵)=2.Theanalogousquestionofexistenceofasymplecticresolutioninthe setting of Nakajima quiver varieties is settled in [BS16b,Theorem1.6],whereit is shown that, for generic ✓,blowinguptheidealsheafdeﬁningthesingularlocus gives a symplectic resolution of singularities. This is achieved by showing that, ´etale locally, the variety is isomorphic to the product of C4 with the closure of the cone given by the six dimensional adjoint orbit closure to Sp(V ), for V a4-dimensional symplectic vector space, see [BS16b,Theorem5.1]andthereferencestherein.Given this, one might conjecture that an analogous result holds for multiplicative quiver

71 varieties, and such a result should be proved by studying the ´etale local structure of the variety: in fact, by Artin’s approximation theorem [Art69], it would be sucient to give a description of the formal neighbourhood of a point. This will be discussed in a future work. For more details, see Section 4.5.

3.4 Combinatorics of multiplicative quiver varieties

In this section we study some combinatorial problems which are related to the geometry of multiplicative quiver varieties. Indeed, an interesting problem is to classify all the possible “(2, 2)-cases”: these are the main q-divisible cases for which we conjecture that there exists a symplectic resolution. In the next subsection we carry out these computations in the case of crab-shaped quivers, i.e., we classify all punctured character varieties that can, conjecturally, be symplectically resolved. We shall see how most of the (2, 2)-cases occur in the case of star-shaped quivers, i.e., when the surface has genus 0. It is also important to point out that the classiﬁcation below yields a classiﬁcation of the crab-shaped quivers for which the factoriality and terminality results apply: assume Q is crab-shaped. As we will explain in Section 3.5,wecanreducetothecasethat↵ (Q). Moreover, if ↵ is q-indivisible, it 2F is actually in ⌃q,✓ (by Theorem 3.5.16 below), hence Theorem 3.0.7 implies that (Q, ↵) admits a projective symplectic resolution. Suppose, on the contrary, Mq,✓ that ↵ = n for ⌃ q-indivisible and n 2. Then one of the following holds: 2 q,✓

(a) is one of the cases described below (p()=2);thenforn =2,asymplectic resolution of (Q, ↵)isconjecturallyobtainedbyvarying✓ and blowing up Mq,✓ the reduced singular locus; while for n>2, (Q, ↵)isfactorial,terminal, Mq,✓ and singular for generic ✓;

(b) The support of ↵ is ane Dynkin, and is an imaginary root ( = m for m 1); (c) We are in another case. As we will explain in the proof of Corollary 3.5.27, (Q, ) = . Thus, by Theorem 3.0.7, (Q, ↵)isfactorial,terminal, Mq,✓ 6 ; Mq,✓ and singular.

Finally, it is possible to have ↵ (Q) q-divisible but not a multiple of an element 2F of ⌃q,✓.Inthiscase,asexplainedinTheorem3.5.16 in the next section, ↵ must have

72 a very special form. Namely, ↵ = p for p 2prime,andQ is a quiver obtained from an ane Dynkin one (of types A0, D4, E6, E7,orE8)byaddinganadditional vertex and arrow connecting it to an extending vertex of the ane Dynkin quiver. e e e e e Furthermore, the dimension vector =(m, 1), with notation meaning putting m times the primitive imaginary root of the ane Dynkin quiver and then attaching a new vertex to an extending vertex of the quiver. The extending vertex of the former quiver has dimension m,whereasthenewvertexhasdimensionone.Inthiscase, by expectation (*), we do not anticipate a stable representation of dimension ↵ to exist, so the reasoning above likely does not apply. We therefore leave the question of existence of symplectic resolutions open in this case.

Thus, modulo some general results we will explain in Section 3.5,thecombinatorial work in this section completes the classiﬁcation of dimension vectors in the crab- shaped case which either admit symplectic resolutions, do not admit them, or for which the question is left open (with an expectation or a conjecture given).

3.4.1 (2,2) cases for crab-shaped quivers

The analysis is based on some standard numerical arguments and the constraints on the dimension vector ↵ for it to satisfy the conditions of Section 3.2,i.e.ithas to represent the multiplicities of the eigenvalues in the prescribed conjugacy class.

Theorem 3.4.1. There are exactly 13 pairs (Q, ↵), where Q is a star-shaped quiver as in Section 3.2 and ↵ (Q) is in the fundamental region, such that ↵, ↵ = 1. 2F h i Such pairs are depicted as follows, where a vertex is substituted by the corresponding entry of the dimension vector:

1 1 1

(3.3) 1 2 1

1 2 1

(3.4) 1 3 2 1

73 2 2

(3.5) 1 4 3 2 1

3 2 1

(3.6) 1 2 4 3 3 1

2 2

(3.7) 2 4 2 1

4 3 2 1

(3.8) 2 5 4 3 2 1

3 (3.9)

1 3 5 4 3 2 1

(3.10) 1 2 3 4 5 6 4 2 1

4 (3.11)

2 4 6 4 2 1

74 4

2 5 8 7 6 5 4 3 2 1

(3.12) 4

(3.13) 1 2 4 6 8 6 4 2

(3.14) 1 4 7 10 8 6 4 2

4 8 12 10 8 6 4 2 1

(3.15)

Remark 3.4.2. It is important to highlight that quivers (3.7), (3.11), (3.13), (3.15)are the ane Dynkin quivers D˜ 4, E˜6, E˜7, E˜8, respectively, with dimension vector given by (2, 1), where is the minimal isotropic imaginary root of the corresponding quiver. See Remark 3.4.5.

Proof of Theorem 3.4.1. In order to prove this theorem, we ﬁrst calculate explicitly the value of ↵, ↵ ,for↵ a general dimension vector. The general star-shaped quiver h i has g loops and k legs, each of which has li arrows, i =1,...,k.Wehave

k k l 1 i ↵, ↵ =(1 g)n2 + ↵2 n ↵ ↵ ↵ . h i i,j i,1 i,j i,j+1 i,j i=1 i=1 j=1 X X X X Assume now that ↵ (Q)andthat ↵, ↵ = 1; then, given that ↵, ↵ = 2F h i h i ↵ ↵, e = ↵ e ,↵ ,thisimpliesthattherecanonlybetwopossibili- i Q0 i i i Q0 i i 2 h i 2 h i ties:P P

a) there exists a unique vertex i Q such that either ↵ =1and(↵, e )= 2, or 2 0 i i

75 ↵ =2and(↵, e )= 1, with (↵, e )=0forj = i;thisimpliesthat,denoting i i j 6 by Adj(i)thesetofverticeswhichareadjacenttoi,onehas j Adj(i) ↵j =5 2 for ↵i =2and j Adj(i) ↵j =4for↵i =1; P 2 P b) there are two distinct vertices i and i0 such that (↵, e )=(↵, e )= 1and i i0

↵i = ↵i0 =1,with(↵, ej)=0forj = i, i0.Inthiscaseonehas j Adj(k) ↵j =3 6 2 for k = i, i0. P

In this case, if i or i0 are the central vertex, then the only possibility is given by the quiver (3.3) in the statement of the theorem. Otherwise, we have that, if v is the central vertex, then (↵, ev)=0,whichimpliesthat j Adj(v) ↵j =2n,where 2 ↵v = n:indeed, P

0=(↵, e )= ↵, e + e ,↵ = n ↵ + n ↵ =2n ↵ . v h vi h v i k l j k v v l j Adj(v) X! X! 2X

Now, ﬁx a branch along which none of the special vertices i and i0 appear, let l be its length and let 0 = n, 1,...,l the components of the vector ↵ along such a branch.

Then, using that (↵, e )=0forj = i, i0,wegettherecursiveformula j 6

2j = j 1 + j+1, for j =1,...,l 1, and also l 1 =2l,whichimpliesthat

=(l +1 j) . j l

Therefore, such a branch has the form

n n c n 2c ... c, ! ! ! ! where c is a positive integer such that c n.Moreover,inorderforconditiona)tobe | satisﬁed there has to be one branch ending with one of the following

5 2, 4 2 1, 4 1, ! ! ! ! and, thus, having the form

76 n 3 ..., 5 2, ! ! ! n 2 ..., 4 2 1, ! ! ! ! n 3 ..., 4 1(§) ! ! ! respectively; for condition b), there have to be two branches ending as

3 1, ! having the form n 2 ... 3 1. (§§) ! ! !

Therefore, we are left to consider a star-shaped quiver where all but one or two branches are as follows:

a1 ai al

......

n a n a n a 1 i l

Moreover, if the quiver satisﬁes condition a), then l = k 1andthereisanadditional branch having one of the forms in (§); on the other hand, if the quiver is as in case b), then l = k 2andtherearetwoadditionallegsoftheformdescribedby(§§). We shall now use some numerical arguments to prove that, among all such possibilities, only the ones listed in the statement of the theorem can actually occur. First, let us spell out how the equality 0 = (↵, ev)canberephrased:onehasthat

k k 0=(↵, e ) 2n = (n a ) a =(k 2)n. v () i () i i=1 i=1 X X Therefore, one has the following possibilities:

77 a) in the cases of a branch ending with 5 2or4 1theequality0=(↵, e ) ! ! v reads as k 1 3 1 + = k 2, n n/a i=1 i X where n 2(mod3)andn 1(mod3),respectively,andn>a 2, a n ⌘ ⌘ i i| for every i;theseshallbementionedinthefollowingascasesa.1anda.2.On the other hand for a branch ending with 4 2 1wehave ! !

k 1 2 1 + = k 2, (3.16) n n/a i=1 i X where n has to be even and a n;thisisrenamedascasea.3. i| b) there are two branches 3 1and0=(↵, e )isequivalentto ! v

k 2 4 1 + = k 2, n n/a i=1 i X and a n for every i and n has to be odd, and a

In cases a.1 and a.2 one has that n 4whichforcesk 4: indeed, one has that for  n 4 n/a 2and,therefore, i

k 1 3 1 3 k 1 k +1 + + < , n n/a  4 2 2 i=1 i X which implies that k +1 k 2 < , 2 and thus k 4. For k =4andn =4itiseasilycheckedthatquiver(3.5)inthe  statement of the result is the only possibility. If k =3,thenthefollowinginequality holds: 1 1 3 + n 1 n = n 3, 2 4 n ✓ ◆ ✓ ◆ which forces n 12; one can check that case a) cannot be realised for n =4, 7, 11  and that the cases n =5, 8, 10 give quivers (3.8), (3.12)and(3.14)respectively. Next, for case a.3 one has: n even and k 4: indeed, since n 4, from equation  (3.16), one has that 2 k 1 + k 2, 4 2 78 which implies that k 2. 2  If k =4andn =4thenonegetsquiver(3.7). If k =3,thenn 12: indeed, from  equation (3.16), we have 2 1 1 + + 1 n 2 3

This leads to: quiver (3.6)forn =4,quivers(3.10)and(3.11)forn =6,quiver (3.13)forn =8,andquiver(3.15)forn =12.

We turn now to case b): n is odd and 3; this implies that k 4: indeed, in the  same way as the previous case, equation (3.16)gives

4 k 2 8 2k + k 2= . 3 3 ) 3 3

Therefore, setting k =4forcesn = 3, which leads to quiver (3.4). When k =3 one has that n 6, which implies that n =3orn =5.Onechecksthatn =3  is impossible, whereas n =5givesquiver(3.9). Since we have dealt with all the possible cases, the proof is complete.

Theorem 3.4.3. Assume that g 1. Then, the only pairs (Q, ↵), where Q is a crab-shaped quiver and ↵ (Q) is such that ↵, ↵ = 1 are the following: 2F h i

1 (3.17)

2 1 (3.18)

Remark 3.4.4. Parallel to Remark 3.4.2,inthesecondcaseabove,thequiverand dimension vector are also of the form (2, 1) where =(1)istheprimitiveimaginary root of ane type A0 (the Jordan quiver with one vertex and one arrow).

e Proof of Theorem 3.4.3. As in the arguments of the previous theorem, we see that, if v is the central vertex and ↵ ,thenthereare3possibilitiesforthevalueof 2F 79 (↵, e ), i.e. (↵, e )canbeeither0, 1or 2. In general one has that v v

k (↵, e )=2(1 g)n ↵ . v i,1 i=1 X If k = 0, then one must have n =1andg =2,whichgivestheﬁrstquiverofthe statement of the result. Thus, one is left to show that for k 1, there are no crab- shaped quivers satisfying the mentioned conditions other than the second quiver in the statement of the theorem. If k 1, n>1, which implies that either (↵, e )=0 v or 1. In the ﬁrst case, we get that

k 2(1 g)n = ↵ ; i,1 i=1 X but g 1, which gives k ↵ 0, a contradiction. If (↵, e )= 1, then one i=1 i,1  v must have n =2and↵i,1P=1fori =1,...,k and g =1.Thisimpliesthat

k =5 4g, which forces k =1.Therefore,wegetthequiver

2 1.

Since we dealt with all the possible cases the proof is complete.

Remark 3.4.5. Note that, to get a list of all the (2, 2) cases one has to take each of the pairs (Q, ↵)drawnaboveandconsiderthepair(Q, 2↵). Moreover, given

Q0 Q0 q (C⇤) , ✓ Z it follows from Theorem 3.5.16 below that the given ↵ are in 2 2 ⌃ (since they are already in (Q)) if and only if the following are satisﬁed: (a) q,✓ F they are in N ,i.e.,q↵ =1and✓ ↵ =0,and(b)inthe(2, 1) cases (mentioned q,✓ · in Remarks 3.4.2 and 3.4.4), /N ,i.e.,q =1or✓ =0. 2 q,✓ 6 · 6

3.5 General dimension vectors and decomposition

One fundamental tool in the classiﬁcation theorem [BS16b,Theorem1.4]isthe canonical decomposition of a dimension vector of a quiver variety into summands

80 which lie in ⌃,✓,whichistheadditiveversionoftheset⌃q,✓ deﬁned in this paper (one just needs to replace the condition q↵ =1with ↵ =0).Thisappears · in Crawley-Boevey’s canonical decomposition in the additive case (extended to the case ✓ =0in[BS16b]). Combinatorially, it says: 6 + Lemma 3.5.1. [Cra02, Theorem 1.1], [BS16b, Proposition 2.1] Let ↵ NR . 2 ,✓ Then ↵ admits a unique decomposition ↵ = n (1) + +n (k) as a sum of elements 1 ··· k (i) ⌃ such that any other decomposition of ↵ as a sum of elements from ⌃ 2 ,✓ ,✓ is a reﬁnement of this decomposition.

Geometrically, the statement (together with the consequence for symplectic resolutions) is:

Theorem 3.5.2. [Cra02, Theorem 1.1], [BS16b, Theorem 1.4] The symplectic va- 1 ✓ ss riety M,✓(Q, ↵)=µ () // GL(↵) is isomorphic to the product

k ni (i) M,✓(Q, ↵) ⇠= S M,✓(Q, ). i=1 Y

(i) Moreover, it admits a symplectic resolution if and only if each M,✓(Q, ) admits a symplectic resolution.

For multiplicative quiver varieties, the combinatorial statement still holds, but it is not clear that such a geometric decomposition holds. We instead prove a weaker statement, which gives a decomposition into factors which might not be minimal, but still has all of the needed properties. Moreover, the resulting classiﬁcation of symplectic resolutions is the same statement as if the canonical decomposition as above held. As a result we are able to generalise Theorem 3.0.7 to the case of general dimension vectors (Theorem 3.5.26), and give its specialisation to the crab-shaped case (Corollary 3.5.27). To complete the proof, we need to establish that (Q, ↵) (Q, ↵)isasymplecticresolutionformany↵ not in ⌃ Mq,✓0 !Mq,✓ q,✓ (Theorem 3.5.22), in order to handle such factors appearing in the decomposition. In the additive case, such resolutions include the Hilbert schemes of points in C2 and in hyperk¨ahler ALE spaces (i.e., minimal resolutions of du Val singularities).

81 3.5.1 Flat roots

In order to write a product decomposition in the multiplicative setting, the dimension vectors for the factors need to be more general than those in ⌃q,✓.The dimension vectors turn out to include all of the ﬂat roots,whicharethoseforwhich the moment map is ﬂat – this is true for roots in ⌃q,✓. This condition is also very important in order to have a geometric understanding of the varieties.

Definition 3.5.3. Avector↵ N is called flat if, for every decomposition ↵ = 2 q,✓ ↵(1) + + ↵(m) with ↵(i) R+ ,wehavep(↵) p(↵(1))+ + p(↵(m)). Let ⌃ ··· 2 q,✓ ··· q,✓ be the set of flat roots. e Remark 3.5.4. As in [Cra01, Theorem 1.1], we could alternatively have made the definition only requiring ↵(i) N .Indeed,itfollowsfromtheproofofthede- 2 q,✓ composition theorem (Theorem 3.5.17) below that if ↵ N ,thenthereisade- 2 q,✓ (1) (k) (i) iso + composition ↵ = + + with ⌃q,✓ N 2 ⌃q,✓ Rq,✓ such that ··· 2 [ · ✓ p(↵) p((1))+ +p((k)). Hence, if we know the inequality when the ↵(i) R+ ,  ··· e 2 q,✓ we also know it when the ↵(i) N . 2 q,✓ The definition has the following interpretation. Let SL(↵):= g GL(↵) { 2 | det(g )=1 GL(↵). Note that factors through the inclusion SL(↵) i Q0 i ↵ 2 }⇢ ! ✓ ss GL(↵); let :Rep(Q, ↵) SL(↵)bethefactoredmap. Q ↵ !

Proposition 3.5.5. If ↵ is a ﬂat root, then ↵ is ﬂat over a neighbourhood of 1 ✓ ss q. In particular, ↵ (q) is a complete intersection and is equidimensional of dimension g↵ +2p(↵).

Proof. The second statement follows from the argument of Proposition 3.1.22 (following [Cra01,Theorem1.11]):allargumentsgothroughwiththestrictinequality replaced by the non-strict one, except that no statement can be deduced about the stable (or simple) representations forming a dense subset. For the first statement, 1 ✓ ss ✓ ss concerning flatness, note that dim (q) =dimRep(Q, ↵) dim SL(↵). ↵ Then the statement follows from the following general considerations. Given a morphism of varieties f : X Y ,byuppersemicontinuityofthefibredimension,the ! minimum fibre dimension is dim Y dim X and the locus in Y where the fibres have this minimal dimension is open. Next, it is a standard fact that a morphism from a Cohen-Macaulay variety to a smooth one is flat if and only if the dimension of every

82 1 ﬁbre equals dim X dim Y .Therefore,givenanyy Y such that f (y)attains 2 this minimum dimension, there exists an open neighbourhood U of y over which f is ﬂat.

Putting this together with Proposition 3.1.21,weconcludethefollowinganalogue of the last statement of Proposition 3.1.22:

1 ✓ ss Corollary 3.5.6. A dense subset of ↵ (q) is given by the union of preimages (1) (r) r (i) of strata of types (1, ; ...;1, ) with p(↵)= i=1 p( ). P Observe that ⌃ ⌃ .Theoppositeinclusiondoesnothold:forinstance,with q,✓ ✓ q,✓ (q, ✓)=(1, 0), one can take the quiver with two vertices and two arrow, one a loop e at the ﬁrst vertex, the other an arrow to the second vertex. Then the dimension vector (m, 1) is ﬂat for all m,butin⌃q,✓ only for m =1.

Remark 3.5.7. Note that, for every pair q, ✓,therealwaysexistsq0 such that Nq0,0 =

Nq,✓,andhence⌃q,✓ =⌃q ,0 and ⌃q,✓ = ⌃q ,0.Indeed,letz C⇤ be a multiplicatively 0 0 2 independent element from the qi (i.e., z,qi / qi is inﬁnite cyclic, where , e eh i h i h i ✓i denotes the multiplicative group generated by the given elements). Set qi0 := qiz .

Then q0 has the desired properties.

Remark 3.5.8. Similarly, given any parameters in the additive case, CQ0 ,✓ 2 2 Q0 Q0 a a Z , we also can construct q0 (C⇥) such that N = Nq ,0 where now N denotes 2 ,✓ 0 a a the additive construction; hence ⌃,✓ =⌃q0,0 and ⌃ ,✓ = ⌃q0,0.

Q0 Recall that the fundamental region is the locus (fQ):= e↵ N (↵, ei) 0, i . F { 2 |  8 } We recall, following [Cra01]and[Su06], how to classify ﬂat roots in terms of this region.

Deﬁnition 3.5.9. We say that the transformation ↵ s (↵)isa( 1)-reﬂection 7! v if s (↵)=↵ e . v v

We point out a useful geometric consequence of this deﬁnition:

Proposition 3.5.10. Suppose that ↵ s (↵) is a ( 1)-reﬂection and that q =1 7! v v and ✓ =0. Then there is a reﬂection isomorphism (s ↵) ⇠ (↵). v Mq,✓ v !Mq,✓

Proof. There is an obvious map q,✓(↵ ev) q,✓(↵), given by ⇢ ⇢ Cv, M !M 7! where Cv is the trivial representation (all arrows act as zero). We claim that it is an

83 isomorphism. In the decomposition of any ✓-polystable representation of dimension ↵ into stable representations, at least one factor must have dimension vector which has positive pairing with ev.By[CS06,Lemma5.1](inthecase✓ =0,whichextends to every ✓-stable representation the general case by replacing simple representations by ✓-stable ones), this summand must be Cv itself. Therefore, the direct sum map gives an isomorphism (Q, ↵ e ) (Q, e ) ⇠ (Q, ↵). Finally, note Mq,✓ v ⇥Mq,✓ v !Mq,✓ that, since there cannot be any loops at v, (Q, e )isjustapoint. Mq,✓ v Definition 3.5.11. Given ↵ R+ ,callasequencev ,...,v Q areflecting 2 q,✓ 1 m 2 0 sequence if, setting (q(i),✓(i),↵(i)):=(u u (q),r r (✓),s s (q)), we vi ··· v1 vi ··· v1 vi ··· v1 (m) (i) (i 1) have (a) ↵ (Q) e v Q ,and(b)↵v <↵v for all i. 2F [{ v | 2 0} i i Lemma 3.5.12. A reflecting sequence always exists.

Proof. By deﬁnition, ↵ is a root if and only if there exists a sequence of reﬂections at loopfree vertices taking ↵ to either the fundamental region or to an elementary root ev (it is imaginary in the former case and real in the latter case). Now, given

Q0 + ↵ N ,letN↵ := R real (↵, ) > 0 .Theneachreflectionsatisfying 2 |{ 2 | }| (b) decreases N↵ by one, and a nontrivial reflection not satisfying (b) increases N↵ by one. Now assume that ↵ R+.ThenN < .Sinces (R+ e ) R+, 2 ↵ 1 v \{ v} ✓ an arbitrary sequence of reflections satisfying (b) will remain in NQ0 .Thus,if↵ is real, an arbitrary N 1 reflections satisfying (b) will send ↵ to e for some v Q , ↵ v 2 0 and if ↵ is imaginary, then an arbitrary N↵ reflections satisfying (b) will take ↵ to (Q). F

As in [Su06,Theorem1.2],wehavethefollowing.

Theorem 3.5.13. Let ↵ R+ . Pick any sequence of vertices v ,...,v Q 2 q,✓ 1 m 2 0 such that, for (q(i),✓(i),↵(i)):=(u u (q),r r (✓),s s (↵)), we have vi ··· v1 vi ··· v1 vi ··· v1 (m) (i) (i 1) (m) ↵ (Q) and ↵v <↵v . Then ↵ is flat if and only if (a) ↵ (Q) is flat, 2F i i 2F and (b) for every i, either (b1) (q(i),✓(i),↵(i)) is an admissible reflection (Definition (i 1) (i 1) (i 1) (i 1) (i 1) (i) 3.1.9) of (q ,✓ ,↵ ) (i.e., qv =1or ✓v =0), or (b2) ↵ is a ( 1)- i 6 i 6 (i 1) reflection of ↵ .

As in [Su06], we will show below that ↵(m) (Q) is ﬂat if and only if it is not of 2F the form m for the minimal imaginary root of an ane Dynkin subquiver and m 2. 84 The theorem actually gives an algorithm to determine if a root is ﬂat, by playing a variant of the numbers game [Moz90](withacuto↵inthenon-admissiblecaseasin [GS11]).

Proof. Under an admissible reflection the condition of being flat does not change + + + since svi : R (i 1) (i 1) R (i) (i) is a bijection (as evi / R (i 1) (i 1) ), cf. [Cra01, q ,✓ ! q ,✓ 2 q ,✓ Lemma 5.2]. Here we let q(0) := q and ✓(0) := ✓.Ifweapplya( 1)-reflection, we claim that the condition of being flat does not change. We only have to show that, if (i) (i 1) ↵ ⌃ (i) (i) ,thenalso↵ ⌃ (i 1) (i 1) , since the converse follows immediately 2 q ,✓ 2 q ,✓ (i 1) (1) (k) from the definition of flat. Suppose on the contrary that ↵ = + + is e e ··· (j) + (i 1) (1) (k) adecompositionwith R (i 1) (i) and p(↵ )

(i) (i 1) (1) (k) (1) (k) p(↵ )=p(↵ )

(i) so that ↵ / ⌃ (i) (i) .Wehaveprovedthecontrapositive. 2 q ,✓ (i 1) (i 1) It remains to show that, if (↵ ,e ) > 1and(q ,✓ )=(1, 0), then ↵ / e vi vi vi 2 ⌃ (i 1) (i 1) In this case there can be no loops at v ,sothate is a real root. q ,✓ i vi (i 1) (i 1) (i 1) (i 1) Moreover, ↵v 1. Then, p(↵ ev )=p(↵ ev )+p(ev ) >p(↵ ). So e i i i i (i 1) ↵ is not ﬂat.

Remark 3.5.14. The same theorem as above applies in the additive case, to charac- terise the analogous set ⌃,✓ of flat roots. Also, note that when ✓ =0,theabove proof simplifies the proof of [Su06,Theorem1.2],sinceitdoesnotrequiretheclas- e sification [Cra01,Theorem8.1]ofrootsin (Q) ⌃ . F \ ,0 Remark 3.5.15. Thanks to [GS11,Theorem3.1],theconditionthatany(orevery) reflecting sequence consists only of admissible and ( 1)-reflections is equivalent to the condition that, for every real root R+ , we have (↵, ) 1. 2 q,✓ 

3.5.2 Fundamental and flat roots not in ⌃q,✓

To complete the characterisation, we need to determine the set (Q) ⌃ .This F \ q,✓ follows from [Cra01,Theorem8.1],whichcomputes (Q) ⌃q,✓.Westateasharper F \ e

85 and more general version:

Theorem 3.5.16. A root is not in ⌃q,✓ if and only if applying a reﬂecting sequence as in Theorem 3.5.13, either one of the reﬂections is not admissible, or the resulting element of (Q) is one of the following: F

(a) m` with the indivisible imaginary root for an ane Dynkin subquiver, m 2, and ` R+ ; 2 q,✓ (b) The support of ↵ is J K for J, K Q disjoint subsets with exactly one t ✓ 0 arrow from a vertex j J to a vertex k K, ↵ =1, and either: 2 2 k (b1) ↵ =1, and ↵ R+ , or j |J 2 q,✓ (b2) Q is ane Dynkin and ↵ = m for some m 1, with R+ . |J |J 2 q,✓

Moreover, the root is not in ⌃q,✓ if and only if either one of the reﬂections which is neither admissible nor a ( 1)-reﬂection, or the resulting element of (Q) is in case e F (a).

Proof. First, it is clear that if a inadmissible reflection is applied in the sequence, p(↵) p(s ↵)+p(e ), which shows that ↵ is not in ⌃ .Sowecanassumeall  i i q,✓ reflections are admissible. By [Cra01,Theorem8.1]intheadditivecase(with✓ =0), there is a sequence of admissible reflections taking to one of the given cases, not assuming that the result is in (Q). The proof extends verbatim to this case, F replacing N by Nq,✓. Although the additive statement has ` =1(orincharacteristic p has ` = p), for us we note that if m R+ has the property that k / R+ for 2 q,✓ 2 q,✓ k m,thenactuallym ⌃ , since the only roots with entries dominated by m | 2 q,✓ are real. Note that in [Cra01,Theorem8.1],itisrequiredincase(b2) that R+, 2 although this is not in the statement. We claim that in fact we can take the result to be in (Q). Note that, in [Cra01], it is not required in conditions (a),(b) that ↵ F be in (Q). However, we already know that, in order to have ↵ ⌃ , there must F 2 q,✓ be an admissible reflection sequence taking ↵ to (Q). Thus we may assume that F ↵ (Q)andmoreoverthatitissincere.Then,applying[Cra01,Theorem8.1], 2F there is a further sequence of admissible reflections taking ↵ to one of the forms above. After this we can apply an admissible reflection sequence to get back to an element of (Q), necessarily ↵ again. It is clear that doing so will not change the F

86 form as above, since ↵ was assumed to be sincere, so the reﬂections cannot shrink the support of ↵;notethat,incases(a)and(b2),thismeansthatnoreﬂectionswill be applied to the coecients of the multiple of .

For the converse, it remains to verify that cases (a) and (b1–2) are not in ⌃q,✓.In case (a) p(m)=1

3.5.3 Canonical decompositions

Let ⌃iso ⌃ be the subset of isotropic imaginary roots. Let M⌃iso := m↵ m q,✓ ✓ q,✓ q,✓ { | 2,↵ ⌃iso . 2 q,✓} Theorem 3.5.17. (i) Given ↵ N , if (Q, ↵) = , then there exists a unique 2 q,✓ Mq,✓ 6 ; decomposition ↵ = ↵(1) + + ↵(m) with ↵(i) ⌃ such that any other such ··· 2 q,✓ decomposition is a reﬁnement of this one.

(ii) There is also a unique decomposition ↵ = (1)+ +(k) with (i) ⌃ M⌃iso, ··· 2 q,✓[ q,✓ satisfying the properties: e

(i) (a) Every real root is in ⌃q,✓;

(b) Every other element (i) has the property ((i),) 0 for every real R+ ;  2 q,✓ (c) Any other decomposition into ⌃ M⌃iso satisfying (a) and (b) is a reﬁnement q,✓[ q,✓ of this one. e

Also, ((i),(j))=0if (i),(j) are imaginary and i = j. 6 (iii) The decomposition in (i) is a reﬁnement of the one in (ii). The real roots coincide. If in the reﬁnement (i) = ↵(i,j), then each pairing (↵(i,j),↵(i0,j0)) j 2 0, 1 , and the intersection graph for the ↵(i,j) (putting in edges for pairings 1) is { } P a forest, with connected components precisely the (i). In particular, (ii) is recovered from (i) by iteratively combining together pairs of imaginary roots with pairing 1.

87 (iv) The direct sum map produces an isomorphism (of reduced varieties), using the decomposition in (ii),

k (i) (Q, ) ⇠ (Q, ↵). Mq,✓ !Mq,✓ i=1 Y

Notice the following immediate consequence (of the decomposition in (ii)), giving a weakened version of (*):

Corollary 3.5.18. If ↵ is the dimension of a ✓-stable representation of ⇤q, then ↵ ⌃ M⌃iso. Moreover, ↵ is related by admissible reﬂections either to e (i.e., 2 q,✓ [ q,✓ v ↵ is a real root in ⌃q,✓), or to an element of the fundamental region. e

Note that, in the real case ↵ ⌃ ,itisobviousfromthepropertiesofadmissible 2 q,✓ reﬂections that ↵ is the dimension of a ✓-stable representation (see [CS06,Theorem 1.9] for a stronger statement). In the imaginary case, following (*), we only expect a stable representation if ↵ ⌃ ,butitisnotatallclearhowtoproveitsexistence. 2 q,✓ In the proof of the theorem, we will produce also an algorithm for constructing the (i) ,byasequenceofreﬂectionsandsubtractingrootsei,withtheendresultan element in (Q) whose connected components give the imaginary (i).Forthereal F roots, we obtain the unique decomposition into real roots in ⌃q,✓ (as a real root which is the sum of multiple real roots cannot be in ⌃q,✓). Observe that the same result and proof holds in the additive case.

Remark 3.5.19. We added the condition of reduced varieties in (iv) because it is not completely clear that the reﬂection isomorphisms in [Yam08,Theorem5.1]are deﬁned scheme-theoretically.

Proof. We prove the statements by induction on the sum of the entries of ↵.Ifthere is a vertex v Q at which (↵, e ) > 0andeitherq =1or✓ =0,thenwecanapply 2 0 v v 6 v 6 an admissible reflection. Since admissible reflections preserve the set of flat roots, the statements follows from Theorem 3.1.17 and the induction hypothesis. So suppose that there is no such vertex. Instead, suppose that v Q is such that (↵, e ) > 0 2 0 v but qv =1,✓v =0.Theneverydecompositionof↵ into elements of ⌃q,✓ must have an element having positive Cartan pairing with ev.Thiscannothappenbydefinition e for the imaginary roots. Since ev is the only real root in ⌃q,✓ with positive pairing

88 with ev,thisimpliesthatev must appear as a summand of every decomposition of types (i) and (ii). Moreover, as explained in [CS06,Lemma5.1],everyrepresentation in (Q, ↵)haseitherasubrepresentationoraquotientofdimensione .Since Mq,✓ v there is a unique such representation up to isomorphism, and it is stable, the direct sum map gives an isomorphism (Q, ↵ e ) (Q, e ) ⇠ (Q, ↵). We Mq,✓ v ⇥Mq,✓ v !Mq,✓ can apply the induction hypothesis to ↵ e . v This reduces to the case that ↵ (Q). In this case we can decompose ↵ into 2F its connected components. By Theorem 3.5.16,alloftheseareﬂatroots,except for elements of the form m` with ` ⌃iso and m 2. This yields the desired 2 q,✓ decomposition.

Let us prove that the decomposition is unique. The uniqueness in (ii) is clear, since roots have connected support. For the uniqueness in (i), we can prove this by induction, since every decomposition into roots in ⌃q,✓ of each of the types (a),(b1),(b2) of components described in Theorem 3.5.16 must reﬁne a particular one. Namely, in type (a), one can directly apply the argument of [Cra02,Lemma3.2],replacing there by `. Next, in type (b1), one applies the argument of [Cra02,Lemma 5.3]: every decomposition into elements of ⌃ must reﬁne ↵ + ↵ ,sinceoth- q,✓ |J |K erwise if ⌃ appears and = + is a nontrivial decomposition, then 2 q,✓ |J |K p()=p( )+p( ) is a contradiction. Finally, in type (b2), again the result |J |K follows from the argument of [Cra02, Lemma 5.4]. We omit the details.

Finally, let us see that the properties in (iii) are satisﬁed. By construction, the distinct imaginary roots (i),(j) pair to zero. By the inductive procedure of the preceding paragraph, the reﬁnement into ↵(i) have the stated property (the pairings are in 0, 1 and the intersection graph is a forest whose connected components { } are the (i)).

As a consequence, we obtain the following description of divisibility criteria for elements of ⌃q,✓ and ⌃q,✓:analogousto[BS16b,Theorem2.2]:

Corollary 3.5.20. Let ↵ = m for R+ q-indivisible and imaginary, and e 2 q,✓ m 2.

(i) ↵ ⌃ if and only if ⌃ is anisotropic and the decomposition of Theo- 2 q,✓ 2 q,✓ rem 3.5.17.(ii) is trivial for (i.e., a reﬂecting sequence involves only admis- e e sible reﬂections).

89 In particular, for ⌃q,✓, every rational multiple r Nq,✓ for r Q 1 is 2 2 2  also in ⌃q,✓. e (ii) ↵ ⌃ eif and only if is anisotropic and either ⌃ , or is equivalent 2 q,✓ 2 q,✓ via admissible reﬂections to one of the forms (b1) or (b2) in Theorem 3.5.16.

In particular, a rational multiple r Nq,✓ for r Q 1 is in ⌃q,✓ unless 2 2  it is equivalent to one of the forms (b1) or (b2). This can only happen if r =1/gcd() and qr =1.

Proof. (i) This follows from the classification of flat roots in Theorems 3.5.13, 3.5.16, and 3.5.17.Observesimplythatifareflectionsequencefor involves an inadmissible ( 1)-reflection, then the same sequence for ↵ involves an inadmissible ( 2)-reflection (which is not allowed). There are no inadmissible reflections if and only if the decomposition of Theorem 3.5.17.(ii) of has no real roots; then if is itself a root the decomposition is trivial. Finally observe that if has the trivial decomposition, then m also gives a flat root unless is isotropic.

(ii) Assume that ⌃ . After applying admissible reﬂections, we can assume 2 q,✓ (Q). Then ↵ = m will be in ⌃ unless is isotropic, by Theorem 3.5.16. 2F Conversely, if ↵ ⌃ ,againbyapplyingadmissiblereﬂections,wecanassume 2 q,✓ ↵ (Q). Then by Theorem 3.5.16,weseethat is either itself in ⌃ or of one 2F q,✓ of the forms (b1) and (b2).

Remark 3.5.21. Although we are working with reduced varieties throughout the chapter, we emphasised this in Theorem 3.5.17.(iv) because it is not completely clear that the reflection isomorphisms in [Yam08,Theorem5.1]aredefinedscheme- theoretically. On the other hand, this is the only obstacle here. That is, if these isomorphisms are defined scheme-theoretically, then the proof would appear to extend to this case, i.e., to not-necessarily-reduced multiplicative quiver schemes.

3.5.4 Symplectic resolutions for q-indivisible flat roots

Theorem 3.5.22. Suppose that ↵ ⌃ is q-indivisible and (Q, ↵) is non- 2 q,✓ Mq,✓ empty. Then for suitable ✓0 ✓, q,✓ (Q, ↵) q,✓(Q, ↵) is a symplectic resolu- M 0e !M tion.

90 Remark 3.5.23. Observe that the theorem also holds in the additive setting, where the result is also interesting. Indeed, it explains and generalises the technique of framing used to construct resolutions such as Hilbert schemes of C2 or of hyperkähler ALE spaces. In the former case, the quiver is again the framed Jordan quiver (with two vertices and two arrows, a loop at the first vertex, and an arrow from the second to the first vertex). The dimension vector is ↵ =(m, 1). The theorem recovers the well-known statement that taking ✓ = 0 gives a symplectic resolution 6 m m of the singularity Sym C2 in the additive case; this identifies with Hilb C2.Inthe multiplicative case for the same quiver, by Theorem 3.2.6 and Remark 3.2.7,after localisation, we obtain a resolution of the character variety of the once-punctured torus in the multiplicative case for rank m local systems with unipotent monodromy A satisfying rk(A I) 1. 

Proof of Theorem 3.5.22. In view of Lemma 3.1.16,weonlyhavetoshowthatwe can ﬁnd ✓0 ✓ such that (Q, ↵)issmoothand (Q, ↵) (Q, ↵)is Mq,✓0 Mq,✓0 !Mq,✓ birational. We will make use of the combinatorial analysis of [Cra01,Section8]. Note that, if ↵ ⌃ ,thentheresultfollowsfromthediscussioninSection3.3.1, 2 q,✓ so we can assume that this is not the case.

It will be useful to generalise from integral to rational stability conditions: note

Q0 that if ✓0 Q , then the sign of ✓ equals the sign of m✓ for all m 1, 2 · · and hence this added generality does no harm, as one can always multiply by the greatest common denominator of all the ✓i.

Q0 Allowing rational stability conditions, we claim that for ✓0 Q in a small enough 2 neighbourhood of ✓ (intersected with ✓0 ↵ =0),then✓0 ✓ under the ordering of · Deﬁnition 3.1.11. Indeed, ✓0 ✓ follows if ✓ >0 ✓0 >0forall<↵, · ) · which is an open condition. Moreover, we can guarantee that ✓ <0 ✓0 <0, · ) · i.e., that every ✓-stable representation is also ✓0-stable. If, moreover, we choose ✓0 so that ✓0 =0with<↵,then (Q, ↵)willbesmooth,sothatweonlyhave · 6 Mq,✓0 to show that (Q, ↵) (Q, ↵)isbirational. Mq,✓0 !MQ,✓ By Corollary 3.5.6,foreachconnectedcomponentof (Q, ↵), there is a dense Mq,✓ (1) (r) r (i) stratum of the form (1, ; ...;1, )withp(↵)= i=1 p( ). We need to show that each representation ⇢ in such a stratum is in theP boundary of a unique GL(↵)- ✓0 s q orbit in Rep (⇤ (Q),↵). Equivalently, we must show that there is a unique ✓0- stable representation ⇢0 up to isomorphism such that ⇢ is the ✓-polystabilisation of

91 ⇢0.

We will prove the statement by induction on ↵,withrespecttothepartialordering . First, applying admissible and ( 1)-reflections, we reduce to the case that ↵ is  in the fundamental region. Indeed, it is clear from Theorem 3.1.17 and Proposition 3.5.10 that applying these reflections causes no harm. Note that each ( 1)-reflection will modify stratum types by removing a real root from the type; once we are in the fundamental region no real roots will appear.

We ﬁrst show uniqueness. If ⇢0 is as above, then suppose that there is an exact sequence of ✓-semistable representations of the form 0 ⇢0 0. By ! ! ! ! our assumptions, the dimension vectors of and are sums of complementary subsets of the (i).ItfollowsfromtheproofofTheorem3.5.16 (using [Cra02, Section 8]) that there is a corresponding decomposition of ↵ as in Theorem 3.5.16, of type (b1) or (b2), with the following property: in type (b1), ↵(1) := ↵ and |J ↵(2) := ↵ are the dimension vectors of and ,ineitherorder;or,intype |K (b2), ↵(1) := and ↵(2) := ↵ are these dimension vectors, again in some order. Note that the ordering of the ↵(i) is ﬁxed by the conditions that dim ✓<0and · dim ✓>0. Next, since (↵(1),↵(2))= 1, it follows from Proposition 3.1.3 that · 1 1 dim Ext ( , )=dimExt(, )=1.Sotheextension⇢0 is uniquely determined, up to isomorphism, from and .

We claim that and are uniquely determined from their dimension vectors up to isomorphism. We give the argument for ;theonefor is symmetric. Let (i) C 1,...,r be a subset of indices such that dim = i C .(Thissetis ✓{ } 2 unique except in case (b2) with dim = ↵(2) = ↵ .) There exists a unique P (i) (1) (2) i C such that ( , dim )= 1(since(↵ ,↵ )= 1). Now, define ✓00 := 2 ✓0 (✓0 dim )e ,wherev is the unique vertex in supp dim which has |supp dim · v non-zero Cartan pairing with supp dim (so (dim ) =1).Byconstruction,✓00 v · (j) (j) dim =0=✓00 dim .Moreover,✓00 = ✓0 for j C i .Weclaim · · · 2 \{} that is ✓00-stable. By definition of ✓0-stability, every nonzero submodule ⌘ of (i) satisfies ✓0 dim ⌘<0. Now, if dim ⌘,then✓00 dim ⌘ = ✓0 dim ⌘<0. On · 6 · · the other hand, if ⌘ is a proper submodule of with (i) dim ⌘,then /⌘ is  anonzeroquotientmodulewith(i) dim( /⌘). Then (dim( /⌘), dim )=0. 6 By Proposition 3.1.3,Ext1(, /⌘)=0.Therefore,wehaveanexactsequence

0 ⌘ ⇢0 ( /⌘) 0. As a consequence, /⌘ itself is a quotient module ! ! ! !

92 of ⇢0.Itfollowsthat✓0 dim( /⌘) > 0. Therefore, ✓00 dim( /⌘)=✓0 ( /⌘) > 0. · · · Therefore, ✓00 ⌘<0. We conclude that is ✓00-stable, as desired. By induction on · ↵, is then uniquely determined up to isomorphism.

This completes the proof of uniqueness. We move on to existence, which is similar. Begin with a decomposition given by Theorem 3.5.16 of type (b1) or (b2). Let us keep the notation ↵(1),↵(2) deﬁned above. The same construction as above yields (1) (2) (i) (i) (i) modiﬁcations ✓ ,✓ of ✓0 such that ✓ ↵ =0.Byinductionwecantake✓ - · stable representations , of dimension vectors ↵(i).Thensince(↵(1),↵(2))= 1, 1 1 dim Ext (, )=dimExt( ,) = 1. Assume that ✓0 <0, otherwise swap and · .Thenformanon-trivialextension0 ⇢0 0. The same computation ! ! ! ! as above guarantees that ⇢0 is ✓0-stable.

Corollary 3.5.24. In the situation of the proposition, the normalisation of (Q, ↵) Mq,✓ is a symplectic singularity.

Proof. This follows since we have constructed a symplectic resolution (see Proposi- tion 3.3.5 or Remark 3.3.6).

Remark 3.5.25. Note that the main step of the proof is to show that (Q, ↵) Mq,✓0 ! (Q, ↵) is birational for suitable ✓0 ✓.Forthis,wedidnotneedthehypothesis Mq,✓ that ↵ is q-indivisible. On the other hand, by Theorems 3.5.13 and 3.5.16,when ↵ ⌃ ⌃ , ↵ is actually indivisible (not merely q-indivisible). For ↵ ⌃ , 2 q,✓ \ q,✓ 2 q,✓ the birationality statement is Corollary 3.1.24,whichiseasy.(Moreover,thefull g statement of Theorem 3.5.22 was established for ↵ ⌃ in Section 3.3.1.) So 2 q,✓ it does not really add anything to state the birationality property without the q- indivisibility hypothesis.

3.5.5 Symplectic resolutions for general ↵

Theorem 3.5.26. Assume that (Q, ↵) is non-empty and that the decomposition Mq,✓ of Theorem 3.5.17.(ii) has no elements (i) of the forms (a) (i) =2 for N 2 q,✓ and p()=2, or (b) (i) = m for m 2 and ⌃iso. Then: 2 q,✓

• The normalisation of q,✓(Q, ↵) is a symplectic singularity; M (i) (i) • Each factor q,✓(Q, ) with / ⌃q,✓ admits a symplectic resolution; M 2

93 • If for any factor (i) there exists a ✓-stable representation of dimension (i) = 1 (i) with m 2, then (Q, ↵) does not admit a symplectic resolution. In m Mq,✓ fact, it has an open, singular, terminal, factorial subset.

Proof. The first statement follows if we show that the normalisation of each factor (Q, (i))isasymplecticsingularity.Forthefactorssuchthat(i) is in ⌃ , Mq,✓ q,✓ this is a consequence of Theorem 3.0.7.For(i) / ⌃ ,afterapplyingadmissible 2 q,✓ reflections it follows from Theorem 3.5.16 that it is indivisible. Hence (i) is itself indivisible. By our assumptions, (i) is flat. The result then follows from Theorem 3.5.22.Thisalsoprovesthesecondstatement.

We proceed to the third statement. Under the hypotheses, since we have excluded the isotropic and (2, 2)-cases, an open subset of (Q, (j))isfactorialterminal Mq,✓ singular by Theorem 3.0.7 (see Theorem 3.3.17). Hence so is an open subset of (Q, ↵), which therefore does not admit a symplectic resolution. Mq,✓

3.5.6 Classifications of symplectic resolutions of punctured character varieties

Here, we combine the results of this section and Theorem 3.4.1 to get a classiﬁcation of all the character varieties of punctured surfaces which admit a symplectic resolution, modulo the conjectural results of the (2, 2)-cases. As explained in Section 3.2, in order to get such a result, it suces to consider multiplicative quiver varieties of crab-shaped quivers, where the parameter q and the dimension vector ↵ are chosen in an appropriate way; see Theorem 3.2.6.

Q0 Let Q be a crab-shaped quiver, q (C⇥) and ↵ Nq,✓ and consider the corre- 2 2 sponding quiver variety (Q, ↵). Then, if ↵/ (Q), we can apply the algorithm Mq,✓ 2F of Theorem 3.5.17 and obtain a decomposition where the dimension vectors of the factors are in the fundamental region (such dimension vectors are the connected components of the reﬂection of ↵). Moreover, note that, in the crab-shaped case, all the vector components not containing the central vertex are Dynkin quivers of type A:therefore,theassociatedmultiplicativequivervarietyisjustapoint.This implies that we can assume, without loss of generality, that ↵ be sincere (↵i > 0for all i) and in the fundamental region. After having performed this reduction, we can prove the following.

94 Corollary 3.5.27. Let Q be a crab-shaped quiver and ↵ N a sincere vector 2 q,✓ in the fundamental region. Further assume that (Q, ↵) is not one of the following cases:

(a) = 1 ↵ N and (Q, ) is one of the quivers in Theorem 3.4.1 and Theorem 2 2 q,✓ 3.4.3;

(b) Q is ane Dynkin (of type A˜0 (i.e., the Jordan quiver with one vertex and

one arrow), D˜ 4 or E˜6, E˜7, E˜8) and ↵ is a q-divisible multiple of the indivisible imaginary root of Q.

Then:

• The normalisation of q,✓(Q, ↵) is a symplectic singularity; M

• If ↵ is q-indivisible, q,✓(Q, ↵) admits a symplectic resolution; M • If ↵ = m for m 2 and there exists a ✓-stable representation of ⇤q(Q) of dimension , then (Q, ↵) does not admit a symplectic resolution (it Mq,✓ contains an open singular factorial terminal subset);

• In the case that ↵ is q-divisible, the condition of the preceding part is always satisﬁed, except possibly in the case: (c) Q = Qe is an ane Dynkin [ {⇤} quiver Qe of type A˜ , D˜ , E˜ , E˜ , or E˜ together with an additional vertex 0 4 6 7 8 {⇤} and an additional arrow from this vertex to one with dimension vector 1 in , and ↵ has the form (p, p`) for p 2 a prime, ` q-indivisible. Here denotes the indivisible imaginary root of Qe.

Remark 3.5.28. In the ﬁnal part of the corollary, expectation (*) from the introduction predicts that the exception indeed fails to satisfy the conditions of the preceding part. Nonetheless, we believe that, also in this case, there should not exist a projective symplectic resolution; this would be implied by Conjecture 4.5.1 in the appendix together with the consequence that follows it, by [BS16b,Theorem1.5].

Proof of Corollary 3.5.27. By Theorem 3.5.17,weknowthat↵ is ﬂat unless it is a positive integral multiple of an isotropic root, excluded in case (b). Then the ﬁrst and third statements follow from Theorems 3.0.7 and the second from 3.5.22.Forthe fourth statement, we apply Theorem 3.5.16.toseethat,inthecrab-shapedcase,the

95 dimension vector can only be in the fundamental region but not in ⌃q,✓ if the quiver is a framed ane Dynkin quivers of types A˜0, D˜ 4, E˜6, E˜7,andE˜8,andthedimension vector is (1,`). Thus only prime multiples of this vector can be q-divisible but have no factor in ⌃ .Itthenremainsonlytoshowthat (Q, ) = for ⌃ . q,✓ Mq,✓ 6 ; 2 q,✓ In the star-shaped case, this is a consequence of [CS06,Theorem1.1].Inthecrab- shaped case with g>0 loops at the central vertex, it suces by Theorem 3.2.6 and Remark 3.2.7 to show that, for all conjugacy classes 1,..., m GL(n, C) C C ⇢ with product of determinants equal to one, there exists a solution to the equations [A ,B ] [A ,B ]=C C for C .Thisfollowsbecausethereisasolution 1 1 ··· g g 1 ··· m i 2Ci to the equation [X1,Y1]=C for arbitrary C SL(n, C)(by[Tho61,Theorems1, 2 2]).

Remark 3.5.29. The assumptions made in the above theorem relate to the fact that in cases (a)and(b), it is still unknown whether a symplectic resolution exists, as this problem seems to be solvable only through a deep understanding of the local structure of the variety. Nonetheless in case (a), we expect such symplectic resolutions to exist and to be constructible by using analogous techniques to the ones used by Bellamy and Schedler in [BS16b,Theorem1.6](seeRemark3.0.10).

3.5.7 Proof of Theorems 3.0.1 and 3.0.4

Theorems 3.0.1 and 3.0.4 follow from Corollary 3.5.27,togetherwithTheorem3.2.6, as follows.

First, we claim that q-divisibility for the collection of conjugacy classes coincides C with the same-named property for the dimension vector ↵ of the corresponding crab- shaped quiver. To see this, ﬁrst note that m indeed corresponds to m ↵.So ·C · we only have to show that the condition that det =1isequivalenttoq↵ =1. i Ci This is true by construction. Q Next, we claim that, for g =0,thecondition` 2n of Theorem 3.0.1 is equivalent to the condition that ↵ (Q), whereas for g>0, we have ↵ (Q)unconditionally. 2F 2F By the chosen ordering of the ⇠ ,wehave(↵, e ) 0foralli Q except possibly i,j i  2 0 the node. There, the condition ` 2n is equivalent to (↵, e ) 0. On the other i  hand, when g 1, then as there is a loop at the node, it is automatic that (↵, e ) 0 i  for i Q the node, and hence in this case ↵ (Q)automatically. 2 0 2F

96 We now claim that the dimension of (g, k, )equals2p(↵)whenthequiverisnot X C Dynkin or ane Dynkin and moreover ` 2n or g 1. This follows from Theorem 3.5.16 and Theorem 3.2.6 (see also Remark 3.2.7), provided that ↵ is not both q- divisible and isotropic. However, the latter conditions, for ↵ (Q), are equivalent 2F to saying that the graph is ane Dynkin and ↵ is q-divisible. C With the preceding claims established, we proceed to the proof of the theorems. Note that applying reﬂections as earlier in this section preserves the property that adimensionvectorisonecorrespondingtoacharactervariety(byTheorem3.2.6). So we can always reduce to case that ↵ (Q), unless we end up with something 2F with Dynkin support (hence (g, k, )isapoint)orsomethingwhere↵ becomes X C negative (hence (g, k, )isempty).ThisprovestheﬁrstpartofTheorem3.0.1. X C The remaining assertions of the theorems follow from the above claims, which allow us to translate Corollary 3.5.27 into the given results via Theorem 3.2.6.

97 4 Conclusion and future directions

The last chapter of this thesis contains some open questions which naturally arise from the study carried out in Chapters 2 and 3:someofsuchquestionsconcernthe geometry of multiplicative quiver varieties for dimension vectors not in the region

⌃q,✓; others are more strictly related to the moduli spaces of the nonabelian Hodge correspondence and to a conjectural generalisation of the Isosingularity theorem to parabolic Higgs bundles. On a more speculative basis, at the end of the chapter we outline a general setting and pose a number of questions which should generalise the work of carried out in this thesis and that of other authors, e.g. [AS15; BS16b; KL07]: indeed, it seems that many of the techniques exploited in these works are particular instances of theorems which conjecturally hold in the context of moduli spaces of semistable objects in 2-Calabi–Yau categories, under suitable hypotheses. This assertion is motivated also by the work of Bocklandt, Galluzzi and Vaccarino, [BGV16], who studied moduli spaces of representations 2-Calabi–Yau algebras and proved that such varieties is singular at strictly semisimple representations, provided that the simple locus is non-empty, see also [Boc08; Van15].

Before getting into these issues, we recall what is known about non-emptiness of multiplicative quiver varieties, which is unfortunately not much: although we do

98 not have a lot to say about this issue, it is an obvious and relevant open question which we have already mentioned throughout the previous chapter.

4.1 Non-emptiness of multiplicative quiver varieties

As explained in the previous chapter, one of the subtleties of in the study of multiplicative quiver varieties is represented by the fact that it is not known in general when they are non-empty (nor how many connected components they have). On the other hand, there are special cases in which non-emptiness can be shown. For example, when q =1,then,foranyquiverQ and any vector ↵ NQ0 ,thezero 2 representation is a suitable element of Rep(⇤q,↵), since the invertibility condition is automatically satisﬁed as well as the multiplicative preprojective relation. Thus (Q, ↵) = .Moregenerally,foreveryrealroot R+ ,thenbyapplyingre- M1,0 6 ; 2 q,✓ ﬂection sequences as in Section 3.5.1 (see also the discussion after Corollary 3.5.18), we conclude that (Q, ) = . As a result, if ↵ can be expressed as a sum of real Mq,✓ 6 ; roots in R+ (not necessarily coordinate vectors) then (Q, ↵) = . Another, less q,✓ Mq,✓ 6 ; trivial, case in which we are guaranteed that (Q, ↵) is non-empty is when Q is Mq,✓ star-shaped and ↵ N :then,inordertoconstructarepresentationofdimension 2 q,✓ ↵,onecanusethecorrespondenceofbetweenmultiplicativequivervarietiesand character varieties of punctured surfaces and, further, use parabolic bundles and monodromy, see [Cra13](and[Yam08]for✓ =0).Inthecrab-shapedcase,wethen 6 also know that (Q, ↵) = for ↵ N ,usingThompson’sthesis[Tho61], as Mq,✓ 6 ; 2 q,✓ explained in the proof of Corollary 3.5.27.

We can also ask when the stable locus s (Q, ↵) = . Note that an answer to this Mq,✓ 6 ; question for all ↵ also answers the question of non-emptiness of the entire locus, since every point in (Q, ↵)isrepresentedbyapolystablerepresentation.More Mq,✓ explicitly, (Q, ↵) = if and only if ↵ can be represented as a sum of roots Mq,✓ 6 ; ↵(i) for which s (Q, ↵(i)) = . Note that, when ↵ ⌃ ,thennon-emptinessof Mq,✓ 6 ; 2 q,✓ (Q, ↵)isequivalenttothatof s (Q, ↵), by Proposition 3.1.22.Ourexpec- Mq,✓ Mq,✓ tation (*) says that s (Q, ↵) = implies ↵ ⌃ . Mq,✓ 6 ; 2 q,✓

99 4.2 Refined decompositions for multiplicative quiver varieties

Recall Crawley-Boevey’s canonical decomposition in the additive case (Theorem 3.5.2,Lemma3.5.1). It is useful to ask to what extent such a decomposition holds in the multiplicative setting, reﬁning the one of Theorem 3.5.17.Let(i),↵(i,j) be as in Theorem 3.5.17,andgrouptogetherthe↵(i,j) that are equal, yielding distinct (i,j) each occurring r 1 times. Note that, when (i,j) is anisotropic, then r =1, i,j i,j since r (i,j) ⌃ , by the uniqueness of the decomposition in Theorem 3.5.17.(i). i,j 2 q,✓ Question 4.2.1. Do we have a decomposition as follows:

(Q, ↵) = Sri,j (Q, (i,j))? (4.1) Mq,✓ ⇠ Mq,✓ i,j Y

The following proposition partly answers this question modulo expectation (*).

Proposition 4.2.2. If (*) holds, then the decomposition of Theorem 3.5.17.(iv) reﬁnes to one of the form

(Q, ↵) = (Q, r (i,j)). (4.2) Mq,✓ ⇠ Mq,✓ i,j i,j Y Moreover, in this case, the direct sum map

Sri,j (Q, (i,j)) (Q, r (i,j))(4.3) Mq,✓ !Mq,✓ i,j is surjective.

Proof. It suces to decompose each of the (Q, (i)). By Proposition 3.5.16,the Mq,✓ ﬁrst statement follows by the arguments of [Cra02, Section 5] verbatim, replacing simple representations by ✓-stable ones. For the second statement, if ri,j > 1, (i,j) (i,j) then is isotropic. Then, the canonical decomposition of ri,j appearing (i,j) in Theorem 3.5.17.(i) is just as a sum of ri,j copies of .Thusthestatement follows from Theorem 3.5.17.(i) and expectation (*), since every representation in (Q, r (i,j))isrepresentedbyapolystableone. Mq,✓ i,j

Therefore, modulo (*), Question 4.2.1 reduces to the following one:

Question 4.2.3. Is the natural map (4.3) an isomorphism?

100 Example 4.2.4. Suppose that (i) is the following dimension vector supported on aframedtypeE6 quiver:

e n

2n (4.4)

n 2n 3n 2n n 1

Then, by the star-shaped case of Theorem 3.2.6 (proved in [CS06,Section8]),the variety (Q, (i))isisomorphictothecharactervarietyofrank3n local systems M1,0 1 on the three-punctured sphere ⌃0,3 = P 0, 1, with unipotent monodromies: \{ 1} about the first two punctures, there should be n Jordan blocks of size three (or some refinement), and about the third puncture, there should be n 2 Jordan blocks of size three, one Jordan block of size four, and one of size two (or some refinement). On the other hand, (Q, )isthecharactervarietyofrank3localsystemson M1,0 ⌃0,3 with arbitrary unipotent monodromies. Question 4.2.3 then asks whether the first variety is isomorphic to the n-th symmetric power of the second; it does not seem so obvious that this should be the case.

In the additive case, Question 4.2.3 has a positive answer. One of the diculties in trying to adapt the proof of this in [Cra02,Section3]isthat,inthemultiplicative case, it is no longer guaranteed that one of the components of (i,j) equals one (since (i,j) need only be q-indivisible, not indivisible). However, if (*) holds and Conjecture 4.5.1 below holds, then it would follow from the ´etale-local identiﬁcation of the multiplicative quiver variety with an additive one that (4.3)isindeedan isomorphism. Thus we expect a positive answer to Question 4.2.3 as well.

Note that the proof of Theorem 3.5.2 ([BS16b,Theorem1.4]),intheadditivecase, relied on hyperk¨ahler twistings, for which one needs to assume that the parameter is real. In fact, some of the issues we face are not yet resolved in the general additive case where both /R and ✓ =0. 2 6

101 4.3 Symplectic resolutions and singularities

In view of our results and the ﬂexibility of symplectic singularities, as well as the relationships between multiplicative and additive quiver varieties, we propose the following:

Conjecture 4.3.1. Every multiplicative quiver variety is a symplectic singularity.

Note that a product of Poisson varieties is a symplectic singularity if and only if each of the factors is (because normality and being symplectic on the smooth locus have this property, and in the deﬁnition of symplectic singularity it is equivalent to check the extension property for one or all resolutions of singularities). Therefore the conjecture reduces to the case of factors appearing in Theorem 3.5.17,andif iso (*) holds, to the case ↵ ⌃q,✓ N 2⌃q,✓ by Proposition 4.2.2.IfQuestion4.2.3 2 [ additionally has an armative answer, (e.g., in the presence of Conjecture 4.5.1), then we can furthermore reduce to the case ↵ ⌃ . 2 q,✓ Next, we ask to what extent the property of having a symplectic resolution is equivalent to the same property for the factors.

Question 4.3.2. (i) Is it true that (Q, ↵) admits a symplectic resolution if and Mq,✓ only if each of the factors (Q, (i)) does? Mq,✓ (ii) Suppose that (*) holds. Is it true that (Q, ↵) admits a symplectic resolution Mq,✓ if and only if each of the factors (Q, r (i,j)), appearing in Proposition 4.2.2, Mq,✓ i,j does?

Note that, when r > 1, then (i,j) ⌃iso ,andhenceisq-indivisible. Therefore, in i,j 2 q,✓ this case, (Q, (i,j))hasasymplecticresolutionbyvarying✓,byLemma3.1.16 Mq,✓ and Corollary 3.1.24;sinceitisasurface,sodoesitsri,j-th symmetric power, by the corresponding Hilbert scheme. Therefore, if Question 4.2.3 has a positive answer (e.g., in the presence of Conjecture 4.5.1), we can ignore these factors in (ii) above, and only consider the ones with ri,j = 1. Also notice that all q-indivisible factors, including those with (i,j) / ⌃ , admit symplectic resolutions. Also, if any factor 2 q,✓ (i,j) appears which is a 2multipleofthedimensionvectorofa✓-stable representation, there can be no symplectic resolution of (Q, ↵), nor of (Q, (i,j)). Mq,✓ Mq,✓ So again, for the question (ii), it is enough to consider only the factors (i,j) which

102 are anisotropic, q-divisible, and not a 2 multiple of the dimension vector of a ✓-stable representation.

4.4 Moduli of parabolic Higgs bundles and the Isosingularity The- orem

In this section, the aim is to pose some conjectures on possible generalizations of the main theorems of Chapter 2 to the case of parabolic Higgs bundles, in the light of the results of Chapter 3 and, in particular, those regarding the correspondence between character varieties of punctured surfaces and multiplicative quiver varieties for crab-shaped quivers: indeed, it is a natural question to ask whether an analogue of the main theorems of Chapter 2 holds for the Dolbeault moduli spaces defined on complex curves with punctures, which turn out to be the moduli spaces of parabolic Higgs bundles. This question fits well into the framework of this thesis also because of the parabolic version of the nonabelian Hodge correspondence, which we briefly describe below.

Motivated by the work of Simpson, [Sim90], we recall ﬁltered local systems, following [Yam08, §4], which gives a slightly di↵erent but nonetheless equivalent deﬁnition from the one given in Simpson, [Sim90].

Definition 4.4.1. [Yam08,Definition4.5]LetX be a compact Riemann surface and let D X be a finite subset. Let L be a local system on X D.Foratuple ⇢ \ of non-negative integers l =(lp)p D,afiltered structure on L of filtration type l is a 2 tuple (Up,Fp)p D,whereforeachp D: 2 2

(i) U is a neighbourhood of p in X (we set U ⇤ := U p ); and p p p \{ }

(ii) Fp is a ﬁltration

L = F 0(L) F 1(L) F lp (L) F lp+1(L)=0 |Up⇤ p p ··· p p

by local subsystems of L . |Up⇤

Two ﬁltered structures (Up,Fp)p D, (Up0 ,Fp0)p D of the same ﬁltration type are equiv- 2 2 alent if for each p D,thereexistsaneighbourhoodV U U 0 of p such that 2 p ⇢ p \ p Fp and Fp0 coincide on Vp⇤.AlocalsystemL together with an equivalence class of

103 filtered structures F =[(Up,Fp)p D]iscalledafiltered local system on (X, D)of 2 filtration type l.

From [Yam08]onehasalsothefollowingdeﬁnitionof(semi-)stability.

Definition 4.4.2. [Yam08, Definition 4.6] Let (L, F )beafilteredlocalsystemon (X, D)offiltrationtypel.Let =(j p D, j =0,...,l ) be a tuple of rational p | 2 p i j numbers satisfying p <p for any p and i

Yamakawa established a correspondence between semistable filtered local systems and multiplicative quiver varieties of star-shaped quivers. This is, as mentioned, aparticularcaseofthecorrespondenceoutlinedinSection3.2 of Chapter 3.To shed more light on this correspondence, we spell out the correspondence between the parameters q, ↵, ✓ defining a multiplicative quiver variety (Q, ↵)ofastar Mq,✓ shaped quiver and the weights of a filtered local system (L, F ): start with a crab- shaped quiver Q with vertex set Q0 = 0, (i, j)i 1,...n ,j 1,...,l –i.e.Q has n legs, { 2{ } 2{ i}} each of which has length li,fori =1,...,n.Moreover,let↵ be a dimension vector,

Q0 Q0 Q0 ↵ N , ⇠ (C⇤) atupleofnon-zerocomplexnumbersand Q atupleof 2 2 2 rational numbers. Then, on one side, one can consider ﬁltered local systems (L, F ) 1 1 on (P , p1,...,pn ), where pi,i =1,...,n are pairwise distinct points in P ,with { } stability parameter and such that:

1. rank(L)=↵0,

j 2. dim Fpi (L)=↵i,j,

j j+1 3. the local monodromy of Fpi (L)/Fpi (L)aroundpi is given by the scalar mul- j tiplication by ⇠pi for all i, j.

On the other side, one can consider the multiplicative quiver variety (Q, ↵), Mq,✓

104 where Q and ↵ are as above and q and ✓ are given as

0 1 j 1 j q0 := (⇠pi ) ,qi,j = ⇠pi /⇠pi i Y

i,j ✓i,j↵i,j j j 1 ✓0 := ,✓i,j = pi pi . P ↵0 The other main concept in the Nonabelian Hodge correspondence on noncompact curves is that of parabolic Higgs bundle, which we recall below (note that, in [Sim90] the term ﬁltered Higgs bundle is used instead).

Deﬁnition 4.4.3. Let X and D be a compact Riemann surface and a reduced divisor on X respectively. Let E X be a holomorphic vector bundle on X.A ! parabolic structure on E is the datum of weighted ﬂags (Ei,p,↵i,p)p D 2

E = E E E =0, p 1,p ◆ 2,p ◆···◆ l+1,p

0 ↵ < <↵ < 1,  1,p ··· l,p for each p D.Amorphism of parabolic vector bundles is a morphism of holomor- 2 phic vector bundles which preserves the parabolic structure at every point p D. 2

Deﬁnition 4.4.4. Given a parabolic bundle (E,(Ei,p,↵i,p)p D), its parabolic degree 2 is deﬁned to be

pardeg(E)=deg(E)+ mi(p)↵i,p, p D i X2 X where m (p)=dimE dim E is called the multiplicity of ↵ . i i,p i,p+1 i,p Remark 4.4.5. Given the notion of parabolic degree, stability and semistability of aparabolicbundlearedeﬁnedasinthecaseofvectorbundles,usingtheparabolic degree in place of the ordinary degree, see [LM10, §2] for more details and an outline on some geometric properties of the corresponding moduli spaces.

Deﬁnition 4.4.6. A parabolic Higgs bundle on (X, D)isgivenbythedatumof aparabolicbundle(E,(Ei,p,↵i,p)p D)togetherwithameromorphicmap:E 2 ! E K with poles of order at most 1 at the points p D.Theresidueofat ⌦ X 2 marked points is assumed to preserve the corresponding ﬁltration.

From the Riemann-Hilbert correspondence, we know that representations of the fundamental group of a punctured surface with ﬁxed monodromies correspond bi-

105 jectively to ﬁltered local systems. Moreover, in [Sim90], the following theorem is proved.

Theorem 4.4.7. [Sim90, Theorem, p. 718] There is a one-to-one correspondence between (stable) ﬁltered local systems and (stable) parabolic Higgs bundles of degree zero.

Even though we will not go into the details of this correspondence, we shall at least explain how it works at the level of parameters ↵ and .Tothispurpose,letX be a compact Riemann surface and D X aﬁnitesubsetofdistinctpointsofX. ⇢ Fixing p D,considerthesets 2

j (, ↵p) C [0, 1) the action of ResponEj,p/Ej 1,p has an eigenvalue , { 2 ⇥ | }

k k+1 k the monodromy of Fp (L)/Fp (L) (⇠,p ) C⇥ R . ( 2 ⇥ | along a simple loop around p has an eigenvalue ⇠ ) Then the correspondence between these two sets is explicitly given by (, ↵) 7! (⇠,), where := ↵ , ⇠ := exp( 2⇡p 1). <

Another fundamental result of Simpson that is crucially used in Chapter 2 is the Isosingularity theorem, which, roughly speaking, states that the moduli spaces of the Nonabelian Hodge Theorem in the compact case, i.e. with no punctures, are ´etale-locally isomorphic at corresponding points. It is still not known whether the same result holds in the noncompact case.

Conjecture 4.4.8. The Isosingularity theorem holds between the moduli space of semistable ﬁltered local systems for ﬁxed parameters and the moduli space of semistable parabolic Higgs bundles of degree zero with corresponding parameters.

A possible approach to the above conjecture would be to study the deformation theory of parabolic Higgs bundles and representations of the fundamental group of punctured surfaces, generalising Simpson’s arguments [Sim94b, §10]: one would have to apply Goldman-Millson deformation theory, [GM88], and compute the dif- ferential graded Lie algebras governing the deformations of parabolic Higgs bundles and representations of ⇡ (X S)–inthenotationofthepreviouschapter–and 1 \ understand whether they are quasi-isomorphic.

106 From the above conjectural result, in combination with the results proved in Section 3.3,onewouldhavethefollowing(conjectural)theorem.

Conjecture 4.4.9. The moduli space of semistable parabolic Higgs bundles with ﬁxed parameters is a symplectic singularity, admitting a symplectic resolution if and only if the corresponding character variety admits a symplectic resolution.

Apossiblestrategytoprovetheresultslistedabovewouldbetostudythelocal structure of moduli spaces of (semistable) objects in Calabi–Yau categories. More details are provided in the next subsection.

4.5 Moduli of representations of 2-Calabi–Yau algebras

As mentioned multiple times in this thesis, the problem of studying the singularities of a variety can be carried out by analysing the local structure of the variety itself around a point. This method is powerful in certain cases, e.g. when one is able to prove the existence of some étale-local isomorphism between the variety of interest and another variety whose singularities are well known: for example, this was carried out by Kaledin and Lehn in [KL07] and later by Arbarello and Saccàin [AS15], where they prove that, given a strictly semistable bundle in the moduli space of semistable sheaves on a K3 surface with a fixed non generic polarization, there exists an étale neighbourhood around that point that is isomorphic to an ane quiver variety, which depends on the point itself. Similar computations, which find their inspiration from [KL07], were also performed by the Bellamy and Schedler in [BS16b]inthecontext of quiver and character varieties.

The fact that such a technique can be used and gives the same results in so many apparently di↵erent situations suggests that these are indeed particular cases of a series of theorems which should apply in much greater generality, namely in the context of 2-Calabi–Yau categories – see the next sections for this topic.

In the work [BGV16]theauthorscarryoutadetailedstudyofthedeformation theory of representation spaces of 2-Calabi–Yau algebras – ﬁrst deﬁned by Ginzburg in [Gin06], see below – and they show that among all semisimple representations, the ones that correspond to smooth points are precisely the simple ones. One may ask the question whether multiplicative preprojective algebras are 2–CY, in order

107 to use such techniques. On the other hand, it is known that, when Q is a Dynkin quiver and q =1,thereisanisomorphism

1 1 ⇤ (Q) ⇠= ⇧ (Q), between the multiplicative preprojective algebra and the additive preprojective algebra, as shown in [Cra13,Corollary1].Moreover,giventhatsomeadditivepreprojec- tive algebras have inﬁnite homological dimension, the above isomorphism suggests that, in general, multiplicative preprojective algebras are not 2-Calabi–Yau. On the other hand, a conjectural statement can be made for the case of non-Dynkin quivers.

Q0 Conjecture 4.5.1. Let Q be a connected non-Dynkin quiver and q (C⇤) . Then 2 ⇤q(Q) is a 2-Calabi–Yau algebra.

We recall the deﬁnition of a homologically smooth algebra and then Ginzburg’s deﬁnition of Calabi–Yau algebra, [Gin06].

Definition 4.5.2. Let A be an associative algebra over a field K. A is said to be homologically smooth if A admits a finite resolution by finitely-generated projective (left) Ae-modules, where Ae = A Aop. ⌦ Definition 4.5.3. Let A be a homologically smooth algebra over K. A is said to be a d-Calabi–Yau algebra (d–CY, for short), if there are Ae-modules isomorphisms

Ai= d, i e ExtAe (A, A ) ⇠= 80 i = d. < 6 : From this deﬁnition, we see that, in the case at hand, one would have to prove that:

• ⇤q is homologically smooth;

• the condition in the above deﬁnition is satisﬁed when d =2.

From [CS06,Lemma3.1],onehasthatthereexistsanexactsequenceofﬁnitely generated projective (⇤q)e-modules

P ↵ P P ⇤q 0. 0 ! 1 ! 0 ! !

108 Then, if ↵ were injective, ⇤q would be homologically smooth. Moreover, the injectivity of ↵ would also ensure that the homological dimension of ⇤q satisﬁes hdim(⇤q) 3.  With such an exact sequence at hand, one could compute the Ext groups and check whether the 2–CY condition is satisﬁed. In order to do that, one can take the q e dual sequence by applying the controvariant functor ( )_ := Hom q e ( , (⇤ ) ) (⇤ ) and obtain the complex,

_ ↵_ 0 P _ P _ P _ 0, ! 0 ! 1 ! 0 ! from which one can conclude that

0 q q e Ext(⇤q)e (⇤ , (⇤ ) )=ker_,

1 q q e ker ↵_ Ext(⇤q)e (⇤ , (⇤ ) )= , Im _ and 2 q q e P0_ Ext(⇤q)e (⇤ , (⇤ ) )= . Im ↵_ 2 q q e q From [CS06,Lemma3.5],onehasthatExt(⇤q)e (⇤ , (⇤ ) )=⇤ and, assuming the in- i q q e jectivity of ↵,onehasthatallthehigherextgroupsvanish,i.e. Ext(⇤q)e (⇤ , (⇤ ) )= 0fori 3. Even though, from [CS06, §3], the maps ↵, and have a very explicit description, trying to prove the 2–CY condition directly for a generic non-Dynkin quiver is highly non-trivial and, at the moment, seems out of reach. An alternative approach to this problem might be to use the induction procedure used by Etingof and Eu in [EE07]: the goal of that paper was to provide a closed formula the Hilbert series of the preprojective algebra ⇧(Q)ofaconnectednon-DynkinquiverQ over any ground ﬁeld K and to prove that, under such hypotheses for the quiver Q,⇧(Q) is a Koszul algebra; their techniques to prove such results involve considering the so-called partial preprojective algebras,⇧(Q): these are algebras such that J I J ⇢ is a subset of the set I of vertices of Q and, by deﬁnition, ⇧J (Q)isthequotient of the path algebra of CQ by the preprojective algebra relations imposed only at vertices not contained in J.Theinductionstepisonthenumberofverticesofthe quiver Q and relies heavily on the use the algebras ⇧J (Q). A possible approach

109 to prove injectivity of the map ↵ and the isomorphisms of Ext groups listed above would be to use this strategy, deﬁning partial multiplicative preprojective algebras and, therefore, proving the base cases given by extended Dynkin quivers of type A,˜ D˜ and E˜ and the induction step.

Assuming the above conjecture, then [BGV16,Theorem6.3,6.6]impliesthefollow- ing for ✓ = 0: given a dimension vector ↵ ⌃ and a point x (Q, ↵), then 2 q,✓ 2Mq,✓ formally locally around x, we have an isomorphism

\(Q, ↵) = M \(Q ,↵) Mq,✓ x ⇠ 0,0 0 0 0 for some appropriate quiver Q0 and dimension vector ↵0.Ifwecangeneralisethisto arbitrary ✓ and prove the conjecture, there would be many interesting consequences. First, it would make it possible to handle the (2, 2)-case, where one may construct asymplecticresolutionbyﬁrstperformingGIT(replacing✓ by suitably generic ✓0) and then performing a blow-up of the singular locus. Moreover, this result would imply normality for (Q, ↵), without any assumption on ↵ and ✓. Mq,✓

4.6 Character varieties and Higgs bundles for arbitrary groups

From Simpson’s work, [Sim94a; Sim94b], we know that the nonabelian Hodge correspondence holds for a general reductive algebraic group G, i.e. when we consider

G-valued representations if ⇡1(X)andprincipalG-Higgs bundle on the Betti and Dolbeault sides, respectively. Moreover, the problems tackled in this thesis for the case when G = GL(n, C)makesensealsoforarbitraryG.Therefore,aninteresting question would be to analyse whether the results of Chapters 2 and 3 can extend to character varieties of (punctured) Riemann surfaces with representations in arbitrary groups and, via the nonabelian Hodge correspondence, to the moduli spaces of (parabolic) Higgs principal bundles.

Since the results on punctured character varieties contained in Chapter 3 are deduced based on the correspondence described in Section 3.2,whichheavilyreliesonthe fact that one considers representations and conjugacy classes inside GL(n, C), it seems unlikely that the techniques used here could be applied in the more general setting of G-representations. Therefore, in order to extend our results, one needs a more general method: a possible solution is outlined in the following section, where

110 we pose some conjectures on the geometry of moduli spaces of (polystable) objects in 2-Calabi–Yau categories.

An alternative approach which is worth pointing out would involve studying in more detail Simpson’s proof of the Isosingularity theorem: in particular, one should look at [Sim94b,Theorem10.5],[Sim94b,Proposition10.5]andtherelatedremarks;indeed, it is from such results that one sees that the spaces (X, G)and (X, G)are MH MB formally locally isomorphic at corresponding points – call them p and q,respectively –becauseitcanbeprovedthatbothformalcompletionsareisomorphictothesame formal scheme, which is essentially the completion at 0 of the categorical quotient Q(G) of a quadratic cone, inside a certain cohomology group in degree one modulo the action of a reductive group. For the Betti moduli space (X, n), it is proved MB in [BS16b, §8.2] that Q(GL(n, C)) is the tangent cone Cq B(X, n)of B(X, n) M M at the point q and, moreover, that it is a conical quiver variety. Therefore, it seems natural to pose the following

Conjecture 4.6.1. The quotient Q(G) is isomorphic to a quiver variety.

If this conjecture were true, then it would be possible to use all the machinery on quiver varieties proved in [BS16b] to study the geometry of the spaces (X, G) M⇤ for = B,dR,H,and,moreover,understandthenatureoftheirsingularitiesand ⇤ the existence of crepant symplectic resolutions.

Note that one possible approach to prove the above conjecture would be to adapt the arguments used in [BS16b, §8.2] and in particular of [BS16b,Theorem8.6],using the general properties of simple G-representations and, for the Dolbeault moduli space (X, G), of stable G-Higgs principal bundles. MH

4.7 Moduli spaces of objects in 2-Calabi–Yau categories

In this last section we describe a setting and a series of conjectures which, if true, would generalise all the results proved in the previous chapters. The underlying idea inspiring such conjectural statements is that it should be possible to extend the techniques used in this thesis to the context of moduli spaces of (polystable) objects in 2–CY categories. Before outlining what we mean by this, we recall the deﬁnition of Calabi–Yau category of dimension d and make a connection with Section 4.5.

111 Essentially, a Calabi–Yau category of dimension d – d–CY category, for short – is the datum of a triangulated category and a special autoequivalence of ,called C C the Serre functor,satisfyingaparticularcondition.d–CY categories are ubiquitous in mathematics and theoretical physics – especially when d =3–and,therefore, there are many references to look at, starting from the original paper [BK90]. We suggest [Kel08]andtherecentwork[Kuz15].

Deﬁnition 4.7.1. ([BK90]) Let be a triangulated category. A Serre functor in T is an autoequivalence of , S : ,suchthatthereareisabifunctorial T T T!T isomorphism

Hom(F, G)_ ⇠= Hom(G, S(F )), for F, G Ob( ). 2 T Deﬁnition 4.7.2. Atriangulatedcategory is a n-Calabi–Yau category if it has T aSerrefunctorS and, moreover, S = [n]forsomen Z.Theintegern is called ⇠ 2 the CY-dimension of . T Remark 4.7.3. An example of a Calabi–Yau category of CY–dimension n is the bounded derived category b(X)ofcoherentsheavesonX,whereX is a projective D Calabi–Yau manifold of dimension n,ifonetakethefunctor

S(F ):=F ! [dim X], ⌦ X as Serre functor, where !X is the canonical sheaf of X and the condition on the Serre functor is satisfied thanks to Serre duality. Remark 4.7.4. It is possible to prove that if A is a n-Calabi–Yau algebra, then the derived category of the abelian category A fmod of finite dimensional modules is n–CY. To see this, consider M and N to be finite dimensional A-modules. Then, what has to be proved is that there is an isomorphism,

i n i Ext (M,N)⇤ ⇠= Ext (N,M) which is natural in M and N.Tothisend,ﬁrstnotethat

i i Ext (M,N) ⇠= HH (A, HomC(M,N)).

i By Van den Bergh duality, for V = HomC(M,N), HH (A, V ) = HHn i(A, V ). ⇠ 112 Now, we dualising and applying adjunction, one has that

Ae n i n i HHn i(A, V )⇤ = HomC(Torn i(A, V ), C) = ExtAe (A, HomC(V,C)) = HH (A, V ⇤). ⇠

n i Now since V ⇤ ⇠= HomC(N,M), as observed before, the last term is Ext (N,M). These operations were all natural in M and N,whichcompletestheproof.

Since the work of Bocklandt et al is on representation varieties of 2–CY algebras, we shall focus on the case in which the CY–dimension of the category is 2. In particular, a generalisation of the arguments presented in this thesis would be to study the (symplectic algebraic) geometry of moduli spaces of polystable objects in aﬁxedabelian 2–CY category. In order to perform such a generalisation, one would have to undertake several steps, among which the following: given an abelian 2–CY category ,oneshouldbeabletoconstructacoarsemodulispace of polystable C M objects; more generally, let be an abelian C-linear category: the scheme should C M satisfy the property that its R-points, for R an arbitrary C-algebra, parametrise isomorphism classes of polystable objects of the category (R mod) ,where is ⇥ C ⇥ the Deligne tensor product of abelian categories. This construction, when applied to the category of Higgs bundles of a smooth projective curve X over C,which is 2–CY by [FSS17], should give back the usual moduli space of polystable Higgs bundles. Even though from our point of view this idea seems quite natural, it is apparent that, in order to make this construction rigorous, there are many issues to be discussed, e.g. what notion of polystability is preferable in this context. Once the scheme is constructed, one would have to study the formal local structure M of around a point, i.e. to prove an analogue of [BGV16,Theorem6.3].Wecan M formalise this statement as the following:

Conjecture 4.7.5. Let be an abelian 2–CY category and the coarse moduli C M space of polystable objects in , as above. Moreover, let x be a point represent- C 2M ing the isomorphism class of a polystable object A, then the formal scheme is Mx formally locally isomorphic to the formal completion of an appropriate ane quiver c variety depending on A.

With the above result, it would be it possible to understand the symplectic algebraic geometry of a moduli space parametrising certain geometric objects from the homological algebra of the geometric objects themselves. In addition, it would be

113 possible to prove a number of results in the spirit of the Isosingularity Theorem mentioned in Chapter 2.

114 Bibliography

[AMM98] A. Alekseev, A. Malkin, and E. Meinrenken. “Lie group valued moment maps”. In: J. Di↵erential Geom. 48.3 (1998), pp. 445–495.

[AS15] E. Arbarello and G. Sacc`a.“Singularities of moduli spaces of sheaves on K3 surfaces and Nakajima quiver varieties”. In: ArXiv e-prints (2015). arXiv: 1505.00759 [math.AG].

[Art69] M. F. Artin. “Algebraic approximation of structures over complete local rings”. In: Publications Math´ematiquesde l’IHES´ 36 (1969), pp. 23–58.

[Bea00] A. Beauville. “Symplectic singularities”. In: Inventiones mathematicae 139.3 (2000), pp. 541–549.

[Bel09] G. Bellamy. “On singular Calogero–Moser spaces”. In: Bulletin of the London Mathematical Society 41.2 (2009), pp. 315–326.

[BS13] G. Bellamy and T. Schedler. “A new linear quotient of C4 admitting asymplecticresolution”.In:Mathematische Zeitschrift 273.3 (2013), pp. 753–769.

[BS] G. Bellamy and T. Schedler. “On the (non)existence of symplectic resolutions for imprimitive symplectic reﬂection groups”. To appear in Math. Res. Lett.

[BS16a] G. Bellamy and T. Schedler. “On the (non)existence of symplectic resolutions of linear quotients”. In: Math. Res. Lett. 23.6 (2016), pp. 1537– 1564.

[BS16b] G. Bellamy and T. Schedler. “Symplectic resolutions of quiver varieties and character varieties”. In: ArXiv e-prints (Jan. 2016). arXiv: 1602. 00164 [math.AG].

[BK16] R. Bezrukavnikov and M. Kapranov. “Microlocal sheaves and quiver varieties”. In: Ann. Fac. Sci. Toulouse Math. (6) 25.2-3 (2016), pp. 473– 516.

115 [BR94] I. Biswas and S. Ramanan. “An inﬁnitesimal study of the moduli of Hitchin pairs”. In: J. London Math. Soc. (2) 49.2 (1994), pp. 219–231.

[Boa07] P. Boalch. “Quasi-Hamiltonian geometry of meromorphic connections”. In: Duke Math. J. 139.2 (2007), pp. 369–405.

[Boc08] R. Bocklandt. “Graded Calabi Yau algebras of dimension 3”. In: J. Pure Appl. Algebra 212.1 (2008), pp. 14–32.

[BGV16] R. Bocklandt, F. Galluzzi, and F. Vaccarino. “The Nori-Hilbert scheme is not smooth for 2-Calabi-Yau algebras”. In: J. Noncommut. Geom. 10.2 (2016), pp. 745–774.

[BK90] A.I. Bondal and M. M. Kapranov. “Representable functors, Serre functors, and mutations”. In: Mathematics of the USSR-Izvestiya 35.3 (1990), p. 519.

[BPW16] T. Braden, N. Proudfoot, and B. Webster. “Quantizations of conical symplectic resolutions I: local and global structure”. In: Ast`erique 384 (2016), pp. 1, 73.

[Bra+16] T. Braden et al. “Quantizations of conical symplectic resolutions II: category and symplectic duality”. In: Ast`erique 384 (2016), pp. 75, O 179.

[CW18] S. Casalaina-Martin and J. Wise. “An introduction to moduli stacks, with a view towards Higgs bundles on algebraic curves”. In: NUS IMS Lecture Note Series on The Geometry, Topology, and Physics of Moduli Spaces of Higgs Bundles (2018), To appear.

[CF17] O. Chalykh and M. Fairon. “Multiplicative quiver varieties and generalised Ruijsenaars–Schneider models”. In: Journal of Geometry and Physics 121 (2017), pp. 413 –437.

[Cor88] K. Corlette. “Flat G-bundles with canonical metrics”. In: Journal of Di↵erential Geometry 28.3 (1988), pp. 361–382.

[Cra02] W. Crawley-Boevey. “Decomposition of Marsden-Weinstein reductions for representations of quivers”. In: Compositio Math. 130.2 (2002), pp. 225– 239.

[Cra01] W. Crawley-Boevey. “Geometry of the moment map for representations of quivers”. In: Compositio Math. 126.3 (2001), pp. 257–293.

116 [Cra04] W. Crawley-Boevey. “Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity”. In: Publ. Math. Inst. Hautes Etudes´ Sci. 100 (2004), pp. 171–207.

[Cra13] W. Crawley-Boevey. “Monodromy for systems of vector bundles and multiplicative preprojective algebras”. In: Bull. Lond. Math. Soc. 45.2 (2013), pp. 309–317.

[Cra03] W. Crawley-Boevey. “Normality of Marsden-Weinstein reductions for representations of quivers”. In: Math. Ann. 325.1 (2003), pp. 55–79.

[CS06] W. Crawley-Boevey and P. Shaw. “Multiplicative preprojective algebras, middle convolution and the Deligne-Simpson problem”. In: Adv. Math. 201.1 (2006), pp. 180–208.

[CHZ14] S. Cremonesi, A. Hanany, and A. Za↵aroni. “Monopole operators and Hilbert series of Coulomb branches of 3d =4gaugetheories”.In:J. N High Energy Phys. 005 (2014). arXiv:1309.2657.

[Deb01] O. Debarre. Higher-dimensional algebraic geometry. Universitext. Springer- Verlag, New York, 2001, pp. xiv+233. isbn:0-387-95227-6.

[DEL97] R. Donagi, L. Ein, and R. Lazarsfeld. “Nilpotent cones and sheaves on K3 surfaces”. In: Birational Algebraic Geometry: A Conference on Algebraic Geometry in Memory of Wei-Liang Chow (1911-1995).Con- temporary mathematics - American Mathematical Society 207 (1997), pp. 51–62.

[Don87] S. K. Donaldson. “Twisted Harmonic Maps and the Self-Duality Equa- tions”. In: Proceedings of the London Mathematical Society s3-55.1 (1987), pp. 127–131.

[Dre91] J.-M. Drezet. “Points non factoriels des vari´et´esde modules de faisceaux semi-stables sur une surface rationnelle”. In: Proceedings of the Collo- quium on Complex Analysis and the Sixth Romanian-Finnish Seminar. Vol. 36. 9-10. 1991, pp. 635–645.

[EL17] T. Etg¨uand Y. Lekili. “Fukaya categories of plumbings and multiplicative preprojective algebras”. In: ArXiv e-prints (Mar. 2017). arXiv: 1703.04515 [math.SG].

117 [EE07] P. Etingof and C.-H. Eu. “Koszulity and the Hilbert series of preprojective algebras”. In: Math. Res. Lett. 14.4 (2007), pp. 589–596.

[FSS17] R. Fedorov, A. Soibelman, and Y. Soibelman. “Motivic classes of moduli of Higgs bundles and moduli of bundles with connections”. In: ArXiv e-prints (May 2017). arXiv: 1705.04890 [math.AG].

[Fle88] H. Flenner. “Extendability of di↵erential forms on nonisolated singularities”. In: Invent. Math. 94.2 (1988), pp. 317–326.

[FGN13] E. Franco, O. Garcia-Prada, and P. E. Newstead. “Higgs bundles over elliptic curves for complex reductive Lie groups”. In: ArXiv e-prints (Oct. 2013). arXiv: 1310.2168 [math.AG].

[FT] E. Franco and A. Tirelli. “O’Grady’s examples of IHS manifolds and Higgs moduli spaces”. In preparation.

[Fu06] B. Fu. “A survey on symplectic singularities and symplectic resolutions”. In: Annales math´ematiquesBlaise Pascal 13.2 (July 2006), pp. 209–236.

[Gam18] B. Gammage. “Private communication”. Feb. 2018.

[GR15] A. Garc´ıa-Raboso and S. Rayan. “Introduction to Nonabelian Hodge Theory”. In: Calabi-Yau Varieties: Arithmetic, Geometry and Physics: Lecture Notes on Concentrated Graduate Courses.Ed.byRaduLaza, Matthias Sch¨utt, and Noriko Yui. New York, NY: Springer New York, 2015, pp. 131–171.

[GS11] Q. R. Gashi and T. Schedler. “On dominance and minuscule Weyl group elements”. In: J. Algebraic Combin. 33.3 (2011), pp. 383–399. issn:0925- 9899.

[Gin06] V. Ginzburg. “Calabi-Yau algebras”. In: ArXiv Mathematics e-prints (Dec. 2006). eprint: math/0612139.

[GM88] W. M. Goldman and J. J. Millson. “The deformation theory of representations of fundamental groups of compact K¨ahler manifolds”. In: Inst. Hautes Etudes´ Sci. Publ. Math. 67 (1988), pp. 43–96.

[GK05] P. B. Gothen and A. D. King. “Homological algebra of twisted quiver bundles”. In: J. London Math. Soc. (2) 71.1 (2005), pp. 85–99.

[Gro14] M. Groechenig. “Hilbert schemes as moduli of Higgs bundles and local systems”. In: Int. Math. Res. Not. IMRN 23 (2014), pp. 6523–6575.

118 [Har10] R. Hartshorne. Deformation theory.Vol.257.GraduateTextsinMath- ematics. Springer, New York, 2010, pp. viii+234.

[HLR11] T. Hausel, E. Letellier, and F. Rodriguez-Villegas. “Arithmetic harmonic analysis on character and quiver varieties”. In: Duke Math. J. 160.2 (2011), pp. 323–400.

[HLR13] T. Hausel, E. Letellier, and F. Rodriguez-Villegas. “Arithmetic harmonic analysis on character and quiver varieties II”. In: Adv. Math. 234 (2013), pp. 85–128.

[Hit87] N. J. Hitchin. “The Self-Duality Equations on a Riemann Surface”. In: Proceedings of the London Mathematical Society s3-55.1 (1987), pp. 59– 126.

[Ish14] S. Ishii. Introduction to singularities.Springer,Tokyo,2014,pp.viii+223. isbn:978-4-431-55080-8;978-4-431-55081-5.

[Jor14] D. Jordan. “Quantized multiplicative quiver varieties”. In: Adv. Math. 250 (2014), pp. 420–466.

[Kal06] D. Kaledin. “Symplectic singularities from the Poisson point of view”. In: J. Reine Angew. Math. 600 (2006), pp. 135–156.

[KL07] D. Kaledin and M. Lehn. “Local structure of hyperk¨ahler singularities in O’Grady’s examples”. In: Mosc. Math. J. 7.4 (2007), pp. 653–672, 766–767.

[KLS06] D. Kaledin, M. Lehn, and C. Sorger. “Singular symplectic moduli spaces”. In: Invent. Math. 164.3 (2006), pp. 591–614.

[Kal09] D. B. Kaledin. “Normalization of a Poisson algebra is Poisson”. In: Tr. Mat. Inst. Steklova 264.Mnogomernaya Algebraicheskaya Geometriya (2009), pp. 77–80.

[Kat96] N. M. Katz. Rigid local systems. Vol. 139. Annals of Mathematics Stud- ies. Princeton University Press, Princeton, NJ, 1996, pp. viii+223.

[Kel08] B. Keller. “Calabi-Yau triangulated categories”. In: Trends in representation theory of algebras and related topics.EMSSer.Congr.Rep.Eur. Math. Soc., Z¨urich, 2008, pp. 467–489.

119 [KY08] Y.-H. Kiem and S.-B. Yoo. “The stringy E-function of the moduli space of Higgs bundles with trivial determinant”. In: Mathematische Nachrichten 281.6 (2008), pp. 817–838.

[Kin94] A. D. King. “Moduli of representations of ﬁnite-dimensional algebras”. In: Quart. J. Math. Oxford Ser. (2) 45.180 (1994), pp. 515–530.

[Kir85] F. Kirwan. “Partial desingularisations of quotients of nonsingular varieties and their Betti numbers”. In: Ann. of Math. (2) 122.1 (1985), pp. 41–85.

[Kuz15] A. Kuznetsov. “Calabi-Yau and fractional Calabi-Yau categories”. In: ArXiv e-prints (Sept. 2015). arXiv: 1509.07657 [math.AG].

[LM10] M. Logares and J. Martens. “Moduli of parabolic Higgs bundles and Atiyah algebroids”. In: J. Reine Angew. Math. 649 (2010), pp. 89–116.

[Maf02] A. Ma↵ei. “A remark on quiver varieties and Weyl groups”. In: Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1.3 (2002), pp. 649–686. issn: 0391-173X.

[Moz90] S. Mozes. “Reﬂection processes on graphs and Weyl groups”. In: J. Combin. Theory Ser. A 53.1 (1990), pp. 128–142.

[MFK02] D. Mumford, J. Fogarty, and F. Kirwan. Geometric Invariant Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge. Springer Berlin Heidelberg, 2002.

[Nak94] H. Nakajima. “Instantons on ALE spaces, quiver varieties, and Kac- Moody algebras”. In: Duke Math. J. 76.2 (1994), pp. 365–416.

[Nak16] H. Nakajima. “Towards a mathematical deﬁnition of Coulomb branches of 3-dimensional =4gaugetheories,I”.In:Adv. Theor. Math. Phys. N 20.3 (2016), pp. 595–669. issn:1095-0761.

[Nam01a] Y. Namikawa. “A note on symplectic singularities”. In: ArXiv Mathe- matics e-prints (Jan. 2001). arXiv: math/0101028.

[Nam01b] Y. Namikawa. “Extension of 2-forms and symplectic varieties”. In: J. Reine Angew. Math (2001), pp. 123–147.

[Nit91] N. Nitsure. “Moduli Space of Semistable Pairs on a Curve”. In: Pro- ceedings of the London Mathematical Society s3-62.2 (1991), pp. 275– 300.

120 [O’G99] K. O’Grady. “Desingularized moduli spaces of sheaves on a K3 surface”. In: J. reine angew Math. 512 (1999), pp. 49–117.

[Ses77] C. S. Seshadri. “Desingularisation of the moduli varieties of vector bundles on curves”. In: Proceedings of the International Symposium on Al- gebraic Geometry (Kyoto Univ., Kyoto, 1977).1977,pp.155–184.

[Sim90] C. T. Simpson. “Harmonic bundles on noncompact curves”. In: J. Amer. Math. Soc. 3.3 (1990), pp. 713–770.

[Sim92] C. T. Simpson. “Higgs bundles and local systems”. In: Publications Math´ematiquesde l’IHES´ 75 (1992), pp. 5–95.

[Sim94a] C. T. Simpson. “Moduli of representations of the fundamental group of asmoothprojectivevarietyI”.eng.In:Publications Math´ematiquesde l’IHES´ 79 (1994), pp. 47–129.

[Sim94b] C. T. Simpson. “Moduli of representations of the fundamental group of asmoothprojectivevarietyII”.eng.In:Publications Math´ematiquesde l’IHES´ 80 (1994), pp. 5–79.

[Su06] X. Su. “Flatness for the moment map for representations of quivers”. In: J. Algebra 298.1 (2006), pp. 105–119.

[Tay02] J.L. Taylor. Several Complex Variables with Connections to Algebraic Geometry and Lie Groups. Graduate studies in mathematics. American Mathematical Soc., 2002.

[Tho61] R. C. Thompson. “Commutators in the special and general linear groups”. In: Trans. Amer. Math. Soc. 101 (1961), pp. 16–33.

[Tir18] A. Tirelli. “Symplectic resolutions for Higgs moduli spaces”. to appear in Proceedings of the American Mathematical Society. 2018.

[Van15] M. Van den Bergh. “Calabi-Yau algebras and superpotentials”. In: Se- lecta Math. (N.S.) 21.2 (2015), pp. 555–603.

[Van08a] M. Van den Bergh. “Double Poisson algebras”. In: Trans. Amer. Math. Soc. 360.11 (2008), pp. 5711–5769.

[Van08b] M. Van den Bergh. “Non-commutative quasi-Hamiltonian spaces”. In: Poisson geometry in mathematics and physics.Vol.450.Contemp.Math. Amer. Math. Soc., Providence, RI, 2008, pp. 273–299.

121 [Wel07] R. O. Wells. Di↵erential analysis on complex manifolds.Vol.65.Springer Science & Business Media, 2007.

[Yam08] D. Yamakawa. “Geometry of multiplicative preprojective algebra”. In: Int. Math. Res. Pap. IMRP (2008), pp. 1–77.

122