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Group quantum cohomology Mizerski, Maciej 2007

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Group quantum cohomology by Maciej Mizerski B . S c , M c G i l l University, 1998 M . S c , M c G i l l University, 2000 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F Doctor of Philosophy in The Faculty of Graduate Studies (Mathematics) The University Of British Columbia June 3, 2007 © Maciej Mizerski 2007 Abstract Given a finite group G acting on a smooth projective variety X, there exists a G-algebra qA* (X, G) whose structure constants are defined by integrals over moduli spaces of G-equivariant stable maps of Jarvis-Kaufmann-Kimura. It is a deformation of the Fantechi-Gottsche group cohomology, and its invariant part qA*(X,G)G is canonically isomorphic to the Abramovich-Graber-Vistoli orbifold quantum cohomology of the quotient stack [X/G]. We provide the technology to study the associativity of the above algebra, and we prove it for some special cases. i i i Contents Abstract • • • u Contents u i Acknowledgements v i 1 Introduction 1 1.1 Overview 1 1.2 Note on terminology 1 1.3 Structure of the thesis 2 2 Moduli of G-curves 4 2.1 Admissible and balanced G-curves 4 2.2 Moduli of marked admissible G-curves 5 2.3 Forgetful morphism 6 2.4 Properness 6 2.5 Universal families 7 2.6 Stabilization morphism 7 3 Admissible G-maps 9 3.1 Spaces of morphisms 9 3.2 Morphisms of specified degree 10 3.3 Properness 10 3.4 Group actions 10 3.5 Monodromies 11 3.6 Evaluation maps H 4 Fundamental class 13 4.1 Obstruction theory 13 4.2 Expected dimension 14 5 Inflation 16 5.1 Inflation of spaces 16 5.2 Inflation of admissible curves 17 5.3 Inflation of morphisms 19 5.4 Compatibility of virtual fundamental classes under inflation 20 Contents iv 6 Groupoid actions, inertia varieties and stacks 22 6.1 Quotient by a product of cyclic subgroups 22 6.2 Actions of G ( 7 ) 23 6.3 Inertia variety and rigidified inertia stack 24 6.4 A note on evaluation morphisms 25 7 Presentation of the moduli space of admissible 3-marked genus 0 curves 26 7.1 Canonical element of M3 26 Q 7.2 Canonical presentation of 28 7.3 Compactification of the moduli space of G-equivariant morphisms from a G-cover of P 1 with 3 ramifications 29 7.4 Inflation : . 30 8 Group quantum cohomology 32 8.1 Group cohomology 32 8.2 Quantum parameters 33 8.3 Construction of the group quantum cohomology 33 8.4 G-graded G-algebra and the braid group action 35 8.5 Inflation is a homomorphism 37 8.6 Degree 0 specialization and Fantechi-Gottsche group cohomology 38 8.7 Orbifold quantum cohomology 39 8.8 Invariant part of the group quantum cohomology 41 9 Associativity 42 9.1 Reducing the problem 42 9.2 Gluing 44 9.3 Compatibility of the virtual fundamental class under the cutting edges mor-phism 46 9.4 Compatibility of the virtual fundamental class under the gluing morphism . 48 9.5 Associativity in group quantum cohomology • • 51 9.6 Smooth locus of 9J t£ A mg,A 53 9.7 Z f e-covers of P 1 by P 1 55 9.8 Z 2-covers of P 1 by genus 1 curves 56 A Equivariant Riemann-Roch for Nodal Curves 57 A . l Introduction 57 A.2 Pushforward and pullback 57 A.3 Cohomology 58 A.4 Representions of finite groups and their characters 59 A.5 if-theory : . • • 60 A.6 The smooth case 61 A . 7 Normalization of a G-curve 64 A.8 Admissible action 65 A.9 Balanced action 66 A . 10 Equivariant Riemann-Roch for nodal curves 66 Contents v B Galois Covers • 69 B . l Some Conventions and Notations 69 B.2 Pointed Principal G-Bundles 69 B.3 BG\C • 7 0 B.4 Principal G-Bundles Over Curves 73 B.5 Inflation 7 6 B.6 Monodromies ' 77 Bibliography 7 ^ vi Acknowledgements Thanks to my supervisor Dale Peterson for his time and help. Many thanks to K a i Behrend, J im Bryan and J im Carrell for their support. Thanks to Adam Clay, Alexandre Girouard, Izak Grguric, Samuel Hikspoors, Dagan Karp, Olivier Rousseau, Yinan Song and Yunfeng Jiang for their friendship. 1 Chapter 1 Introduction 1.1 Overview Gromov-Witten invariants are defined as integrals over moduli spaces of curves, and maps from these curves to a given target space. They give rise to an extremely rich theory, and in particular if one restricts to genus 0, one gets the so called quantum cohomology. One can extend the theory of Gromov-Witten invariants to study target spaces being orbifolds or stacks. In the symplectic category this study has been initiated by Chen and Ruan in [CR02] and [CR04], and in the algebraic category by Abramovich, Corti, Graber and Vistoli in [AV02], [ACV03] and [AGV02]. In this thesis, we will study a natural extension of the classical Gromov-Witten invari-ants to the category of G-objects where G is a finite group. More precisely, one replaces curves by G-curves and maps by G-equivariant maps, and formally proceeds as in the classical theory (see [JKK03]). By analogy with the classical Gromov-Witten theory, one constructs the moduli spaces of marked G-curves and G-maps, the virtual fundamental classes and integrate cohomology classes. The genus 0 case is of particular interest for this paper. We construct a certain G-ring called the group quantum cohomology and show that its invariant part is the orbifold quantum cohomology of the quotient stack. This result is a generalization of a result in [FG03] on group cohomology and orbifold cohomology. We show that the degree 0 specialization of the group quantum cohomology is the group cohomology of [FG03]. We also show that the group quantum cohomology have functorial properties with respect to inclusions of groups. We develop the theory necessary to prove the associativity of the group quantum cohomology (or in otherwords W D V V type equations). More precisely we relate it to equivalences of certain divisors on moduli spaces of marked G-curves. We prove these equivalences in certain cases. Proving them in general is still an open problem. The problem is that these moduli of marked G-curves are disconnected Deligne-Mumford curves with components of arbitrary genus. To prove the associativity, one probably needs a better understanding of the structure of those moduli. 1.2 Note on terminology We use the names "quantum cohomology", "group cohomology" and "orbifold cohomol-ogy" for objects constructed based on the intersection theory (Chow rings). This seems like a bad choice, but the term "cohomology" is commonly used in the Gromov-Witten literature, even though the main references usually cited ([Ful98], [Kre99] and [Vis89]) are about the intersection theory. Also, we found it difficult to find good references about the cohomology of stacks. Therefore we will stick with intersection theory, and note that it seems that the same methods would work for cohomology. Chapter 1. Introduction 2 1.3 Structure of the thesis In the first few sections of Chapter 2, we make the standard constructions necessary for a Gromov-Witten type theory. We review the definitions of balanced and admissible marked G-curves, of their moduli stacks and of their basic properties. In Chapter 3, we describe the moduli stacks of G-equivariant maps from marked G-curves into a smooth projective G-variety. We relate the monodromies of the markings and the evaluation maps to the inertia variety. In Chapter 4, we describe natural obstruction theories on the above moduli stacks, construct their virtual fundamental classes using the approach of the intrinsic normal cone of Behrend and Fantechi ([BF97]), and compute their dimensions in terms of the ages, which is a set of numerical invariants associated to a variety with an action of a finite group. In Chapter 5, we begin studying a purely equivariant phenomena, namely the behavior of the above objects under the inflation (sometimes called the induction), which is a functorial way of constructing a G-space from an J?-space when i f is a subgroup of G. We show that there are inflation morphisms defined between the moduli stacks of equivariant maps, in such a way that the virtual fundamental classes are compatible. We devote the Chapter 6 to the study of inertia varieties associated to G-varieties, of rigidified inertia stacks of global quotients and of their twisted sectors in a general context of groupoids. In particular we explain why the natural target spaces for the evaluation maps from the moduli stacks of twisted stable maps are rigidified inertia stacks (as opposed to the inertia stacks). In Chapter 7, we specialize our study of admissible and balanced G-curves to the case where we have 3 markings and the quotient curve is P 1 . These are smooth G-covers of P 1 with branching above 3 fixed points, and whose automorphisms are G-equivariant deck transformations. Denote their moduli stack by . It is a Deligne-Mumford stack of dimension 0, and we construct it in an explicit way. We consider the set A 7 ^ of triples in G whose product is the identity. There exists a canonical G-cover of P 1 whose monodromies at the 3 markings are given by any such triple. It can be constructed by taking a quotient of the product of G and the upper half plane by an appropriate action of the principal congruence subgroup of level 2 (which can be identified with the fundamental group of 3-punctured sphere). This gives a morphism J7^ —>.~M% which turns out to be a cross section for an action of G 2 by translation of any two of the three markings, hence providing a presentation of At 3 . Let M^(X) denote the stack of stable G-maps defined in Chapter 3, we will consider the fiber product JT^(X) of M^(X) by J7^ over M^. This is a compactification of the scheme of G-equivariant morphisms from G-covers of P 1 into X. Note that we are compactifying an actual scheme, as we don't allow any G-equivariant deck transformations of the G-cover as isomorphisms of our moduli problem. The Chapter 8 is dedicated to the construction of the group quantum cohomology, which is a Q-graded and G-graded algebra whose structure constants are given by inte-grals over J7^(X). We prove that it behaves functorialy under the inclusion of groups, that degree 0 specialization is the group cohomology of Fantechi-Gottsche, and that the subalgebra of the G-invariants is the orbifold quantum cohomology of the quotient stack [X/G]. Chapter 1. Introduction 3 Finally in the Chapter 9, we verify two important properties of the virtual fundamental classes of the moduli stacks M^n(X), namely their compatibility under the cutting edges and the gluing morphisms. We link the associativity of the group quantum cohomology to equivalences of divisors on M04, obtained by gluing G-curves in two different ways. This is completely analogous to the classical case, except that is far from being P 1 in general. 4 Chapter 2 Moduli of G-curves In this chapter we will define moduli stacks of marked G-curves with non-trivial isotropy at the nodes and markings. First we review the definitions of admissible and balanced G -curves (see [ACV03]). We then define the categories fibered in groupoids parameterizing stable admissible and balanced G-curves, and show that they form moduli stacks (with universal properties) by relating them to the moduli stacks of twisted stable curves. We also construct the Ar t in stack of (non-necessary stable) admissible and balanced G-curves. 2.1 Admissible and balanced G-curves Let G be a finite group. Given a G-scheme X and a geometric point Q, denote by GQ the stabilizer of Q in G. A morphism / : Y —> X of G-schemes induces a canonical monomorphim of stabilizers G Q —* GpQ) for every Q E.Y. Definition 2.1. A morphism / : Y —> X of G-schemes is called fully faithful if for every point Q e Y the canonical monomorphism of stabilizers GQ —> G^Q) is an isomorphism. A normalization of a G-scheme is a G-scheme by the universal property of normaliza-tion. Definition 2.2. A G-action on a curve E is called admissible if the normalization n : E —> E is a fully faithful morphism. Let E be a possibly nodal G-curve over C. Suppose the G-action on E is admissible. Let Q\ and Q2 be the points on E lying above a node P G E. Then for any locally free G-C^-module T of finite rank, let XQX^ (resp. XQ1<JT) be the character of the C-representation of GQ1 = Gp on FQ1 (resp. FQ2), the fiber at Q i (resp. Q2) of the G-vector bundle F corresponding to T. Let be the G-O^-sheaf of Kahler differentials on E. Definition 2.3. A n admissible G-action on a nodal curve E is balanced at a node P € E if XQ1IQ. and XQ2tn- are dual characters of Gp. A G-action is balanced if it is balanced at all nodes of E. The admissibility of a G-action can be understood as a property of the action of the stabilizer of a node in an analytic or etale neighborhood, namely that it preserves each irreducible component of the curve. Balancing says that the action on the two components are compatible in the sense that they induce dual representations on the tangent spaces. In fact these two conditions imply that the nodal G-curve is a degeneration of a flat family of smooth G-curves, and conversely a flat degeneration of smooth G-curves is admissible and balanced (see [ACV03]). Chapter 2. Moduli of G-curves 5 2.2 M o d u l i o f m a r k e d a d m i s s i b l e G-cu rves To define our moduli problems, we need the notions of category fibered in groupoids (in short C F G ) and stack, whose definition can be found in [FanOl], [LMBOO] and [Beh97b]. Roughly a C F G is a pair consisting of a category and of a functor to the category of schemes of finite type over C, having two properties: existence of pullbacks and unique factorization of pullback diagrams. Given two objects of a C F G over the same scheme, we can construct the presheaf of morphisms from one to the other (in the flat Grothendieck topology, see [Beh97b]). If this presheaf is a sheaf, then we call the C F G a prestack. A prestack is a stack if one can glue objects (every descent datum is effective). A stack X is called Artin if there exists an affine scheme U and a smooth epimorphism p : U —> X (any such epimorphism is called a presentation of X) and if the diagonal morphism X - t X x X is representable and of finite type. A n Art in stack is called Deligne-Mumford if it has an etale presentation. Let G be a finite group, A be a finite set and g be a non-negative integer. Let Sch /C be the category of schemes of finite type over C. Definition 2.4. Let m^A be the C F G over Sch /C defined as follows. A n object of VJlfA over a scheme T is a triple e := (Ee, S e , CTc), where (1) Ee is a flat and projective (in the sense of [Gro61a][5.5]) G-scheme over T, (that is it has a G-invariant structure morphism pgt : Et —> T), whose fibers above geometric points are reduced nodal G-curves, (2) E c is a disjoint union of G-subschemes {S e , i}ieA of Et over T, such that for each i e A the structure morphism £ e , i —> T is etale, and induces an isomorphism E c , i / G ~ T, (3) cre is a set {u^i^A of sections of £ C ) j —> T indexed by A. This data has to respect the following conditions: (a) E e doesn't intersect the nodes of the fibers of pt : Ee —> T (which is equivalent to requiring that its sheaf of ideals is invertible) (b) for each t : SpecC —> T, the G-action on Ee>t is admissible, balanced and free away from the nodes and E C i t , (where Ee<t and E C i t denote the fibers of respectively Et and E c above t), (c) for each t : SpecC —> T the dimension of H 1 ( i ? e j t , OE,^)G 1S 9-Let e (resp. f) be an object of - O t ^ over T (resp. T'). A morphism e —> f of -OT^ is a pair (a,t) consisting of a G-equivariant morphism a : Et —* and a morphism t :T —>T' such that is cartesian and such that the sections of c pullback from sections of f, that is a o cre,i = o^i o t for each i e A. Chapter 2. Moduli of G-curves 6 Definition 2.5. A n objects e of 97t^ 4 is called a marked admissible G-curve; Et is called the underlying G-curve of e; £ e , i is called the ith twisted section of e; and <jtti is called the ith section of e. If we don't require the trivialization of the etale covers £C )$ —> T (point 3 in the definition), we will obtain the moduli of twisted stable maps into the quotient stack (see [ACV03]). Definition 2.6. Given an object e of 971^ over SpecC, a point in the underlying G-curve of e is called special if it is either a nodal point, or it belongs to a twisted section. The C F G Tl^A is a prestack: given two objects e and f of 97r^A over T , the presheaf Zsom(e, f) is representable by a scheme over T (which implies that it is a sheaf): it is a closed subscheme of the scheme of G-equivariant T-isomorphisms Isonij. (Ee,Ej) cut out by the condition that the morphisms preserve the sections. 2.3 Forgetful morphism Let VJlg}A be the Ar t in stack of A-marked nodal curves (see [Beh97a]). When G is the trivial group, then 971^ is isomorphic to VJlgtA- For a general G , there is a morphism from 97f^ to 9Rg,A, sending each object to its quotient by G , which is defined as follows. Definition 2.7. Let e be an object of 971^. Let Et := Ee/G be the geometric quotient of Et by G over T. For each i G A, taking the quotient of E c , i induces a morphism atti : T ~ E e , i / G —> Et into the smooth locus of Et. Let e := (Et,atti), which is an object This construction is functorial and commutes with base change. The following propo-sition follows. Proposition 2.8. There is a morphism %ftftA —+ ^Hg,A that sends an object etoe. 2.4 Properness To avoid repetitions, we use the following terminology. Definition 2.9. Let 971 be a prestack with a representable diagonal. Given an object e of 9Jt over T, let Auty(e) be the T-group of automorphisms of e. Then the object e is called stable if Autr(e) is of relative dimension 0 over T. Note that if 971 is a prestack with a representable diagonal, then Autx(e) = Isomx(e, e) is representable, and we can talk of relative dimension of Auty(e) over T. Let Autgj; be the universal automorphism group over 971. Then the subprestack M. of stable objects of 971 is the locus where the morphism Autgjt —> 971 is of relative dimension 0. It follows that A t is open in 971. Definition 2.10. Let Mg<A be the prestack of stable objects of 97?^. Chapter 2. Moduli of G-curves 7 Next result follows from [ACV03] by interpreting A4gA as a fiber product of the universal twisted sections over MG A{BG) (see [JKK03]). Theorem 2.11. Suppose that 2g — 3 + O(J4) > 0, then the prestack "M^A is a proper and smooth Deligne-Mumford stack of pure dimension 3g — 3 + o(A). Corollary 2.12. The prestack A is a smooth Artin stack, and if 2g — 3 + o(^4) > 0, the substack MgA is open and dense in A . Proof. That A ^ y t is open follows from the fact that the stability is an open condition. That it is dense follows from Theorem 2.11 and the observation that the complement of M^A is of proper codimension in Ti^ A . The first statement is proven by an argument analogous to the one proving the same statement about Mg,A and %Rg,A, and which can be found in [Beh97a]. The key to constructing a presentation of Tl^A is to consider the morphism TT : 9Jlg xu{*} ~~* ®^g,A that forgets the section *, where Tl^'^T^ is the locus in ^R^Aui*} w n e r e the section <r* has no non-trivial stabilizer. Then SDT^yT^} is a smooth G-curve over A , namely it is the complement of the nodes and twisted sections in the universal G-curve. In particular TT is a smooth morphism. Restricting to the stable locus, we get a morphism from a Deligne-Mumford stack M^^AU^I} to Tl^A. Notice that this morphism is not necessarily surjective (and it is possible that M.^'A^y is empty), but we can remedy to this problem by adding more non-stabilized sections. By adding enough such sections we can cover a neighborhood of any point in Tl^ A . It follows that SDt^ is an Ar t in stack. • 2.5 Universal families The construction of universal structures over a stack is a simple exercise, and reflects the fact that stacks have been created precisely for the purpose of serving as moduli spaces with the right universal properties. Therefore we leave out the details of the constructions involved in the following definitions. Definition 2.13. There is a stack € ^ over Wl^ A such that the fiber over a marked ad-missible G-curve c is isomorphic to the underlying G-curves of e. It is called the universal G-curve over Tl^A. F ° r a n y i £ A, there is a substack ^ Ai — ^ g A such that the fiber over e is identified under the above natural isomorphism with the ith twisted section of e. This is called the ith universal twisted section over Tl^ A . Each universal twisted sec-tion has a universal section denoted a^A i with the obvious universal property. Denote by £gA the restriction of (E '^ A to MG A . 2.6 Stabilization morphism Denote ~M^tA> A> A a n d ^A by A4, <£ and £ respectively. The stabilization morphism is a retraction s : Tt —• M, together with a G-morphism c : (E —> s*£ over 9Jt Chapter 2. Moduli of G-curves 8 compatible with the universal sections, such that given an object e of 971 over T, then c induces Ee —> Es^ which is universal in the following sense: given an object f of M. over T, and a G-morphism Et —> E^ compatible with the markings, there exists a unique arrow a : s(e) —» f in M such that Ee commutes. One can prove the following Proposition using the existence of stabilization morphism for nodal curves and its universal property. Proposition 2.14. Stabilization morphism exists for 97t = 97?^ and M = M^tA-9 Chapter 3 Admissible G-maps We define the moduli stacks of G-equivariant maps from a (admissible and balanced) G -curve into a fixed G-variety X. We describe different structures found on these moduli stacks, namely the universal G-curves and maps, the evaluation maps into the inertia variety, and the G-actions by translation of the sections. 3.1 Spaces of morphisms Given two G-schemes E and X over a base scheme T, let Morlf (E, X) be the scheme over T of G-equivariant T-morphisms from E to X. Its functor of points is defined as follows: a morphism S —> MorG(E,X) is a pair (s,<p) where s : S —> T and ip is a G-equivariant S-morphism s*E —> s*X. This definition generalizes to the case were E —> T and X —> T are representable morphisms of algebraic stacks. Then Mov^(E, X) is an algebraic stack over T with the property that the structure morphism MovG (E, X).—> T is representable. Let X be a G-variety over C . We want to construct a space whose points are G -equivariant morphisms from G-curves into X. Consider the diagram arc? where <E^A is the universal G-curve over STJt^ t and X x is the product of X and 2Jt^A (with the G-action on the first factor). These are both G-stacks representable over Definition 3.1. Let X be a G-variety over C. Let mfA(x) := MoigoJ^,x x a n £ t ) , and let Mg<A(X) be the open substack of dJl^A(X) consisting of stable objects (see Def-inition 2.9). The Art in stack mfA(X) comes with a representable morphism to 971^. Given a geometric point c : SpecC —> SDt^, the pull back of DJ\^A(X) by e is the space of G -equivariant morphisms MoiG(Ee,X): MorG(Ee,X) ^mGA(X) SpecC ^2)1^4 Chapter 3. Admissible G-maps 10 In general an object e of iXHGA(X) is a quadruple e := (Ee, E c , cre, / c ) where (Ee, E c , <7e) is an object of 9RG<A and / c : £?c —+ X is a G-equivariant morphism over C. A morphism e —> f of WlGA(X) is a morphism (a,t) : {Et,He,at) —> (£?j, Ef, <7f) of 971^ compatible with / f and / e , that is such that ff° a = fe. Definition 3.2. Let <BGA(X) (resp. SGA{X)) be the universal G-curve over TtGA(X) (resp. ^M^^A(X)) that is the pullback of the universal G-curve over 2DlfA; let T,GAi{X) be the ith universal twisted section over VJlGA(X); let crGAi(X) be the ith universal section over WlGA(X); and let fGA(X) : <£GA(X) -* X be the universal morphism over<mGA(X). 3.2 Morphisms of specified degree Let A i ( X ) := H o m z ( A 1 ( X , Z ) , Z ) , the dual of the Picard's group P i c (X) = A1(X,Z), and let A + ( X ) C Ai(X) be the semi-group of linear functions whose values on positive line bundles are non-negative. There is a natural G-module structure on P i c ( X ) , and thus on A\(X), and since any automorphism of X induces an isomorphism of P i c ( X ) that preserves the property of being a positive line bundle, the G-action preserves A^(X). Given a curve E and a morphism / : E —> X, let d e g / G Af (X) be the linear function sending the class of a line bundle to the degree of its pullback under / . If E is a G-curve, and / is G-equivariant, then deg / € A f (X)G. Definition 3.3. Given j3 G A+(X)G, let DJlGA(X,(3) (resp. M^tA(X,f3)) be the substack of 971^(X) (resp. M^:A(X)) whose geometric points are morphisms of degree (3. 3.3 Properness Next result follows from [AV02] by interpreting M^A(X,/?) as a fiber product of the universal twisted sections of MbgalA([X/G],J3) (see [JKK03]) where J3 is the pushforward of f3toA1(\X/G}). Theorem 3.4. Given (j G A^{X)G, the stack M^A{X,(3) is a proper Deligne-Mumford stack, and has a projective coarse moduli space. 3.4 Group actions For each i G A, there is a G-action on WlGA(X, (3) (and on M^A(X,(3)) obtained by translating the ith section: given an object e over T, and g G G , let g-^ e := (Ee, E e , g-jcre> / e ) where fcre j if i ^ i, (g -i Ot)j = \ ., . . [g-CTe,i r f « = J -This is called the G-action by translation of the ith marking. Chapter 3. Admissible G-maps 11 3.5 Monodromies Given a generically free action of a finite group on a Riemann surface, the stabilizer of a point is cyclic and acts faithfully on the tangent space. The monodromy at this point is an element of the stabilizer whose action on the tangent space is by rotation by a smallest possible angle, in counterclockwise direction. The monodromies classify faithful one dimensional representations of a given cyclic group. Definition 3.5. Let £ be a G-curve with a generically free G-action, and let p G E be a smooth point. Define the monodromy of E at p as the unique element mp & Gp such that the induced action of m p on the tangent space TPE is by multiplication by exp ( 0 ^ ) ) , where o(G p ) is the order of the stabilizer at p. Note that a monodromy of a point is always a generator of the stabilizer of this point. We decompose the stack !0t^'A into a finite disjoint union of substacks, by specifying the monodromies at each section. This can be done as follows. Let w be the relative dualizing sheaf of the universal G-curve over 9JlfA, and let be the dual of the restriction of <~> to the ith universal twisted section Ej . Then TJ is a line bundle, and its fiber over a point p is the tangent space to the G-curve containing p. Given m G G , let E ^ be the locus of points p in E , fixed by m and where m acts on r, by multiplication by exp (^Q2Q^J • Let OT^f=m be the inverse image of E ^ under the iih universal section. In general, given m G GA let Wl^'A be the intersection of the substacks 9 3 l ^ i = m i where i runs over A. The definitions of this decomposition for Mg <A, Wl^A(X) and MgA(X) are similar. 3.6 Evaluation maps Given an object e of MgtA(X), the composition of the G-equivariant map fe with a section <7e,j defines a morphism from the base scheme into X, called the evaluation at the ith section. We can be more precise about the target space of the evaluation: since the section is invariant under the action of its monodromy and the morphism / e is G-equivariant, the evaluation maps into the subvariety of X whose points are fixed by this monodromy. It is therefore useful to define a space which is the union of the fixed points sets of group elements. Definition 3.6. Define the inertia variety of the G-variety X as X^ := ( J {g} x X9 C G x X. gee The diagonal G-action on G x X where G acts on itself by left conjugation induces a G-action on X^G\ Definition 3.7. For each i G A, let e V i : m%A{X) - X^ Chapter 3. Admissible G-maps 12 be the morphism whose projection on G is the monodromy /ij of the i section and whose projection on X is the composition of the ith universal section OgAi(X) with the universal morphism f^A(X). It is a G-equivariant morphism where G acts on DJt^A(X) by translation of the ith section. Chapter 4 Fundamental class 13 As in the classical Gromov-Witten theory, the moduli stacks M^A(X, (3) are expected to have dimensions given by a Riemann-Roch formula, but their actual dimension is in general different. Therefore one needs to construct a homology class of the expected dimension and having properties of a fundamental class. We will use the intrinsic normal cone construction of [BF97], with an appropriate obstruction theory. 4.1 Obstruction theory In this section we generalize the ideas of [Beh97a] to the G-equivariant setup. To simplify, we will use the following notation for the moduli spaces 97t := MGA M{X):=MGgA{X,(3). Let 7r : £(X) —> M(X) (resp. € —> 9Jt) be the universal G-cover over M(X) (resp. 9Jt), and let / : £(X) —* X the universal G-morphism over M(X). The fibers of the forgetful morphism p : M{X) —> DJl are spaces of G-equivariant morphisms from a fixed G-cover to X. Therefore p is representable, and one can construct the relative cotangent complex Lrj^^X)/m u s m S [11171] as an object of FKCyvfrx)^)' t n e derived category of sheaves of modules on the etale site of M(X). A perfect relative obstruction theory is an element E of D(Oj^^x^t) concentrated in degrees 0 and —1, together with a morphism <pE '• E —> LM(X)/<JJI which induces an isomorphism on H° and a surjection on H _ 1 (see [BF97]). As X is smooth over C , the complex consisting of D,x in degree 0 is quasi-isomorphic to the cotangent complex of X over C. Define the complex E:=RTvG(u;®f*rix)[l} where u is the relative dualizing sheaf of £(X) over A4(X), which has a natural structure of a G-sheaf, and where TTG is the invariant pushforward (see [Gro57]). We construct tf>E '• E —• Ljj^Xym as follows. By the functoriality of the cotangent complex we have f*ttx -*• L£(x)/m ->• L£(X)/£ - ^*LM(x)/m-Tensoring with u, we get a morphism of T){Oj^^x^t) ui ® f*Qx w ® Tr*Lj4(x)/m ~ n-Lj?[{x)/m{-l\. And finally as 7T is the right adjoint of RnG, we get 4>E:E = RnG(u; ® fnx)[l] -> R T T ^ 7 r ! L ^ ( X ) / O T -> Ljz{x)/m. Chapter 4. Fundamental class 14 Proposition 4.1. The morphism <pE • E —> Lj^(x)/m *s a Perfect relative obstruction theory. Proof. One adapts [BF97, Section 6] to the G-equivariant setup. First we show that </>£ is a relative obstruction theory. Denote by p : M(X) —> 971 the forgetful morphism. Let t : T —> T" be an infinitesimal deformation of T by an ideal J, let e be an object of M(X) over T and let f be a deformation of f := p(e) over t: f - f We calculate: E x 4 ( e * £ , J ) = E x t ^ R T T ^ K , ® / c *fix)[l] , J ) = Extg i(/ e*_. J C,<_7'). By [11171], Ext^(f*flx, T^tJ) contains a canonical element ot which is 0 if and only if the deformation f' of f lifts to a deformation of e, in which case the set of all such deformations is a torsor under E x t G , 0 ( / * Q x , K J)- B y a relative version of [BF97, Theorem 4.5], E —> Lj^^xygx is a relative obstruction theory. That the complex E is of perfect amplitude contained in [—1,0] follows from the fact that TT is of codimension 1. • The following is a consequence of Proposition 4.1, and of [Kre99, Theorem 5.2.1] which allows avoiding a technical hypothesis in [BF97, Section 5] about global resolutions. Definition 4.2. We have a class \M(X),E] £A*(M(X)) as constructed in [BF97, Section 5], called the virtual fundamental class of M(X) with obstruction theory E. When there is no confusion about the obstruction theory used, we will denote the virtual fundamental class by [ A 4 ( X ) ] m r . 4.2 Expected dimension We recall the definition of age (see [FG03]). Given a complex vector space V, and an idempotent automorphism v of V, the age is a numerical invariant associated to this pair, defined as follows. Let C„ the one dimensional representation of the cyclic group (v) with monodormy v at 0 (recall from Section 3.5 that it means that v acts by multiplication by exp (^7)))- Let My,j, V) be the multiplicity of (<£.„)& in V. The age of v on V is the following weighted average of the above multiplicities oM-i a ( i ^ ) : = o M £ ^(u,3,V). K ' j=o Chapter 4. Fundamental class 15 Equivalently, if V = v\ © • • • © Vn is the decomposition of V into irreducible representa-tions of (v) and fci,..., kn e { 0 , . . . , o(^) — 1} are such that v acts by multiplication by e x P (l&f) ° n Vj' t h e n 1 " v ' j=i Let flX be the G-vector bundle of Kahler differentials on X. The number m(^, j, fixX) depends only on the connected component Z of Xv containing x, hence we can write &(v,Z,X) for a,(v,flxX). Call a(v, Z, X) the age of v in Z. The following is proven in Appendix A . Theorem 4.3. Let e be an object of A4(X) over SpecC. Let vt^ be the monodromy of Et at atyi. For each i £ A, let Z,,^ be the connected component of X"'* containing the point /c ° cre.i : SpecC —> X. Then the rank of E at e is rk e E = (1 - g)dimX + ^— deg f*tTX - £ a ( i / e i i . * ) w/iere deg f*TX = /3(detTX), tfte value of P on the class of the highest wedge power of TX. The rank rkE of a bounded complex of vector bundles E, is the alternating sum of the ranks of each degree rkE := J2i(~l)*rk._?. Definition 4.4. The number rk e E is called the relative expected dimension of M(X) over 9Jt at e. By [BF97], the rank of the class \M. (X) ,E] at the component containing e is rk c \M(X),E] := rkeE + dimOT = Sg - 3 + o(A) + (1 - g)dimX + deg ftTX - £ a ( i / M , Z h i , X ) ^ ' i€A and it is called the expected dimension of M(X) at e. Chapter 5 Inflation 16 5.1 Inflation of spaces Suppose that i f is a subgroup of G . There is a functor Inf^ = G x H — from the category of if-schemes to the category of G-schemes which takes an if-scheme X and sends it to the quotient GxHX of the product G x X by the right if-action (g, a;) • h i — • (g • h, h _ 1 a;). The scheme Inf^ X = G x# X has a natural left G-action induced by the multiplication on G . If / : X —* Y is a morphism of if-schemes, then the induced morphism G x X —> G x Y is if-equivariant under the above right ff-action, hence taking the quotient induces a morphism Inf^ / : Inf 0, X —> Inf^ Y which is G-equivariant. Consider the functor Res^ from the category of G-schemes to the category of H-schemes, which restricts the action of G to i f . If X is an if-scheme, there is a natural if-morphism i^X : X —> Res^ Inf% X induced by the if-morphism i f x X —• G x X and the natural identification X ~ i f x# X. If X is an if-scheme over T (that is the if-action preserves the fibers of X —> T) , then Inf^ X has a unique structure of a G-scheme over T such that L°J : X —• Res^ Inf ^ X is a morphism of if-schemes over T. Lemma 5.1. The functor Res^ is a right adjoint o/Inf^. Proof. The natural transformation id —> Res^ o ln f^ is given by and the natural trans-formation Inf 0, o ResQ —> id is given by the G-action G x# F —• F for a G-space F . • Lemma 5.2. T/ie functor Inf^ commutes with base change, that is if X is an i f -scheme over a scheme T and t : S —> T is a morphism of schemes, then there is a natural G-isomorphism G x# £ _ 1 X ~ t _ 1 ( G x# X ) over 5. More precisely it means that „ Inf£ „ Sch / T — 5 - S c h G / T t- 1 t- 1 „ InfS * SchH/S S c h G / 5 is a commutative diagram of categories (where two functors are said to commute if they are naturally isomorphic). Proof. We construct the natural isomorphism G x H t~xX ~ t~1(G x# X ) as follows. By applying the base change to the if-morphism 1°, : X —> Res^f Inf^ X we get an i f -morphism t~lX —> t~l Resg Inf^ X over S. It is clear that Resg commutes with base change, hence we get Inf^ t~lX —> Inff, Resg t~l Inff, X —• £ _ 1 Inf 0, X as Res^ is a right adjoint of Inf G f . • Chapter 5. Inflation 17 Lemma 5.3. The functor Inf# is faithful. Proof. Given a morphism / : X —• Y of .ff-schemes, we have a commutative diagram fix X ^ R e s g l n f g X Resg Infg / — ^ R e s g l n f g Y The result follows from the observation that the morphism i°,X is injective. • Lemma 5.4. Given an H-scheme X, the projection morphism G x X —> G, induces a morphism I n f ^ X —> G/H such that X ^ R e s g l n f g X {idH} Resg G/H is a cartesian diagram in the category of H-schemes, where the bottom arrow is induced by the inclusion of the identity element in G. Moreover the G-scheme Inf^ X together with the projection onto G/H is uniquely defined by the above property, up to a unique isomorphism commuting with the projection onto G/H. Proposition 5.5. The functor Inf^ induces an equivalence of categories between SchH and SchG/(G/H) whose objects are G-schemes with a G-morphism to G/H and arrows are morphisms of G-schemes commuting with the projection onto G/H. Proof. Construct the inverse functor using Lemma 5.4. Corollary 5.6. The functor Inf 6, preserves cartesian diagrams. Proof. Follows from Proposition 5.5 and the fact that the forgetful functor Sch G /(G/H) Sch G preserves cartesian diagrams. 5.2 Inflation of admissible curves Suppose that H is a subgroup of G. Given /? £ Al[(X)H, we will define morphisms InfS • fflH —> WlG xillH • •J-"-g,n g,n • • g,n J lg,n where a = InfGr( /S) :— J2seG/H& ' @> that innate the underlying i?-curves. Chapter 5. Inflation 18 Definition 5.7. Given an object e of 9JT^„, let f = Infg(e) be an object of 9Jt G „ defined by £ f := Infg Et E ( ) j = Infg E C ) j for each i. This construction is functorial and commutes with base change. Therefore it defines a morphism Infg : 9Jt^ n —> 9Jt G „ called the inflation morphism. Proposition 5.8. The inflation Infg : 9JC^n —* 9H G ) 7 l is a representable etale morphism, and if in addition n > 1, £/ien i£ is an embedding (that is fully faithful). Proof. First, that it is representable is a direct consequence of Lemma 5.3. Next we show that the inflation is etale. We need to study the if-equivariant infinites-imal deformations of Et over a fixed G-equivariant infinitesimal deformation of G x# Et. But ig(£'c) : Et —> G Xfj Et is a closed and open if-equivariant embedding, and the if-equivariant relative cotangent complex LEt/GxHB, IS 0- This implies that there is no obstructions to lifting the deformations and the deformation space is trivial, which implies the etaleness. Finally we show the full faithfulness. Since it is etale, we only need to show that it is fully faithful when restricted to the open dense substack where the underlying if-curves are smooth. Moreover since it commutes with the projections onto Mg,n, we only need to show it is fully faithful on the fibers, which is a result in Appendix B . • The inflation being etale preserves stability, and therefore induces a morphism Infg : M"n -> Mgitl. Corollary 5.9. Suppose n > 1. Then the following diagram a H Infg - w ; is cartesian, where the vertical arrows are the stabilization morphisms. Proof. Since n > 1, using the inflation morphism, we identify 9JlH (resp. MH) with a an open substack TtH' (resp. MH ) of S0TG (resp. M°), that is characterized by the property: an object e of STJTG (resp. M°) is in TlH' (resp. MH ) if and only if there exists a G-equivariant morphism h : Et —> G / i f such that all the markings lie in the fiber above id i f G G / i f . Given an object e of 9JTG such that the stabilization ts is an object of MH , we have a G-equivariant morphism hs : Ets —> G / i f such that the markings of es lie in the fiber of id i f . Composing the G-equivariant stabilization morphism Et —> Ecs with hs, we get a G-equivariant morphism h : Ee —> G / i f such that the markings of e lie in the fiber of id i f , which implies that e is an object of TlH . • Chapter 5. Inflation 19 5.3 Inflation of morphisms Def in i t i on 5.10. Continuing with the notation of Definition 5.7, if we are given a G-scheme X and an if-equivariant morphism /,:_.,—> Resg X, define / f as the composi-tion of the morphism Inf^ / c : Inf^ Et —> Inf 0, Resf X with the action Inf^ Res G X —> X. This defines Infg : 9Jl^ n (Resf X ) -> 0JlGn{X), and again it induces Infg : AT^JResg X) M^n(X) as it preserves stability. We wil l write M."n(X) (resp. 9Jlf>n{X)) for A T ^ R e s g X) (resp. Wl£n(Res» X)). L e m m a 5.11. The diagram M H n { X ) ^ ! t m G n { X ) Infg • 2TtG is cartesian. Proof. For simplicity write 9 K H (resp. WlG) for 9Jt^„ (resp. 9 J T G J . Let <_G -> 9JtG be the universal curve over 9JtG, and let £ G —» 97tff be the pullback of € G under Infg : 9JcH —> 2JtG. B y definition of spaces of morphisms, the natural diagram M o r G „ ( e ^ , A - x TtH) - M o r G G ( € G , X x 9Jl G) •9JtG is cartesian. By definition of inflation, we have a cartesian diagram I n f g ( £ „ -<_ G Infg which implies that € G __ Infg <BH- Moreover, the morphism M o r g ^ I n f g <EH,X x TtH) - • M o r g a n , X x SPTH) induced by the inclusion -2H —> Resg Infg is an isomorphism by Lemma 5.1, which shows the result. • P r o p o s i t i o n 5.12. Suppose that n>l. The following diagram Infg M H -AC. w/iere ifte disjoint union is over all (3 e A^(X)H such that Infg 0 = a, is cartesian. Chapter 5. Inflation 20 Proof. This follows from Lemma 5.11, Lemma 5.9 and the observation that if /3 G A+(X)H is such that Infg maps m^n{X, (3) into MGn(X, a) then Infg /? = a. • Corollary 5.13. Suppose that n > 1. The following diagram H InfS r UpM"n(X,P)-^M^<n(X,a) where the disjoint union is over all $ e {X)H such that Infg/3 = a, is cartesian. In —J-f Ct particular the inflation morphism Infg : A4gn(X,P) —» Mgn(X,a) is an open and closed embedding of Deligne-Mumford stacks. Proof. By Corollary 5.12 and the observation that an object of VJl^n(X, P) is stable if and only if its image in WlGn(X, a) is stable. • 5.4 Compatibility of virtual fundamental classes under inflation Let H be a subgroup of G. To prove the compatibility of the virtual fundamental classes under the inflation Infg morphism, we only need to show that the obstruction theories are compatible (see [BF97]). More precisely if EG (resp. EH) is an obstruction theory for M^A(X,P) (resp. MgA(X,a)), then we want to show that ( I n f g ) * £ G = EH in Y)g(0—H . ) (where P = Infg a). The following Lemma is straightforward. Lemma 5.14. Let E be an H-scheme over T, with structure morphism p : E —> T. Let q : Infg E —> T be the structure morphism o/Infg E over T, and let t = tg(J5). Then the diagram ModG(0ln{a E) - i l ^ ModH(0E) Mod(Or ) commutes, that is there exists a natural isomorphism p^ o C ~ . Proof. The result follows from the observation that if U is an //-invariant open in E, and T is an object of M o d G ( C I N F G E), then (Infg U)G = T(U)H. • The following shows that the obstruction theories of MgA(X) and MgA{X) are com-patible under the inflation. Proposition 5.15. Let wp (resp. wq) be the relative dualizing complex of p (resp. q). Suppose f : E —> Res^ X is an H-equivariant morphism where X is a G-variety. Then we have Rp"H ® i*f*Slx) = R < / G H ® f*tox) in the derived category of OT-modules. Chapter 5. Inflation 21 Proof. First notice that up = t*toq as u is a closed and open embedding over T. Thus by Lemma 5.14, R p f K ® t*f*nx) = R p j V (w, ® = Rg?(u ; ,® • 22 Chapter 6 Groupoid actions, inertia varieties and stacks Suppose that G acts on a variety X. We've seen in Section 3.6 that there is a variety X^ called the inertia variety that is a natural target for the evaluation maps. In this section we will construct the inertia variety using the action of ( G ) , and describe the natural subvarieties corresponding to conjugacy classes of G . Also we describe the rigidified inertia stack as the quotient of the inertia variety by the groupoid G ( Q ) , as opposed to the inertia stack which is the quotient by G . The rigidified inertia stack is the natural target space for the evaluation maps from moduli spaces of twisted stable maps of [AV02], and hence appears in the construction of the orbifold (quantum) cohomology. 6.1 Quotient by a product of Cyclic subgroups Let G be a finite group, and 7 C G be a subset invariant under conjugation (for example a conjugacy class, or the whole group G ) . Let (7 ) be the subset of G x 7 given by ( 7 ) = {(g,m) G G x 7 | g G (m)} = {(g,m) e (m) x {m} | m G G } . It is a subgroup of G x 7 (as groups over 7 or 7-groups). The fiber of the projection (7) —• 7 above m G 7 is the subgroup generated by m. Let G ( 7 ) be the quotient of G x 7 by the right action of (7 ) over 7 . More concretely G ( 7 ) is the quotient of G x 7 by the relation (g, m) ~ (h, n) O m = n, h - 1 g G (m) and denote by [g, m] the equivalence class of (g, m). Given m G 7 , we have a cartesian diagram G / ( m ) * G ( 7 ) {m} ^ 7 The space G ( 7 ) has a natural structure of a groupoid over 7 : 1. the source map is [g,m] 1—• m, 2. the target map is [g, m] 1—> g m g - 1 , 3. the inverse is [g,m] h-> [ g - 1 , g m g - 1 ] , 4. the unit is m 1—> [id, m], Chapter 6. Groupoid actions, inertia varieties and stacks 23 5. and the product is ([g, m], [h,n]) i—> [gh,m] for m = hnh 1 . Denote this groupoid by BG^, where it is understood that the set of objects is 7 . The following result shows that if 7 is a conjugacy class, then B G( 7) is isomorphic as a groupoid (that is equivalent as a category) to a group. Recall that any groupoid has a canonical functor from the set of object (which is a groupoid with only trivial arrows) to itself. Hence we have a functor 7 —> B G( 7 ) . Lemma 6.1. Suppose that 7 is a conjugacy class in G. Choose an element m e 7 , and let Z( m ) be the quotient group Zc (m) / (m) where Z G (m) is the centralizer of m in G. We have a canonical equivalence of categories B Z( m ) c_ B G( 7 ) . More precisely the diagram Z(m) ^ {m} {m} *BGh) is cartesian, where the morphism {m} —> B G( 7) is the composition of the inclusion {m} —> 7 with 7 —> B G( 7 ) . Proof. The groupoid G( 7 ) acts transitively on 7 (as 7 is a conjugacy class), and it follows that {m} —+ B G( 7) is essentially surjective. The result follows from the observation that the arrows of BG^ fixing m form a group isomorphic to Z( m ) . • 6.2 A c t i o n s o f G( 7 ) Let 7 be an invariant subset of G (under conjugation). We present how natural actions of G( 7) arise from G-varieties. Lemma 6.2. Suppose Y is a G-variety, and a : Y —> 7 is a G-equivariant morphism (where G acts on 7 by conjugation) such that for all m € 7 , m acts trivially on Ym := a _ 1 ( m ) . Then the groupoid G( 7 ) acts on Y with anchor a. Proof. The group (7 ) acts trivially on Y over 7 , hence the canonical action of the groupoid G x 7 on Y induces an action of G( 7 ) . • In particular we have a cartesian diagram Y A- - 7 [ y / G ( 7 ) ] — v £ G ( 7 ) where [F/G( 7 )] denotes the quotient stack of the G( 7)-action on Y. Lemma 6.3. Suppose a\ : Y\ —• 7 and a2 '-Y2 —> 7 are like in the Lemma 6.2, and suppose f :Yi —> Y2 is a, G-equivariant morphism commuting with the morphisms to 7 . Then f is G^yequivariant with the G^-action defined in Lemma 6.2. Proof. The action of [g,m] e G( 7) defines a morphism Y™ —> y . g m g ; which is the same as the morphism defined by the action of a representative g of g (m), and the result follows since / is G-equivariant. • Chapter 6. Groupoid actions, inertia varieties and stacks 24 6.3 Inertia variety and rigidified inertia stack Suppose that G acts on a variety X. Definition 6.4. The group (7 ) acts on 7 x X over 7 , and denote by X^ its fixed point set, that is the fiber product X(i) *• 7 x X X *X x X where the rightmost vertical arrow is the product of the action with the projection, and the bottom arrow is the diagonal. When 7 is the whole group G , the variety X^ is called the inertia variety. When 7 is a conjugacy class, the variety X^ is called the 7-twisted sector of the inertia variety X^G\ Given a group homomorphism h : H —> G, there is a natural closed and open morphism X ^ —> X^G\ which is injective when h is injective, and which sends twisted sectors into twisted sectors. Since the image of Xm under the action of an element g G G is X g m g 1 , the diagonal G-action on 7 x X induces a G-action on X^ such that the projections XW —» 7 and XM X are G-equivariant. By Lemma 6.2, G( 7) acts on X^ over 7 . Lemma 6.5. Suppose f : Z —• X is a G-equivariant morphism. Then there exists a G(7y equivariant morphism such that {m} x Zm Z ( 7 ) {m} x Xm > j ( 7 ) is cartesian for all m G 7 . Proof. The morphism / being G-equivariant, it sends the fixed point set Zm onto Xm for all m G 7 . It follows that above cartesian diagrams define the desired morphism, which is G(7)-equivariant. • Lemma 6.6. Suppose a : Y —> 7 is as in Lemma 6.2. Then the projection Y^ —> Y is G^y equivariant and has a canonical G(7y equivariant section. Proof. Given m G 7 , the fiber a _ 1 ( m ) is fixed by m and hence a subspace of Ym. Compose this inclusion with the natural identification of Ym with the fiber of Y^ above m. This defines a G-equivariant section of Y^) _> Y, which is G(7)-equivariant by Lemma 6.3. • Proposition 6.7. Suppose a : Y —> 7 is as in Lemma 6.2, and suppose f : Y —» X is a G-equivariant morphism. Then we have a G'^y equivariant morphism Y -* J fW Chapter 6. Groupoid actions, inertia varieties and stacks 25 given by composing the morphism —> constructed in Lemma 6.5 with the section of y (7) _> Y constructed in Lemma 6.6. In particular we have an induced morphism of stacks [GM\Y] - [ G ( 7 ) \ l W ] Definition 6.8. The stack [G( G ) \X( G ) ] is called the rigidified inertia stack of [ G \ X ] . Given a conjugacy class 7 , the closed and open substack [G^\X^] of [G^\X^] is called the 7-twisted sector of [ G \ X ] . Lemma 6.9. The quotient X^ [G{y)\X^] is a morphism of degree for any m e 7 . Proof. The diagram X^) »• 7 [ G ( 7 ) \ X ( 7 ) ] _ [ G ( 7 ) \ 7 ] is a cartesian, therefore, the degree of the morphism is equal to the degree of the quotient 7 —> [G( 7 ) \7 ] which is o(Z G (m)) o(G) y l > o(m) o(m)' • 6.4 A note on evaluation morphisms Suppose that E —> T is a G-curve over T with an admissible and balanced action, and suppose that £ C 75 is a twisted section, which means that the morphism E —• T is etale and E / G ~ T. Notice that the monodromy morphism E —> G is an anchor map (see Lemma 6.2) for the action of the groupoid G^ on E . This action has no fixed points, and the stack quotient [G(<5)\E] is isomorphic to T . Suppose that / : E —> X is a G-equivariant. morphism. By Proposition 6.7, there is natural morphism E ^ X<& into the inertia variety, and a morphism T ^ [ G ( G ) \ E ] ^ [ G ( G ) \ X ( G ) ] into the rigidified inertia stack. Chapter 7 26 Presentation of the moduli space of admissible 3-marked genus 0 curves The G W type integrals over A 4 3 ' (X, /?) do not have good properties analogous to the ordinary 3-pointed genus 0 GW-invariants. To get the G W type invariants that are struc-ture constants of a ring, we will construct a compactification of the moduli space of maps from a canonical marked G-cover of P 1 ramified over 0, 1 and oo associated to any triple of elements of G whose product is the identity. In what follows we will use the notation: := M^t{otit00} and -M^(X,p) = Q Mo >{0,l,oo} 7.1 Canonical element of To any triple m = (mo, mi,moo) £ G3 such that momimoo = id, we construct a canonical object of A 4 ^ ' m over SpecC, as the quotient of the cartesian product of G with the upper half plane by an appropriate action of a free group on 2 generators (the fundamental group of P 1 \ { 0 , l , o o } ) . • Let i := {z 6 C 9(z) > 0} be the upper half plane in C , endowed with the hy-perbolic metric. It is a Riemann surface isomorphic to the unit disk. The holomorphic automorphism group of EI is PSL(2, R) , and acts on H by a b c d az + b z = ;• cz + d Let B. := H U Q U {oo}. The action of the subgroup PSL(2,Q) of PSL(2 ,R) extends naturally to an action on H given by the same formula (with the obvious convention about dividing by 0, and acting on oo). The space H has the coarsest topology such that the inclusion H —> HI is continuous, the sets ML. := {z e H | 9(z) > r} U {oo} for r > 0 are open, and PSL(2, Q) acts by continuous morphisms. Note that the action of the subgroup PSL(2, Z) on <Q> U { o o } is transitive. Let T(2) be the principal congruence subgroup of level 2, which is defined as the kernel 1 -> r(2) PSL(2 ,Z) -> PSL(2 ,Z /2Z) -> 1 of the morphism reducing the matrix coefficient modulo 2. The reason we consider this group is that the quotient r (2) \H is conformally isomorphic to P 1 \ {0,1, o o } , and the quo-Chapter 7. Presentation of the moduli space of admissible 3-marked genus 0 curves 27 tient morphism H —> P 1 \ {0,1, oo} is a universal cover. It extends to a unique continuous T(2)-invariant morphism u : H —-> P 1 . Consider the 3 orbits of F(2) on Q U {oo}: { GVG7L 1 — - I odd G 2Z + 1, even G 2Z } odd J : = / ^ i , o d d . € 2 Z + 1 (oaa2 00 odd even odd G 2Z + 1, even G 2Z \ {0} > U {oo}. Hence after composing with an appropriate automorphism of P 1 \ {0,1, oo} we can assume that u :. . P1 I H I oo H-• oo. For each i G {0,1, oo} denote by Fj(2) the stabilizer of i in T(2). These are cyclic groups generated by the parabolic transformations OJQ 1 0 - 2 1 e r 0(2), a i 1 - 2 2 - 3 G ri(2), a o o : = 1 2 0 1 e rQ O(2). These are chosen so that OOOIOQO = id. The group F(2) is a free group generated by any two elements of { 0 0 , 0 : 1 , 0 : 0 0 } . The automorphisms 0 0 , 0 1 and Ooo £ T(2) are called the monodromies of respectively 0, 1 and 0 0 . This terminology is justified by the following construction. Let G be a finite group, and let m« G G for i G { 0 , 1 , 0 0 } be such that mominioo = id. We will construct a canonical object e m of M3'm. Let 6m : T(2) —> G be the homomor-phism given by #m(oj) = mj for i G { 0 , 1 , 0 0 } . Denote by Gm the right F(2)-space G with structure given by 9m. The group T(2) acts on Gm x H b y a - ( g , i ) = {g6m(a)~l,a • x), and let Em = Gm *r(2) ^ be the quotient by this action. Denote by [g, r] the im-age of the pair (g,r) G Gm x HI in Em, and let u m : EI —> Em be the composition -» Gm x 3HI, x '—• (id,x) with this quotient. For i G { 0 , 1 , 0 0 } , let of the inclusion : um(i) = [id, i] € Em, and let £„ Gn cr(2) Lemma 7.1. The triple em := ( E m , S m , < T m ) is an object ofM3'm. Proof. We need to verify the conditions in the Definition 2.4 in Chapter 2. Condition (a) is trivial as Em is smooth. Condition (b) follows from the fact that T(2) acts freely on H and that the union of the three orbits OQ, 0\ and Coo is the complement of EI in HI. For (c) we need the quotient of Em by G to be P 1 which follows from the fact that the quotient of H by the action of T(2) is P 1 . That E m i is a G-orbit of ami follows from the fact that Oi is a T(2)-orbit. Hence we only need to show that the monodromy of am^ in Em is mj = #m(oj) for each i G { 0 , 1 , 0 0 } . Chapter 7. Presentation of the moduli space of admissible 3-marked genus 0 curves 28 We start with i = oo. Consider the open set H i (see Section 7.1). It is an Qoo-invariant neighborhood of oo. Moreover ao • H i n H i = 0. It follows that the image of H i in Em is isomorphic to the quotient ( a ^ m o o ) ) \ H i . But ( a t m i n f t y ) ) \ H i is r o o(2)-equivariently isomorphic to an open disk on which GJoo acts by multiplication by exp (^ol?™ ))• This isomorphism is given by r i—> exp ^ o ^ T for r ^ oo and oo i-> 0. The result follows. The case i = 0 (resp. i = 1) follows from the case i = oo by applying the automorphism z i * ~~ (resp. z i—> to H . It sends 0 (resp. 1) to oo and conjugates ao (resp. a\) into a o o . • Q 7.2 Canonical presentation of M3 Q Recall that the monodromies at the markings give rise to 3 maps fii : A43 —> G, for i G {0,1, oo}. Translation of the markings gives rise to 3 actions of G on M3 , and since the monodromy of a marking stabilizes it, by Lemma 6.2 we have 3 actions of the groupoid G( G ) on M3 over G (where the anchor —> G for each action is the monodromy morphism / i , corresponding to the marking being translated). Let G2G^ act on A43 over G2 by translation of the markings indexed by 0 and 1. Consider the morphism G2 —» , given by (mo, mi) i—> e m where moo = ( m o n i i ) - 1 . For simplicity, denote the element ([go, mo], [gi, mi]) of G?Gs by [g, m]. We have a morphism :G2{G)->M° [g,m] H+ C [ g i m ] :=g- en where g • e m is the translation of the marking 0 by go and the marking 1 by g i . We will show that e_ : G2G^ —> is a stack quotient by a right action of the group T(G) (G) over G. The anchor for this action is given by G ? G 1 —> G , [g, m] i—> momi, and the action is given by ([go, mo], [gi,mi]) -a = ( [g 0 a,a _ 1 m 0 a] , [gia, a _ 1 mia]) for a G (moo)- This action preserves the anchor map, and hence is an action of a group over G . Now let c a : C[g j m] —> «[g,m]-a D e the arrow of M3 given by the G-equivariant isomorphism of G-curves EH*M = ° m xr(2) H - G a ~ l m a x r ( 2 ) H = £ C [ g m ] . a [h,r] i-> [ha,r]. Note that this is a well defined isomorphism as # a - i m a = a - 1 # m a . P r o p o s i t i o n 7.2. The functor e_ is an isomorphism of groupoids Chapter 7. Presentation of the moduli space of admissible 3-marked genus 0 curves 29 Proof. First we prove the essential surjectivity. Let e be an object of M3 . Let E%° be the connected component of <rC)00 in Et. We can assume that Ef3 is a branched cover of P 1 with ramification points over 0, 1 and oo. Let ue : M —» E°° be the completion of the lift of the universal cover u : HI —> P 1 \ {0,1, oo}. After acting by an automorphism of HI we can assume that uc(oo) = a e ) O 0 . Let rm. be the monodromy of ut(i) for i £ {0,1, oo}. For i £ {0,1}, there is a unique g, £ Gj ( ir i j ) such that gi • Uc(i) — (Te,i-We have an element ([go, mo], [gi, mi]) of G2Gy I claim that e[gi in] c± e. The morphism ut : H -> Et induces ve : Eg-img := Gm x r ( 2 ) H -> i? e given by v c([g,r]) = g • ut(r). This is injective as F(2) acts transitively on the intersection of any G-orbit on Et with the connected component E%°. It is surjective as any point of Et can be translated by an element of G into E%°. Moreover ve(o-t[gm,ti) = ve([gi,i}) = gi-ue(i) = ot^ for i £ {0, l ,oo}. This shows that ve is the required isomorphism. We now prove the full faithfulness. Let ip be an isomorphism between ei := e[g,m] and e 2 : = e[h,n]- By definition we have maps u, : HI —> EH such that p e , i ° « i = Pe,2 ° «2 where p c,j is the quotient morphism Etj —> P 1 . As ^  is a G-isomorphism £?C l —> i? C 2 over P 1 , there exits a unique (and hence we have the faithfulness) a £ G such that IP(U\(T)) = a • V,2(T) for all r £ H . The morphism ?/> sends markings onto markings: i>(o~ei/) = cre2}i. For i = oo we have ^(^(oo)) = a-U2(oo) = 1*2(00), which implies that moo = rioo and that a £ (rioo)-For i £ {0,1} we have ^(gi • " i W) = g« • W) = g i a • u 2 ( « ) = hj • u 2 ( i ) and it follows that gjmjgr 1 = hmjh^ 1 and h~ 1gja £ (n,), which shows [g, m] • a = [h, n]. Hence the full faithfulness. • 7.3 Compactification of the moduli space of G-equivariant morphisms from a G-cover of P 1 with 3 ramifications Denote by J7^ the subset of G 3 of triples (mo,mi, moo) such that momimoo = id. Given m £ A / - 0 , we consider the scheme of G-equivariant morphism M3G'm(X) := MorG(Em,X) from Em to X, and the subschemes Af^'m(X,P) of morphism of a given degree 0 £ A+(X)G. We are looking for a compactification of M^'m(X, 0) such that the 3 eval-uation maps extends. Notice that N^,m(X, 0) is the fiber of the forgetful morphism M3'm(X, 0) —> 97t G , m above e m . Hence the fiber above e m of the composition M^'m(X, 0) -9JtG'm —s- A ^ ' m (where the second morphism is the stabilization morphism) is proper and contains A ^ ? , m ( X , 0) as a dense open substack. The evaluation morphisms extend to the morphisms induced by the evaluation morphisms of M3 'm{X,0), and hence it is the desired compactification. Chapter 7. Presentation of the moduli space of admissible 3-marked genus 0 curves 30 Def in i t i on 7.3. Define 77G(X,P) (resp. [X, /?)), by the cartesian diagram 77% (x, p) — - M33'G(x, p) —> M°(X, p) 77° »- G \ G ) MG where 773 -* G2G) is given by (m 0 , m i , m ^ ) ^ ([id, m 0 ] , [id, mi]). Define the virtual fundamental classes [77°(X, p)]vir (resp. \M3°{X, P)]vir) of 77°(X, p) (resp. Ai$ (X,/3)) by pulling back the virtual fundamental class of A 4 3 (X,f3). L e m m a 7.4. Consider the left G2G^ action on M3 (X, p) by translation of the two first sections. Then the composition 77^ (X, P) —• M3 (X,P) —> [G2G<j\M3' (X,P)] is an isomorphism. Proof. The map G2 —* G2G^ is a cross section of the groupoid G2G^ over G2, that is it induces an isomorphism G2G^ xGiG2 —• G2G^ xG2G2G^ —> G2G^ where the second map is the /~<2 , s T(G) ~^ L ° ( G ) \^(GX product map of the groupoid. It follows that the composition G2 —> G2G^ —> [G2G^\G2Gy is an isomorphism, and the result follows from the definitions. • 7.4 Inflation Suppose / f is a subgroup of G. To distinguish between the groups H and G, let tH and e G denote the morphism e of the last section corresponding to the given group. For m €. 77 3 , the inflation of is naturally isomorphic to e G . It follows that the inclusion of H into G induces inclusions M3 —> . M 3 ' and Tv^ -> Tv^ such that the diagram 77" - T v f M3 M •3,G H I < ; I X 7 G 7VI3 ^ A I 3 is cartesian. We pullback the definition of inflation to stack of if-maps. Chapter 7. Presentation of the moduli space of admissible 3-marked genus 0 curves 31 Lemma 7.5. We have a commutative diagram 77° (X) M%(X) A 7 f ( x ) where all faces are cartesian. Let evg :77%(X) -+ (X< G ) ) 3 be the composition of 77%(X) -» J^(X) with the product of the three evaluation maps evj : 7*4%(X) —> X^G\ P r o p o s i t i o n 7.6. Given a € A + ( X ) G , we have a cartesian diagram UP^f(x,p)l^77%(x,a) (x^y where the disjoint union is over all 0 e A±(X)H such that a = $ 2 g e G / H * s cartesian. Proof. The result follows by cartesian diagram chasing from Lemma 7.5, Corollary 5.13 and the following two cartesian diagrams ( x w ) 3 — - (x^y H3 ^ G 3 and H3 ^ G 3 • 32 Chapter 8 Group quantum cohomology In this section we first review the definition of group cohomology of [FG03]. Then we define the algebra of invariant quantum parameters, and build the group quantum cohomology as the tensor product of this algebra with the group cohomology. We define a product on the group quantum cohomology using G W type integrals over the moduli stack constructed in Chapter 7. We then prove a functoriality property of the group quantum cohomology under the inclusion of groups with compatible actions on the variety X. 8.1 Group cohomology The following is an overview of [FG03]. Let G be a finite group acting on a smooth projective variety X. We first define the "cohomology" group. Definition 8.1. Using the intersection theory with coefficients in Q , define the Q-module A*{X,G) :=A*(X^). It is a left G-module under the action given by g - a ^ g - 1 ) * * for g G G and a € A*{X^). Let A*(X,G)m be the direct summand of A*(X,G) cor-responding to the closed and open subvariety {m} x Xm C X^G\ Give A*(X,G)m a Q-grading as follows: given c G Q , let Ac(X,G)m := 0 Ac~<m'z'x\z) C A*(X,G)m zcx™ where Z runs over the connected components of Xm, and where a(m, Z, X) was defined in Section 4.2. Lemma 8.2. The Q-grading on A*(X,G) is invariant under the G-action. Proof. To show that the Q-grading is G-invariant, it suffices to verify that m(m, j, Tp X) = m(gmg _ 1 , j , Tg.p X) for all g G G, which is a consequence of the fact that T X is a G-vector bundle. • Suppose 4>: H —> G is a homomorphism of groups. Then 4> induces an action of H on X. Define A*(X,4>) : A*(X,H) A*{X,G) as the morphism of iJ-modules induced by pushing forward under the open and closed Jf-equivariant morphism X ^ —> X^G\ Proposition 8.3. This defines a functor A* (X, —) from from the category of groups acting on X and homomorphisms compatible with the actions to the category of Q-graded Q-modules. Chapter 8. Group quantum cohomology 33 8.2 Quantum parameters Let G be a finite group acting on a smooth projective variety X. Definition 8.4. Denote by Q*(X,G) the semi-group algebra of the semi-group of G -invariant effective curve classes Af(X)G, that is the algebra on the symbols qG with P e A+(X)G, and relations qG\^ = qG1+p2 for & e A+(X)G. Denote Q*(X,G) by Q*(X) when G is trivial, and its generators by q13 with P e A^(X). The algebra Q*(X, G) is Q-graded by degqG:=j^P(ClTx). where T\ is the tangent sheaf of X. Suppose H is another finite group, and <f> : H —> G is a homomorphism of groups. Then (p induces an action of H on X. Define Q*(X, = ^ G ' * ™ g ' P which induces a homomorphism of algebras Q*(X,<P) :Q*(X,H)^Q*(X,G). This is a graded homomorphism as Ci (Tx) is G-invariant and Tx is a G-sheaf. Proposition 8.5. This defines a functor Q*(X, —) from the category of groups acting on X and homomorphisms compatible with the actions to the category of Q-graded algebras. 8.3 Construction of the group quantum cohomology Definition 8.6. The group quantum cohomology is the Q-graded Q[G]-module de-fined as the tensor product qA*(X,G) : = Q * ( X , G ) ® Q A * ( X , G ) where G acts trivially on Q*(X,G). Given a homomorphism of groups <p : H —> G, the homomorphisms Q*(X, <f>) and A*(X,4>) induce a homomorphism of Q-graded Q[if]-modules qA* (X, <j>) : qA* (X, H) —> q A * ( X , G) . Let i be the involution of X^ sending the component {g} x Xs isomorphically onto { g - 1 } x X&~\ using the identification X% = X^'1. Consider the diagram ^(x , / j )^ (x^) { 0 - w " G , i Chapter 8. Group quantum cohomology 34 where TTG,I is the projection on the ith factor for i G {0,1, oo}. Given pi G A*(X,G) for i G {0,1}, define P0*GPl- = qG(L° ^G.oo)* (^GflPO • 7I"G,lPl 1 ( e V o ) * J7^{X,0) M- »>g,g Lemma 8.7. If 0 ^= 0, then the morphism f forgetting the first marking is the universal G-curve over A4^'s'e (X,0). Proof. This result follows from a widely accepted result of Abramovich and Vistoli (it appears without proof.in [AV02]): the stack of twisted stable maps M2([X/G], 0) (where [X/G] is the quotient stack) has the universal curve identified with a stack that they denote by M~'3([X/G],0). It turns out that this stack pulls back to A f G ' ' d ' g ' s (X,/3) under the morphism M%'e's (X,(3) M2([X/G], 0) that we will construct in Section 8.7. The result follows from the fact that the universal G-curve over Ai2([X/G),0) pulls back to the universal G-curve over M2 .g.g (X,0). • P r o p o s i t i o n 8.8. This makes q A * ( X , G) into a Q-graded algebra with identity. Proof. First of all we prove that the product preserves the Q-grading. This follows from the computation of the dimension of the virtual fundamental class at f: d i m X + d e g / f * T X - a (m 0 , / f (c r f , 0 ) ,X) - a(mi, / f (<r f i i ) ,X) - a (m 0 0 , / f (CT f i 0 0 ) ,X) = -^^degffTX-a(mo,ft(o-jfi),X) - a(mi,/f(o-f,i),X) + a (m- x , / f (cr f i 0 0 ) , X) + dimM<7f ^ Xm~ by using the formula a(g, x, X) + a ( g _ 1 ,x,X) = dim X — d i m x Xs (see [FG03]). We prove now that the identity class on the non-twisted sector id G A* (X) is the identity of qA*(X,G). Let a G qA*(X,G) be a class supported on the sector Xs for some g G G . The coefficient of qG in the product id *a is by definition (4 o ev3)*(ev2 0- • A t ' i d ' s ' s ~ V , / 3 ) = (to 7T2)*(7ri cr • ev* A T ? (x,P) ) where ev ev 2 x e v 3 : A / f ' i d ' g ' g \x,0) X% x X% 1 and TTI : X& x X% 1 -> X% and 7T2 : X& x X&~ Xs 1 are the projections. First notice that J\f^'ld's's (X, (3) — 0. First A i f ' l d ' g ' g \x,0). We will show that if 0 0, ev* [ . M G ' i d ' g , s \x,f3) we note that the morphism ev factors through the morphism / :, A ( G ' l d ' g ' g (X, (3) —* Q „ „ — 1 A42' ' (X, 0) forgetting the first marking. As this morphism is proper of pure codimen-sion 1 by Lemma 8.7, we have / , 0. J^M^\x,0) 0 which implies that ev* A l G ' l d , s ' s \x,0) 7<3,id,g,g '3 the identity on A * ( X s ) . The proposition follows. If 0 = 0, then AT 3 , , u ' 6 ' 6 ( X , 0) = Xs and the above product is just the product by • Chapter 8. Group quantum cohomology 35 8.4 G-graded C7-algebra and the braid group action Definition 8.9. A G-algebra A is called G-graded if there is a decomposition A = 0 g 6 G A 6 compatible with the G-action in the sense that for all g and h s G 1. h • A g C A h g h _ 1 and 2. A6Ah C A g h . It is G-graded commutative if 0g<7h = (g • <rh)ag = CThh_1 • o-g. for all ( T g G As and crn G A h . We will show that the group quantum cohomology is a G-graded G-algebra. That it is G-graded commutative is related to the fact that we can lift the action of the symmetric group on M3 by permutation of the markings to an action of the braid group on 77% := G 2 . We first lift the ^ -ac t ion to a braid group action on 7*4%. Let the symmetric group S3 act on 7^1% by permuting the markings. In general, this action doesn't lift to an action on 77%, but a modification lifts to an action of the braid Q group B3 on two generators 601 and hi^. This /^-action on A t 3 and more generally on M%(X) is defined by 601 :=Poi 0 * 1 (Mo) &I00 : = P l o o 0 ^oo(Ml) where py G S3 is the permutation of the markings i and j, and U(UJ) is the translation of the ith marking by the monodromy of the jth marking. It is straightforward to check the braid relation. This action lifts to J73 as follows. Let 601 and bioo act on G 3 by boi • ( m c m i . m o o ) H-> (momim^^mo .n ioo) hoc • (moj in^moo) H-> (rriojmimoomj^.mi) These two maps preserve 77% C G 3 and respect the braid relation, inducing an action of B3 on 77%. Lemma 8.10. The morphism e_ : 77% —• 7^C% is B3-equivariant. Proof. We will prove equivariance for the action of 601 > the case of &ioo being similar. Let m G 77% and n = 601 ' m - We need a G-equivariant isomorphism <f> : Em ~ En such that <H<7m,o) = C n , i , <Kmo • <7m,i) = o"n,o and <j>{omt00) = a„>00. Recall that Em := G m x F ( 2 ) H and En :— G n x F ( 2 ) H (see Section 7.1). The above conditions are equivalent to 0([id, 0]) = [id, 1], 0([mo, 1]) = ^>([id,Q!o • 1]) = [id,0] and </>([id,oo]) = [id, 0 0 ] . Let 1 1 0 1 G PSL(2 ,Z) Chapter 8. Group quantum cohomology 36 and let <p : Gm x r(2) H —> G n x r ( 2 ) HI be the unique G-equivariant isomorphism such that 0([id,r]) = [id, 0O(T ) ] . That (p is well defined is a consequence of the relations (Poao4>o 1 = a i a n < i (t>oai4>Ql = a r 1 Q ; O Q : I - That it has the above properties is a consequence of the following calculation: <fo(0) = 1, <foao(l) = 0o(—1) = 0 and <fo(oo) = oo. • This induces a unique /^-action on 77% (X) such that the morphisms in the cartesian diagram 77% (X)—*M%(X) 77% **J/% are .B3-equivariant. Lemma 8.11. The class [Al 3 (X, p)]mr is invariant under the action of S3 by permutation of the markings. Proof. Let p be a permutation of the markings. The pull back of the universal curve over M.3 (X, 0) under p is naturally isomorphic to itself, and this isomorphism commutes with the universal morphisms to X. It follows that the obstruction theory is compatible under p, and hence the virtual fundamental class. • The class {7^% (X, 0]mr is invariant under the /^-action as it is invariant under the action of G by translation of any marking and under the action of S3 by permutation of the markings. By B 3-equivariance of 77%(X, 0) - 4 jrf%(X, 0) it follows that [77%(X, 0}vir is _B3-invariant too. The braid group £ 3 has an action on ( X ^ ) ) * 0 ' 1 ' 0 0 * given by boi • ((m0,x0), (mi,a; i) , ( m o o , ^ ) ) i-> ((momimQ 1 , m 0 • xi) , (m 0 , x0), (moo, £00) ) &I00 : ((m 0,a;o), ( m i . n ) , ( m o o , ^ ) ) H-> ( ( m o , i 0 ) , ( m ^ o o m ^ m i • Xoo) , ( m i , i i ) ) making the evaluation ev : 77%(X) —> ( x ^ ) ) * 0 ' 1 ' 0 0 * into a B 3-equivariant morphism. Consider the following G-actions on 77%, M% and 7sC%(X). Given g G G , and m G 77% let <5g-m = (gmog - 1 , gmig _ 1 , gmoog _ 1 ) . On 7*4% and more generally on 7^(%{X) the action is 5g • f = t 0(g) 0 *i(g) 0 *oo(g) • f for any object f of 74% or jrf%{X). Lemma 8.12. The morphism e : 77% —> M% is G-equivariant under the above actions. Proof. Given g G G and m G 77%, we need a natural isomorphism 5g • c m ~ e<sg.m. This is given by the morphism G m x r ( 2 ) H -> Gs^m x r ( 2 ) H , [h, r] h-> [hg" \ r] . • As before, this induces a unique G-action on 77%(X) such that the morphisms to 77% and _/Vf3 (X) are G-equivariant. The virtual fundamental class of 77%{X,0) is invariant under this G-action because the virtual fundamental class of 7^C%(X, 0) is invariant un-der any translation of the markings and because of the G-equivariance of the morphism 77%(x,0^7J%(x,p). Chapter 8. Group quantum cohomology 37 Theorem 8.13. The group quantum cohomology qA*(X,G) is a G-graded commutative G-algebra. Proof. To show that it is a G-algebra we need to show that the G-action is compatible with the product structure. Suppose <JQ and o\ are two classes of A*(X,G). Then .OO [ jvfpr , /?) ] ) = U G . O O ) * (5g • 7rG i 0CT07rG,lO 'l e V f [^/f'(X', /?)] ) = (TTG.OO)* (TTG.O (g • ao) TTG.1 (g • * i ) ev? [ ^ ( X , /?)]) by the invariance of the virtual fundamental class of AT 3 (X,f3), where Sg denotes the diagonal action of g on ( x C 3 ) ) ^ 0 ' 1 ' 0 0 ^ . Suppose uo (resp. a{) is supported on {mo} x X™ (resp. {mi} x X f 1 ) , then the product o~o *G o~i is supported on {momi} x X m ° m i because (ev^)* ao (evf)* <TI is supported on ^ m c m U m o m i ) - 1 m & p s ^ x jjfmoim u n d e r t o e v « That this product is G-graded commutative is a consequence of the £? 3-invariance of . the virtual fundamental class of J7f(X,P) and the B3 -equivariance of ev^: suppose o~i is supported on the twisted sector corresponding to nij for i G {0,1}, then iA7f(X,/?)]) (TTG.OO O b01)* (b*oi {^Gfi^G^i) e v f [ A 7 ^ ( X , / ? ) ] ) (TG.OO)* ( * I ( M O ) * faV^^o) e v f [ A ^ ( X , / ? ) ] ) ( T G . O O ) . (TrG.o^i^G.imr 1 ^ o e v f JJ^(X,/3) ) which shows that <ro *G<7i = c i * G m i 1 • °o- The relation ao * G cri = (mo • <7i) * g -co follows similarly by applying b^ instead of &oi- ^ 8.5 Inflation is a homomorphism Theorem 8.14. For any infective homomorphism <f> : H —> G of groups, the morphism qA* (X, cp) is a homomorphism of Q-graded algebras. Proof. We have to show that for pi € A*(X,H), qA*(X,<p)(pi *H P2) = qA*(X,<j>)(pi) *G qA*(X,(j>)(p2). F ix a e A f ( X ) G . By Proposition 7.6 il [jvf (X,a) /3€A+(X)"\ Infg/3=a A^"{X,I3) where i : ( X ^ ) 3 -> ( x ( G ) ) 3 is the product of the inclusion j : X ^ -> X ( G ) . Since ev f f is proper and i is a regular embedding (ev#)*i ! = i ! (ev G )* (see [Ful98] and [Vis89]). Now Chapter 8. Group quantum cohomology 38 we calculate: qA*(X,cA)(pi *H p2) = = Yl <lA*(X' # ° *H,3). (*H,lPl • **H,2P2 • (eVH)* [jvf / ? ) ] I / 3 e A + ( X ) " 1 V J) q f » 0 A*{X,4>)\(LonH,3)* (n*HtlPl-n*Hi2P2 • (ev*), [jvf (X, / ? ) ] } ( (i 0 7Tff,3)* ""H.IPI • 7T/f,2P2 • (ev#), ] T [X^f (X,/?)] / 3 6 A + P O " Infg fl=a / J = ^ <7GA*(X,0) { ( t o 7 r f f , 3 ) . ( T T ^ X P I • 7r^i2p2 • (ev„)»z ! [ A ^ X ^ ) ] " " ) j = £ QGJ*(,L ° ^ , 3 ) * f ^ i P i ' 7i#,2P2 • i ! ( e v G ) , [ A f ( X , a)] J £ 9G(<- 0 7TG,3)*i*i* f TTG,IJ*PI • •xG,2J*P2 • (evG)* [ A ? (X, a)] a 6 A + ( X ) G = £ 0 ^ G ^ ) * f TT*G,I3*PI • nG:2J*P2 • (evG)* [77°(X, a)] J Q £ A + ( X ) G = qA* (X, 4>) (Pl) * G qA* ( X , 4>)(p2). • 8.6 Degree 0 specialization and Fantechi-Gottsche group cohomology In this section we prove that the degree 0 specialization of the group quantum cohomology is isomorphic to the group cohomology of Fantechi-Gottsche (see [FG03]). Let m € 77°', and suppose we have cr, G A * ( X , G) supported on {m,} x X m * for i G {0,1}. We consider the coefficient of qG in the product ao * G o-\. Notice that ATG'm(x,o)~xmonxm\ and denote this intersection by Xm. Let TT : Em x Xm —> X m be the universal curve over Xm, and let / : Em x Xm —> X be the universal G-morphism . Then / is the unique G-equivariant morphism whose restriction / ° to Em x Xm (where Em is the connected component of Em containing all the markings crm >i) factors through the projection TT° : Em x Xm -y X m and the natural inclusion tm : X™ -> X. The obstruction theory on A " G ' m ( X , 0 ) ~ Xm is RTTg (f*flx <g> and its dual is RTTGJ*TX- The 0th cohomology of the above complex is just the tangent sheaf Tjfm of Chapter 8. Group quantum cohomology 39 Xm which is locally free as Xm is smooth. It follows that R7r G /* is locally free, and by [BF97] [Proposition 5.6], the virtual fundamental class is given by '77%'m(x, o)]m r = cr R 1 ^ / * TX • [Xm] = ( - l ) r Cr 1TG (f*nX ® LOVm) • [Xm] where r is the rank of R 1 ^ 0 / * T ^ - This is precisely the class defined in [FG03], and this proves the following result. Proposition 8.15. Consider the Q-graded ideal qA*(X,G)+ in qA*(X, G) generated by all qG with 0 G A~l(X)G such that 0^=0- Then there is an isomorphism qA*(X,G)/qA*(X,G)+ ~ A*(X,G) of Q-graded G-rings, where A*(X, G) has a ring structure defined in [FG03]. Note that the fiber of R V G / * T X over a point x G Xm is B.Em(f* TX)G, where fx : Em —> X is the unique G-equivariant morphism which is constant on with value x. The rank r of R 1 7 r G / * T x at x can be computed using the equivariant Riemann-Roch formula: r = d i m x Xm — d i m X + a(mi, x, X) + a(m2, x, X) + a(m3,x, X). 8.7 Orbifold quantum cohomology There is a natural inclusion of the orbifold quantum cohomology of the quotient stack into the group quantum cohomology. This inclusion turns out to be a homomorphism of Q-graded algebras, and it identifies the orbifold quantum cohomology with the G-invariant part of the group quantum cohomology. We review the definition of orbifold quantum cohomology (see [AV04]), which we will need only in the case of a global quotient stack. Definition 8.16. Suppose that a finite group G acts on a smooth projective variety X. Let A*orb([G\X}) := A*( [G ( G ) \A- ( G ) ] ) * A*([G\X(G>]) * A*(X^f. Let $X,G be the quotient X^ -> [ G ( G ) \ X ( G ) ] , and let * x , G = Krb([G\X}) = A*( [G ( G ) \X( G >] ) - A*(X^) = A * ( X , G ) be its pullback. There is a unique Q-grading on A*rb([G\X}) making <&*x G into a morphism of Q-graded abelian groups (with the Q-grading on A*(X, G) given in Definition 8.1). Consider the action over G 2 of G2G^ on ~M%(X) by translation of the markings 0 and 1. Let /C 3 (X) := [G2G^\A43 (X)] be the quotient stack. It is a moduli stack whose objects are admissible balanced G-curves with 3 twisted sections where the section oo is trivialized. Hence it is isomorphic to the moduli stack of twisted stable maps into [ G \ X ] , with a trivialization of the twisted section oo (see [AV02] and [AGV02]). It Chapter 8. Group quantum cohomology 40 has a universal G-curve and a universal G-morphism, and hence an obstruction theory and a virtual fundamental class. The morphism .A43'G(X) —> M^(X) constructed in Definition 7.3 is GfiL equivariant and hence induces a morphism JC3'G(X) —> K,G{X) on :3,G, the quotients, where K.3 (X) : \G2G^\M3 (X)}. We pull back the virtual fundamental class of lCx(X) to IC3'G(X). By Lemma 7.4, we have a morphism J7°(X) Ml'G(X) such that the composition with the quotient Ai3 (X) have the following commutative diagram /C 3 ' (X) is an isomorphism. We 77° (X) •M33'G(X) )cl'G(x) •M^(X) •JC°(X) and the virtual fundamental classes are compatible under any morphism in the above diagram as the universal G-curves and G-maps pull back. In particular </>*[A/"G(X, f3)]mr = [icl'G(x,py as <fc is an isomorphism. Definition 8.17. The evaluation morphism ev : M3 (X) ( X ( ° ) ) 3 is G(G)-equivariant and induces ev 2ram i^ -3, G /C 3 (X) -> [ G ( G ) \ X ( G ) ] 2 x X^ on the quotients. In particular the M33'G(X) ( l ( c ) ) 2 x l ( c ) >cl'G(x) [G(G)\X{G)? x is cartesian, where the vertical arrows are quotients by (G( G ) ) 2 . In [AGV02], the G W invariants for the orbifold quantum cohomology are integrated Q over the stack K,3 (X), and then multiplied by the order of the monodromy of the third marking. By integrating over IC3 (X) we will obtain the same numbers as K.3 (X) —> K.3 (X) is an etale morphism whose degree is locally equal to the order of the monodromy 3 Q Q of the third marking. This follows from the fact that M3 (X) —> Ai3 (X) is a stack quotient by the right action of (G) (by Definition 7.3 and Proposition 7.2). Definition 8.18. The orbifold quantum cohomology of [ G \ X ] is the Q-graded algebra qA*([G\X]) := Q*(X,G) ®Q A ^ ( [ G \ X ] ) with the multiplication defined as follows. Given a £ A^ r b ( [G\X ' ] ) for i £ {0,1}, define 7focro -Tf\cfi • (ev), cr0 *' a\ := ^2 1G(L ° ^0°)* (3€A+(X)G . ic3'\x,p) where 7fj : [ G ( G ) \ X ( G ) ] 2 x X^ -> [ G ( G ) \ X ( G ) ] is the projection on the ith factor and 7foo is the projection on X^G\ Note that ao *' o\ is G-invariant, hence it lies in the isomorphic image of A*orb([G\X}) in A*(X-( G)) under <$>*xG. Let ao * a\ be the preimage of ao *' a\ under $*x G . Chapter 8. Group quantum cohomology 41 It was stated in [AGV02] that.qA*([G\X]) is an associative algebra. 8.8 Invariant part of the group quantum cohomology By tensoring ®*X,G with Q* (X, G) we get a Q-graded morphism of Q-modules qA*([G\X])-+qA*(X,G). which will also be denoted by &*XG-Theorem 8.19. The morphism $*XG ^s a homomorphism of Q-graded algebras identifying qA*{[G\X}) with the G-invariant part of qA*(X, G). Proof. For simplicity write lC{0) (resp. 77(0)) for lcl'G(X, 0) (resp. A / f (X , /3 ) ) . Let q the quotient (xM){o,i,°o} [G{G)\X^]i°^ x X^G\ Then (i o Tfoo)* (TTS^O • Tfi^i • e v f \]C(0)]mr^ = (<• ° Tfoo)* (TTO^ O • TFI^I • e v f pv"(/3)p r) = («• o 7foo)*g* (g*7foCTo • 9*7fiCTi • e v f pv"(/3)]w r) - ( to 7r G i 0 0)* ( T r ^ o ^ a o • ^ GAG^I • e v f pv-(/?)]™ r) which is by definition the coefficient of qG in (Q*X QO~O) *G {&X G a i ) - ^ Chapter 9 Associativity 42 In this section we address the problem of proving the associativity of the group quantum cohomology. As in the classical quantum cohomology, this is related to numerical equiv-alences of certain divisors on a moduli space, which in our case is A t 0 4 . We don't know how to prove it for a general G , but we can get partial results for cyclic groups and full associativity for Z 2 . In the classical quantum cohomology, one considers the space .Mo,4, which is P 1 , and hence any two points are rationally equivalent, providing the famous W D W equations. In our case, it is not true (as suggested in [JKK03]) that these ratio-nal equivalences in A - T 0 4 pullback from It is still possible that these divisors are algebraically equivalent, which also has been claimed in [JKK03], but we found the proof unsatisfactory, and hence we will state it as a conjecture. 9.1 Reducing the problem Suppose that a\, a 2 and 0-3 are classes of A*(X,G) supported on {mi} x Xmi, {m 2 } x X™2 and {m 3 } x Xm3 respectively, where m i , m 2 and 1113 are elements of G. Then the associativity of the product in qA* (X, G) can be expressed as the relation (<7i * cr2) * 03 = (<7i * 0-3) * m 3 • <72. (9.1) Indeed [a\ * cr2) * 0-3 = (cr2 * m 2 1 • <7i) * 03 by G-graded commutativity. By the above relation we have (<r2 * 1 • <7i) * 03 = (cr2 * 0-3) * nig 1m.2 1 • o\ which is o\ * (cr2 * o~3) by G-graded commutativity again. We express the triple product (01 * cr2) * 03 as a single integral as follows. Given pa and A, G A+(X)G let 77°(X, pa) (resp. 77°(X, pb)) be the moduli space 77°{X, pa) (resp. 77G(X,Pb)) where the markings are indexed by the set { 0 a , l a , o o a } (resp. {Oft, lft, oo;,}). Let 77%xb(X,pa,pb) be the product o{Af^(X,pa) and 77%{X,pb), and let 77%lb(X, pa, pb) be the fiber product A ^ b ( X , pa, Pb) 77Gaxb{X, Pa, Pb) evoo,, x evo6 X(G)- •XW x X ( ° ) r n -1 vir where A is the twisted diagonal {x,m) H-» ((x,m),(x,m~1)). Let 77^xb{X, pa, Pb) be Chapter 9. Associativity 43 r , G "I vir r i vir the product of Afa(X,pa) and ^{X, /3b) . Consider the fiber product (X<G>) { 0 a , l a , * , l 6 , O O ( , } {0 a , l a ,OO a , 0b , l ( , ,CXD( , } T o o 0 X 7 r 0 t Let V a x 6 : {0a,la,ooa,0(,,li,,oO(,} {0 a , l a , * , l i , ,OO( , } be the natural evaluation maps, and let n : {o„,i„,Mfc,oot} ^ ^ ( G ) b e ^ projec-tions. Then the triple product (01 * 0 2 ) * 0 3 is equal to 0a, 0b where (3a and f3b run over A + ( X ) G . In particular the coefficient of qG is (^ooj* 7 r o a C T i 7 I "L 0 '27ri i ,o-3(ev a | b )*A ! A ^ x b ( X , / ? ) where J7°xb{X,0) is the disjoint union of J7Gxb(X, pa, Pb) over all /?<, and /3b e A j " ( X ) G such that Pa + Pb = P- Consider the forgetful morphism <j> : ( x ( G ) ) { 0 a ' l o ' * ' 1 ' " O O ! , } -> (A.(G )){Oa,i.,U,oo t} ) k t . ^ ( G ) j { 0 a , i o , i 6 l o o t } ^ X ( G ) b e t h e p r o j e c t i o n m o r p h i s m s . Then the above expression is equal to (*oo6)* ^ l a ^ l a ^ H 0 ^ 0 e v a | b ) * A ! 7?aXb(X, p) (9.2) To show associativity (see Equation 9.1) one has to show that the last expression is equal to (^ooi,)* flaG^laa^lbm3 1 ' a2(</> ° eV f l |b)*A ! (9.3) Consider the braid group B4 on four strands generated by Co„,ia, Cia,ii> a n < ^ Cit,,oo(,- It has a unique action on (X( G ) ){° a ' l a ' l i " 0 O | >} such that 'TiCij " x = ^i{x) • Kjx = KjHi{x) -j x ' x = a%x for k $ {i,j}- So for example: Cu,H • ( ( m O a . ^ O a ) , ^ , ^ ) , ^ , ^ ) , ^ , , ^ ) ) = = ( K , % ) , K m i i m i ' a V i 0 -a;ifc) . ( m i ^ z i j . t m ^ z o o j ) Chapter 9. Associativity 44 L e m m a 9.1. On intersection theory we have: Proof. We compute: l a , l 6 (^Oa^U^l^) = 7 r0 a C TlCl a , l i )7ri aCT2Cl o , l bTl i )CT3 = ^ O a O ' l (TflaCla.lb)* CT2 ( ^ l t C l a , l J * °3 = 7ToaCTl ( f r i^ i , , •!(,)* 0-2 (7t"l a )* C T 3 = ^ O a ^ l (Mla-lt)* 7fli)CT2'ri<I0"3 = 7J"0 a c rlm3^1 i ) c r27ri o C r3 • Since 7rooj, is invariant for the action of Ci„,it> t o show the associativity it suffices to t r Q 1 vir show that the class (<£oeva|{,)*A" \Jvaxb(X,(3)\ is Ci a,i 6-invariant too. In next sections we will reduce this to the problem of rational equivalences of some divi-sors on A40 4 . We proceed as follows. First we study the behavior of the virtual fundamen-tal class under the gluing morphism JT^^X, (3) —> M^(X, ft) where A = {O a, l a , 1;,, oo;,}. Then we study the behavior of the virtual fundamental class under the cutting edges mor-phisms TT^^X, (3) —> J7Gxb(X, (3). These two results together with a conjecture about the Q rational equivalence of some divisors on A40 4 yields the associativity. 9.2 Gluing Suppose that E\ and E2 are two flat projective G-curves over T, such that the action on every fiber is admissible and balanced. Let E be an etale G-scheme over T and let i\ : E —* Ei and i2 : E —• E2 be two G-equivariant embeddings of schemes over T such that the images of i\ and i2 lie in the smooth locus of E over T. Let m i : E —> G and 7712 : E —> G be the monodromies of E\ and E2 along the images of t\ and 12- Suppose that m i = m ^ 1 . Proposition 9.2. There exist a flat projective G-curve E overT, such that the action on every fiber is admissible and balanced, and G-equivariant T-morphisms l\ : E\ —> E I2 '• E2 —> E Chapter 9. Associativity 45 such that the following diagram of G-schemes over T E E2 commutes, and having the following universal property: given two G-equivariant mor-phisms f\ : E\ —> X and f2:E2^X such that f\°i\ = f2 o t2, then there exists a unique G-equivariant morphism f : E —> X such that commutes. Moreover the construction of E commutes with base change. Proof. Using the gluing of curves (non-equivariant) and its universal property (see [BM96]), one can glue G-curves and prove the above universal property. • Let Ai and A2 be two non-empty finite sets. Let g\ and g2 be two non-negative integers. For i e {1,2}, let fc* G Ai. Consider the fiber product J J i A i k i \ k 2 A 2 G - G x G where A(m) = (m, m 1 ) . By Proposition 9.2, we have a morphism of stacks Aiki\k2A2 where A = (A\ U A2) \ {ki, k2}. This is called the gluing morphism along sections k\ and k2. Using the universal property of gluing, we also have a morphism MGAlk.\k2A2{X,l3l,p2) ^MG+g2A(X,f3i + P2) where the left hand side is defined as the fiber product M%lkl{k2A2(X,pup2) jtfiiAi(X,Pi) x jrfg\M{X,p2) Chapter 9. Associativity 46 9.3 Compatibility of the virtual fundamental class under the cutting edges morphism Definition 9.3. Define the cutting edges morphism as the inclusion 7^ALKL]K2A2(X,pup2)^MGLTAI(X,p1) x M^A2(X,p2) in the last cartesian diagram. The general construction of the obstruction theory for stacks of the above form is given in the following. Definition 9.4. Given an Ar t in stack 9JT with an admissible balanced G-curve € —» Tl, let M(X) be the stable locus of Morg l(€,SDft x X). Let TT : £ -> M{X) be the pullback of £, and let / : £ —• X be the universal G-map to X. As in Section 4.1, we define the relative obstruction theory of M(X) over SD? as the pair {E,4>E) where £ : = R 7 r G ( u ; T ® / * f i x ) [ l ] and 4>E '• E —> -k^jq/gjt * s * n e natural morphism (see Section 4.1). We've already constructed the obstruction theories and virtual fundamental classes for stack of the form M3IA(X, p) in Section 4.1. Now we can construct them for A4A1XA2(X,Pi,(h) and A4Alk1\k2A2(X,Pi,P2). Using the above definition we just need to interpret MA1*A2{X, Pi,p2) and MALKL\k2A2(X,Pi,P2) as stable loci in a stack of G -equivariant morphisms. The following is straightforward. Lemma 9.5. Let %RA1XA2 be the product o/SJt^ and WIA2 • The universal G-curve £AXXA2 over WIAX XA2 is just the disjoint union of the pullbacks of the universal G-curves over TIA1 andWlA2- The stack 9JlAIXA2(X) := MOTGJ1a ^A (£A1XA2,X) decomposes into a disjoint union ^%XA2{X) = II mGMxA2{x,Pi,p2) Pi, & where the degree of the morphism is given by Pi and p2 on each component. Then MAIXA2(X,pi,p2) is isomorphic to the stable locus ofTlAIXA2(X,Pi,p2). Let 9JIyi1fci|fc2A2 ^e the product of 9RAI and WIA2- The universal G-curve £A1ki\k2A2 over 971,4^ 11*2^ 2 * s obtained by gluing the pullbacks of the universal G-curves over SUl/^ and WIA2 along ki and k2. The stack Tl^kl^^(X) := M o r ^ ^ {£A^\k^A^x) decomposes into a disjoint union OTW^2P0 = II anWA 3(*.A,/%) A , ft where the degree of the morphism is given by Pi and p2 on each component. Then MAlkl\k2A2(X,Pi,pi) is isomorphic to the stable locus ofVJlG^KL^K2A^(X,Pi,p2). Chapter 9. Associativity 47 Using Definition 9.4, we have the virtual fundamental classes [MAlxA2(X,p1,p2)]vir 1 vir •MA1ki\k2A2(X^Pl^2) The following proposition says that the virtual fundamental classes are compatible under cutting edges morphism. Proposition 9.6. We have A! \MAlxA2(X,p1,p2)]mr = p ^ ^ p ^ . A O J . Proof. The proof is a straightforward generalization of the proof of the axiom III in [Beh97a], where we replace locally free sheaves by locally free G-sheaves, pushforward of sheaves by equivariant pushforward, sections by twisted sections, the target space X by the inertia variety X^, etc. • Corollary 9.7. Let A! \Maxb{x,p1,p2)}vir = \Na\b{x,pup2))1 and let 77°^ (X, P) be the closed and open substack of ~\f°\b(X, P) where the monodromies at the sections ooa, c i a , <r\b and o^ are given by m i , m2, m 3 and m 4 respectively. The associativity in group quantum cohomology can be now restated as: Proposition 9.8. Let o\, a2 and 03 be classes in qA*(X,G) supported on the twisted sectors corresponding to m i , m2 and ni3 respectively. If the equality C i 0 l U ( > o e v 0 | 6 ) , [jr3(bm(X,p)]mr = (</>oeva|b), ^b^m(X,P) holds, then we have (a 1 * (72) * cr3 = CTI * [p2 * cr 3). Proof. As the group quantum cohomology is a G-graded G-algebra, note that (o\ * u 2 ) * CT3 is supported on the twisted sector corresponding to m ^ 1 = 01^2^13. It follows that in the equation Equation 9.2 (resp. Equation 9.3) we can restrict the class (</> o ev a | 6)* A! p7°xb{X, /?)] V W to Xmi x X 1 " 2 x Xm* x Xm* (resp. to X m i x Xm3 x I m 3 _ 1 r Q m -j vir X™4) which is equal to (<£°ev a|6)* A a | b {X, P) (resp. to (4>oeva]b by the above corollary. The result follows from Section 9.1. • ' X vir Chapter 9. Associativity 48 9.4 Compatibility of the virtual fundamental class under the gluing morphism We consider the morphisms </»: MA(X,p) ^WlA ip' : MAlk^k2A2{X, Pi,P2) —> 9^Ai*i|FC2>i2 that forgets the G-equivariant map into X. We also have the stabilization morphisms s:WlA^> MA s' : W l A l k i \ K 2 A 2 ~~* • ^ A 1 k i \ K 2 A 2 that stabilizes the G-curve. We have the gluing morphisms gl(X) : MAlH]k2A2(X, Pi,fo) -> MA(X, Pi + p2) gl : MAlkl][k2A2 -> MA il:WlAikl{k2A2^WlA. Lemma 9.9. The diagram MAlkllk2A2(X,pf(-^MA(X,p) Aiki\K2A2 gl WIA is cartesian, where horizontal (resp. vertical) arrows are gluing (resp. forgetful) mor-phisms, and where MAlki{k2A2(X,p):= MAik l\K2A2 /3i+/32=/3 MAlkl\k2A2(X, p) Moreover iX \MA(X,p)]vir Proof. We first prove that the diagram \K2A2 (x, p) — m A ( x , p) L 971 Axki\K2A2 gl is cartesian, where the horizontal arrows are the gluing morphisms and the vertical arrows are forgetful morphisms. Let T be a scheme and let f: T —> WlA(X, p) and (ei , i2) '• T —> ^•A1ki\K2A2 D e m o r p h i s m and let 6 be the natural transformation making the diagram T e-+WlA(X,p) L Alkl\K2A2 Chapter 9. Associativity 49 commute. We need to show that there exists a unique morphism T —> ^IR Axk^k2 A2[X, P) commuting with the gluing morphism and the forgetful morphism. By definition of 3JI j4 ifc1|fc2^2(X", /?), we need morphisms f* : T —• SDT^pQ for z G {1,2} such that evj^ ofx = i o ev^2 of2 where i is the involution on X^G\ and such that locally the degrees of / n and / f 2 add up to p. Let := Eti, £ C i := E C i and cr f i i f c := cr e i i f e. Define / f / by where i$ are the inclusion into the glued G-curves. This is the desired morphism and it is unique with the above properties. The first statement follows from the fact that the gluing morphism is representable and hence preserves stability. The second statement follows from the first and [BF97, Proposition 7.2]. • Consider the diagram MA1ki\k2A2(X>P) ' •21-03 •MAlk1\k2Aa gl •MA(X,p) — i -9JU •MA where every square is cartesian (by Lemma 9.9), and where h and I are uniquely defined by universal property of fiber products and the commutative diagrams MA!kl\k2A2(X>P) L Aiki \ k 2 A 2 ^ A i k i \ k 2 A 2 MA(X,p) WIA •MA Proposition 9.10. We have M A^ki\k2A2\ (x,p) =gmMA{x,p)]mr). Proof. As both MAlk^\k2A2 and MA are smooth of pure dimension, the gluing morphism is a local complete intersection in the sense of [Ful98] and [Vis89] and has a well defined orientation [gl] in bivariant intersection theory (see [Ful98, 17.4] and [Vis89, Section 5]). Chapter 9. Associativity 50 We first claim that one has the following relation in the bivariant intersection theory f, [gl] = s* [gl]. We adapt the proof of [Beh97a, Proposition 8]. This claim follows from the three facts: I is birational, s is flat and gl is a local complete intersection. We first show that I is birational. That the restriction of I to the open dense substack A4Alkl jfc2>i2 of 9RAiki\K2A2 is an isomorphism onto its image is clear as the stabilization morphism restricted to A 4 A l h 1 ^ 2 A 2 is the identity. We only need to show that I is surjective. A n object of A^Aikt-\K2A2 o v e r SpecC is a pair ( e i , e 2 ) where for i G {1,2}, e , is an object of M.Ai such that the monodromy of Etx at oti is inverse of the monodromy of Et2 at <je2. Let f be an object of WIA and suppose we have an isomorphism 4>: gl{t\,t2) — s(f). Hence we have a diagram Etl II Et2 ~ *" Et where e is some object of MA, and where s contracts the unstable components and t identifies the orbits G • <rCi,fci and G • ot2^2. Let p = gl(otlfa) = gl(o~e2tk2) G Et, and let q G Ef be such that s(q) = p. There exists E^ and E^2 C Ef such that Eh U E, E, Eh H Ef2 = G • q s(Eb) = gl(Eti). Indeed choose E^ to be the union of irreducible components of Ef lying above t,(Etl), and let E^2 be the closure of the complement of E^. Induce the twisted sections and the sections on Eft from Ey Thus we have produced objects f i and f 2 of VJIA1 and WIA2 respectively such that f ~ g l ( f i , f 2 ) . Since i is injective when restricted to EH for each i G {1,2}, there is a unique morphism Sj : E^ —> Eti completing the above diagram. It is clear that the Sj's are contracting the components to be contracted. This produces an element ( f i , f 2 ) of T \ A l k 1 ] k 2 A 2 s u c n that i ( f i , f 2 ) is isomorphic to the object ( ( e i , e 2 ) , 4 > , f ) of 03. The claim follows from the birationality of I. The rest of the proof is as in [Beh97a, Axiom V] : gl ! {MA(X,P)]vir = (s* [gl]) {MA(X,P)]vir = (l,[i\\)\MA(X,P)]vir = KiX\MA{X,(3)]vir h* MA lk l \ K 2 A 2 (X,P) where s* [gl] = U |gl] was proved above. The following is a straightforward consequence of the above proposition. • Chapter 9. Associativity 51 Corollary 9.11. We have K £ [A 7 ^ 1 ^|^y l 2 (^/3 i ! /32) ]" r =A7 ( 5 0 ! ( [>l A (X ) / J ) ]^) 01+01=0 where 77(gl) is the composition j7Alk^k2A2 —> MAlk1]k2A2 —> MA, and h! : 21' —> . / V ^ X , / 3 ) is £/ie pullback ofJT(gl). 9.5 Associativity in group quantum cohomology Note that A7^ jj, maps isomorphically onto the subset of G4 consisting of quadruples (mi, m2, m3, m4) such that mim2iii3m4 = id by the map ((mo^mi^mooj , (m0b, m i 6, m^)) ^ (m0a, mi a, m^, m^). The braid group B4 has a natural action action on G 4 defined as follows: Co0,i0 • (m0o,mio,mi i ),mOOi)) = Kmi,\ 1,mo.,mi l,moo l) Cia.it, • (moaImia>mi!,>moo6) = (mo^m^m^m^Smi^mooJ Cib.ooi, • (mo^mi^mi^mooj = ( m 0 ( l , m i a I m i l m O O ! ) m ^ 1 , m l b ) and this action preserves A / ^ b under the above inclusion. It also has a natural action on defined as follows. Given an object e = (E,ooa,aia,aib,o00b) of A ^ ' m , where m = (m 0 o,mi o,mi i ),m o oJ G G 4 , let Coa,ia • e = (E,m0a • (Tia,croa,(rib,eTool>) Cia.ib' c = (E,o-Oa,mia • (7ib,aia,aoob) &b,oob • c = (E,a0a,aia,mib • <Toob,crib) • If a morphism f : ei —> e2 of is given by a G-morphism a of the underlying curves, then the morphism £ • f is given by the same G-morphism a, for any ( e B4. There are obvious natural isomorphisms Ci(C2«) — (GC2)e f ° r 0 £ - B 4 ensuring we have an action of a group on a stack. Even though gl(m) G MA'm, the gluing morphism gl : A 7 ^ 6 —> M°A is not B4-equivariant under these actions. This can be seen in the simplest case where the group G is trivial. The right hand side is isomorphic to MoA. The braid group B4 acts on ./Vfo,4 by permuting the markings, and has no fixed points. The left hand side A7^j6 is just a point, hence the morphism gl cannot be equivariant. On the other hand, it is equivariant up to rational equivalence as MOA — P 1 - We can speculate that gl is equiv-ariant up to rational equivalence for a general G . Unfortunately M A m is far from being P 1 in general. In fact, in general it is not even connected and the connected components aren't necessarily rational curves. To see this, take m = (mi, m2,1113, id), then by a The-orem of Abramovich-Vistoli, the forgetful morphism M°'m —> j^\G'mi'm2'm2 j s a u n i V e r s a l G-curve, and the fibers are curves whose connected components can be of arbitrary genus Chapter 9. Associativity 52 by the Hurwitz formula. In later sections we will show some examples where gl is equiv-ariant up to rational equivalence. For now, we will state the conjecture and show how the associativity follows from it. As before let A / ^ m be the locus where the sections have monodromies given by m, which in this case is just the point ( e (mi,m2,mf2 )' e(mi2,m 3,m 4) ) where m 1 2 = m i m 2 , and where e ^ m i m 2 m - i - j and e ( m i 2 m 3 m 4 ) are points of A / Q as defined in Section 7.1. Let e : Af°\b - » J ^ A m =>1 (e(m1,rti2,m1"21)' e(mi 2,m 3,m4; Conjec ture 9.12. The class g l , [A7^jb] is B^-invariant in A i ( A 4 G ) . More •precisely, given m G G4 such that m i m 2 m 3 m 4 = id, we have C*e* [m] = e* [C 1m] for all C G B4. We will prove this in certain particular cases in the next three sections. For now we will state the important consequence of this conjecture, namely the associativity. The braid group B4 has a natural action on A4A{X, j3), defined in a similar way to the action on M^, such that A4°_(X, p) —> 7/1^ is equivariant. The morphism M^iX,0) —• (x( G ))^ 0 o ' l o ' 1 '" o ° ! ^ is equivariant as well, where the action on (x-(G)^f0a'lo>1'>>00&} j s g j v e n Q -| vir A4A{X,P)\ is invariant under the action of B4 as it is invariant under permutation and translation of the markings. In fact for any ( € £ 4 , c* JtA^{x,p) M M^/~lm(X,P)^MG/"(X,p). ~* I 1m 1 1 as the action of £ restricts to an isomorphism P r o p o s i t i o n 9.13. The above conjecture implies that (<fi o ev a | b )» JT^^X,P)\ is B4-invariant. More precisely it implies that for any £ € B4, C*(0°ev a | 6 ) . [A7S b m (A-, /3)] W r = (c6oev a | b ) , A ^ B C Km{X,P) for any m G G 4 such that m i m 2 m 3 m 4 = id. Proof. First we note that </> o ev Q | b = ev^ og l (X) . Since evA is B4-equivariant, we need to show that c f g l p O , fttff{X,pj gl(X)JA7%bm(X,P) ~^\b~1 m(X,P) g l X [ A ^ b m ( X , / 3 ) ] g i : g l ! [ A ^ ' ' m ( X , / 3 ) ' ^ S * g l , [ A 7 G | b m ] n [ A 4 G ' m ( X , / ? ) We have by Corollary 9.11 Chapter 9. Associativity 53 where gl' is the pullback of gl under sotp, and the last step is by [Ful98] [Example 6.3.4]. by Conjecture 9.12 and We have C*gl* [ A ^ 6 m ] = C*C [m] = e. [C'm] = g l , [iJ^f m C* \A4A' (X,0)\ = MA (X,(3) by the above remark. Moreover ip and s are jE?4-equivariant, which proves the result. • By the last proposition and the Section 9.1 we have: T h e o r e m 9.14. The Conjecture 9.12 implies the associativity of the group quantum co-homology. In particular if m € GA is such that mim2m3ni4 = id and if o\, a2 and a3 are classes of qA*(X,G) supported on the twisted sectors corresponding to mi, m2 and m 3 respectively, then the equality Cia,ibe* [m] = c * [Ci^i,,111] in Conjecture 9.12 implies (CTI * a2) * a3 = ai * (1J2 * 0 3 ) . 9.6 Smooth locus of Tt^A ->• WlgA We begin by studying the ramification locus of the morphism WlGA —> 37ts>J4 that sends a G-curve to its quotient. L e m m a 9.15. Let E —• SpecC be a nodal curve with admissible generically free G-action, and let ITE : E —* C := E be the quotient. Let flE/C : = = ^ f i / ^ ^ C and k = | Stabc;(p)|. Then for any p e sp(E), we have WE/C,p) = * - 1. if p is a smooth point and ifp is a node. In particular TTE '• E —> G is etale except at points with non-trivial stabilizers. Proof. The result is trivial for smooth p e sp(.E). Suppose p is a nodal point, and q = ITE{P) which is also nodal as the action is admissible ( [Miz05]). Let E (resp. G) be the local scheme S p e c 0 £ ] P (resp. SpecOc ) g ) , where C B , p — SpecC[[a:,y]]/(xy) is the completion of the local ring £>E ) P, where x and y are both eigenvectors for the action of StabG(p), and similarly for Oc,q — SpecC[[u, v]]/(uv), where B H I ' and v 1—> yk under the quotient SpecC^p —* SpecOc,?-Then fig ~ C[[a:]] © C[[y]], and the derivation OE —> fig induces the C-derivation dE:C[[x,y]]/(xy)^C[[x]]®C[[y]] x '—> dx :— (1,0) 2/ <-> dy := (0,1). Similarly for fig ~ C[[u] ®C[[v]}, and dc- Hence the morphism fig —> fig of Og-modules, induces C[[u]] © C[[v]] —> C[[x]] © C[[y]], the unique morphism of G[[w,t;]]/(m;)-modules such that du i-> kxk~1dx and dv H - » kyk~1dy. It follows that f i n / c -.= nE/w*End ~ c i M ] / ^ - 1 ) © c i t y ] ] / ^ - 1 ) . • Chapter 9. Associativity 54 Theorem 9.16. Let M G A be the locus in M ^ A °f admissible G-curves whose nodes (if any) have trivial stabilizers in G. Then TT': M . G A —> M G , A is etale. Proof. Let M -.= M G A , M° := M G A and M := M G , A - Let t : T -> T' be a square 0 extension by a quasi-coherent ideal J. Let c be a family of M over T and c be a family of M. over T'. Suppose there is a morphism (a, t) : e —> c over t. Consider the diagram y •r where £ j is the unique deformation of S c to T' (the existence and uniqueness follows from etaleness of E c —• T) . We will study the obstruction to completing the above diagram with the doted lines, such that f : = (E^,pf, £ f , C f ) is an object of M ° , and such that £ * f = e. For simplicity let E := Ee and E := Et. The obstructions to extending the above diagram lie in the group ExtG'2(£lE/E,I) where 1 := v\ J ®oBt Z(E , ) and where I ( S e ) is the ideal sheaf of E e in Ee. Using the Grothendieck spectral sequence applied to the functor H o m G = H G ' ° oliom, to show that the above group is 0, it suffices to show that BG/£xi\SlE/E,l)=Q uG'1£xt\nE/E,i) = o HG>2Hom(flE/E,I)=0. Using Grothendieck flat base change theorem, it suffices to assume that T = SpecC. The group HE'2 Hom(flE/E,I) vanishes as E is of dimension 1, and RE'1 £xt1(n£,^,X) van-ishes as flE/E is concentrated at the nodes and markings. Next we show that £xt2(flE^E, J) 0. The exact sequence of G-O^-modules 0 -> 7 T * % ^ QE - 4 fl E/E where TT : E —> E is the quotient, induces a long exact sequence £xt\TT*Q.Ea) *£xt2(nE/E,l) Chapter 9. Associativity 55 where £x t 2 ( f2# ,Z) = O a s £ - > SpecC is a reduced complete intersection, that is flE has a two step locally free resolution. If p G sp(E) is a node with trivial stabilizer, it follows by Lemma 9.15 that £lE/Ep = 0. If p G sp(E) is a smooth point, q = 7r(p) is a smooth point and irEQE is a free O^-module in a neighborhood of p, and £xt1(Tr*ClE,X)p = 0. The two above cases imply £xt2(flE/E,l) = 0 . • 9.7 Zfc-covers of P 1 by P 1 Let G = Zfc =< r >. Suppose v\ = T, v2 = r _ 1 and v3 — id. Then M%>k$ / — SpecC. Proof. The dimension of is 0. Moreover the objects of M^k3" over SpecC are isomorphic to the fc-fold cover of P 1 with 3-markings where two are totally ramified. This cover has no non-trivial automorphisms, and the result follows. • Suppose v\ = r , v2 = r _ 1 and v$ = i>4 = id. We have an isomorphism MQk4 rz. P 1 such that the projection 7r : MQ^" —> Mo,4 — P 1 induces a degree k morphism P 1 —• P 1 with two totally ramified points. Proof. We know that M^l" is a smooth Deligne-Mumford stack of dimension 1. Let e := (E,J2,a) be an object of . M o 4 " over C, and let e := (E,o) be the projection onto A4o,4- Since E = E/Zk is a genus 0 curve, and the quotient E —> E has two ramification points o 1 and o2 of ramification degree k, by the Hurwitz formula for nodal curves we get that E has arithmetic genus 0. The automorphism group of c is trivial as there is no automorphism of a genus 0 nodal curve with 4-marked points. It follows that has no stacky points and therefore is a smooth curve. Since each smooth stable curve of Mo,4 has k non isomorphic Z^-covers with mon-odromies i/i,...,i>4, the morphism —> MOA has degree k, and it is a degree k cover o f P 1 . _ The action of Z*, on by translation of the 4th section, makes A4Q 4 ' " into a Z f e-cover of P 1 . Next we show that this morphism has 2 totally ramified points. Let T = ( P 1 , 1 , 0 , 0 0 ) be the 3-marked Z^-cover where Z& acts with fixed points 0 and 0 0 of monodromy r and T - 1 respectively. We glue two copies of T along 0 of the first and 0 0 of the second, and we can order the markings in two non-isomorphic ways. We get two point of A4Q 4 " which are fixed under Zk-Finally we show that there can't be more ramification points. By Theorem 9.16 the only possible ramifications are at points c := (E, E , o) where E has a node with a non-trivial stabilizer. Let x G E be a node. Let E — E\ U E2 be the decomposition of the quotient E into its two irreducible components, and let E\ (resp. E2) be the closed subvariety of E above E\ (resp. E2). Suppose <7j and Oj G E\ for i / j, then the monodromy of x on E\ must be ( ^ t ' j ) - 1 . In the case ^ or Uj is id, we get one of the two points of the last paragraph. Otherwise the monodromy of x is id and the curve E has no nodes with non-trivial stabilizer. It follows that MQ K± ~ P 1 . • Chapter 9. Associativity- be T h e o r e m 9.17. Let act on a smooth projective variety X. Let o\, o~2 and 03 be classes of qA*(X, Zfc) supported on the twisted sectors corresponding to id, id and r respectively. Then (a\ * a2) * o3 = o\ * [a2 * 03). Proof. Let m = (id, id, T , T _ 1 ) . Then for any £ G B4, C, • e m and e^.m are both points in A4Q 4'^  m ~ P 1 , so they are rationally equivalent, and the result follows from Theorem 9.14. • 9.8 Z2-covers of P 1 by genus 1 curves Let G = Z2 =< r >, v\ = ... = t/4 = T. The morphism 7r : M Q 2 ^ —> Mo,4 is an etale Z2-gerbe. Proof. We know that M Q 2 ^ is a smooth Deligne-Mumford stack of dimension 1, and that 7r is etale over the locus M^2^ of curves with nodes whose stabilizer is trivial by Theorem 9.16. We will show that M^f = M^f • Let e := (E, E,<r) be an object of M.Q4 over C. Let e := (E,o) be the projection to A4o,4- Suppose that E has two rational components (otherwise E itself is smooth and e is in M^\u), and let E = E\ L)E2 be the decomposition into its two irreducible components. Let Ei (resp. E2) be the closed subvarietes of E above E\ (resp. E2), and let p G E be a node (necessarily lying above the unique node of E). Let i, j 6 {1,2,3,4} such that i ^ j and <Tj, Oj £ S i . As £ 1 is rational, the monodromy of p on i? i must be ( i ^ ) - 1 = id, that is the stabilizer of p is trivial. It follows that n is an etale gerbe. • T h e o r e m 9.18. Let Z2 act on a smooth projective variety X. Then qA*(X, Z 2 ) is asso-ciative. Proof. Let 01, a2 and <r3 be classes of qA*{X, Z2) supported on the twisted sectors corre-sponding to T, T and r respectively. Let m = (T, T, r, r ) . Then for any £ G B4, ( • e m and *Cm = *m are both geometric points in .Mo 4™ which is an etale gerbe over P 1 , so they are rationally equivalent, and {a\ * a2) * 03 = 01 * (a2 * 03) follows from Theorem 9.14. Let < 7 i , o2 and 173 be classes of qA*(X,Z2) supported on the twisted sectors corre-sponding to id, T and r respectively. Let m = (id,r, r, id). Then for any ( G B4, ( • e m and e£.m = e m are both geometric points in M024,m ~ P 1 by the last section, so they are rationally equivalent, and (c i * o2) * 03 = 01 * [a2 * 03) follows from Theorem 9.14. The case where o~\, o2 and 03 are classes of qA*(X, Z2) supported on different sectors is reduced to the case covered in the last section by using the commutativity of qA*(X, Z 2 ) . • 57 Appendix A Equivariant Riemann-Roch for Nodal Curves A . l Introduction Let X be a projective G-variety over an algebraically closed field k. Let T be a G-Ox-module. The k-vector spaces Bl(X,Jr) have natural structures of ^-representations of G, and they define characters Xjp(X r) °f G. Define the G-equivariant Euler character-istic of T to be the virtual character Xx,r = Y^{-^)l^w(x,ry Let E be a projective nodal G-curve over k. Under certain conditions on the G -action on E, there is a Riemann-Roch type formula expressing the G-equivariant Euler characteristic Xx,F m terms of characters of the representations of stabilizers on the fibers of T. A.2 Pushforward and pullback Let X be a G-scheme. We denote by S h G ( X ) (resp. q C o h G ( O x ) , resp. PG{Ox)) the category of G-sheaves of abelian groups (resp. quasi-coherent G-C^c-modules, resp. locally free G-O^-modules of finite rank) on X. When there is no ambiguity about the structure of ringed space on X we will simply write q C o h G ( X ) (resp. PG(X)) for qCohG(Ox) (resp. PG(Ox)). Suppose / : X —> Y is a G-morphism of G-schemes. There is a pullback functor for sheaves f~l : S h G ( Y ) —> S h G ( X ) and a pullback functor for modules / * : q C o h G ( X ) —> q C o h G ( y ) (see [Gro57]). If X is noetherian or / is quasi-compact and separated, then there is a pushforward functor /* : q C o h G ( X ) -> q C o h G ( Y ) . If the G-action on Y is trivial, one also has an invariant pushforward functor fG : q C o h G ( X ) —> qCoh(F) defined on objects by fGT{U) = ^ r ( / - 1 C / ) G for an open U CX. Assume that G is a finite group and X is a G-scheme such that the quotient Y = X/G exists as a scheme. Let IT : X —> Y be the quotient morphism. It is proper, and by the preceding paragraph we have an invariant pushforward TT g : q C o h G ( X ) - • qCoh(y). We will now list without proof some properties of 7r G . Lemma 1.1. The canonical injection ~KGT —> TT^J7 and the canonical morphism 7r*7r* .F —> T induce a morphism of G-Ox-modules 7r*7rG.F —> -K*-K*T —> T that is an isomorphism away from points with non trivial stabilizers, thus injective. Appendix A. Equivariant Riemann-Roch for Nodal Curves 58 Def in i t i on 1.2 ([BorOO], 2.11). A G-scheme X is called loca l ly reduct ive if for each point P G sp(X"), the order of the stabilizer Gp is invertible in Ox,p-L e m m a 1.3 ([BorOO], 2.12 and 2.29). Suppose that X is a locally reductive G-scheme such that the quotient TT : X —> X/G = Y is flat. If T is a locally free G-Ox-sheaf of finite rank, then i r G T is a locally free Oy-module of the same rank and TZ1-KG{!F) = 0 for i > 0. A . 3 Cohomology Let Yx{Ox) (resp. Tx{Ox)) be the ring of regular functions (resp. G-invariant regular functions) on X. Note that Tx{Ox) is naturally a G-ring. We consider the following two functors Yx : qCohG{Ox) - qCohG{Tx{Ox)), TGX : q C o h G ( 0 x ) -> qCoh{TGx{Ox)) where the first one is the functor of global sections and second is the functor of G-invariant global sections. The two functors are left exact and the category qCoh G (C?x) has enough injectives ([Gro57]), hence we have the right derived functors ffx := P J I * : q C o h G ( O x ) -> qCohG {V x{0 x)), H G , i := R ' r g : q C o h G ( C x ) - _ q C o h ( r £ ( 0 j r ) ) . L e m m a 1.4. Suppose that X is a locally reductive G-scheme that admits a quotient Y = X/G. Then H* y(7rf JO ~ T3%\F) for any locally free G-Ox-module T on finite rank and for all i > 0. In fact there is a natural isomorphism between B.x 0 7 r G | p G ( o x ) and H G ' 1 |pG(e>x). Proof. We can write rGx = r Y O i t G and the result follows from the Grothendieck spectral sequence ([Gro57] 5.2.3) and from Lemma 1.3. • L e m m a 1.5. Suppose that X is a G-scheme such that \G\ is invertible in Tx(Ox)- Then In fact there is a natural transformation between T G o and H 0 / . Proof. We can write r G = r G o Tx where TG : qCohG(Tx(Ox)) -> qCoh(rG :((!?x)) (the functor taking G-invariants) is exact, and Rl T G = 0 for i > 0. The result follows from the Grothendieck spectral sequence. • Appendix A. Equivariant Riemann-Roch for Nodal Curves 59 A.4 Representions of finite groups and their characters Let G be a finite group and A; be a field whose characteristic does not divide the order of G. Denote by Rep f c G the category of ^-representations of G and G-equivariant fc-linear maps, and by Irr^ G the set of irreducible representations of G . Given V a representation of G , denote by Xy '• G —> k the corresponding character. The characters of G generate the character ring Chfc G . We have a pairing on Chfc G given by for any X\ and X2 G Chfc G . If V and W are representations of G and V is irreducible, then (XW,XV)Q is the multiplicity of V in W. The inverse i : G —> G induces by composition an involution of Chfc G , denoted by X ^ X Y . Note that ( A i ^ , A r 2 ) = (X1,XVX2). Let i f be a subgroup of a finite group G . Given a representation V of G we denote by Res G V the natural representation of H on V given by restriction. Given a representation W of H, the space G x W is a right fl-representaion with action (g, v) • h = (gh, h~l • v) and the quotient is a representation of G denoted by Indff W. The operations Res^ and Ind^ are functors between the categories Rep f e G and Rep f c H. They induce group ho mo morphisms between the underlying groups of character rings of G and H, that will be denoted by the same symbols. They are adjoint in the following sense: (Xu Indg X2)G = (Resg X\, X2)H (A.1) for X\ e Chfc G and X2 G Chfc H. We will list some properties of characters. • For any representation V of G Xv(id) = d i m K (A.2) • If k[G] denotes the standard representaion of G , then (A.3) Velrrk G • For any X G Ch f c G (A.4) V e l r r f c G L e m m a 1.6. If H is an abelian subgroup of G and X £ Chfc i f , then \ndGH{XvX) = X{id)Xm. V e l r r f c H Appendix A. Equivariant Riemann-Roch for Nodal Curves 60 Proof. The irreducible characters of G form an orthogonal basis for Chfc G under the pairing (—, —}G, hence by Equation A . l we have Y lndG(XvX) = Y £ (IndGr(XvX),Xw)GXw VElTikH W€lTTkGVeliTkH = ^ Y, (XVX, R e s G XW)HXw Welrvk G Velvik H = £ Yl (Xv,Xv Res" XW)HXW • W e l r r f c G V e l r r f e H The irreducible representations of the abelian group H are one dimensional, hence by Equation A.2 to Equation A.4 IndGr(XvX)= (Xy Res% Xw) (id)Xw VeliTk H W&xtk G = A' v ( id ) dimWXw Welvrk G = X{id)Xk[G]. • A. 5 K- theory Let k be a field whose characteristic does not divide the order of G. Let X be a finite dimensional G-scheme, projective over k and such that the structure morphism X —> Specfc is G-invariant. It follows that I is a locally reductive scheme and that | G | is invertible in Tx(Ox) = Tx(Ox) = k. Given an abelian category jrf, let K{srf) be the Grothendieck group of srf. The tensor product of G-Ox-modules induces a ring structure on K (PG(X)). Let T be a locally free G-O^-module of finite rank. B y a theorem of Serre ([Gro61b], 2.2.1) WX(T) are finite dimensional vector spaces over k. Moreover they vanish for i > d i m X by a theorem of Grothendieck ([Gro57], 3.6.5). The same is true for HG '*(J") ~ B. x (T)G. It follows that the two functors Yx : PG(X) - PG(A;) TG : PG(X) - P(fc) induce ring homomorphisms on K-theory K(TX) : K ( P G ( X ) ) - K (PG(fc)) ~ Ch f c G K ( r G ) : K ( P G ( X ) ) ^ K ( P ( f e ) ) ^ Z Appendix A. Equivariant Riemann-Roch for Nodal Curves 61 We define Xx (resp. X^) to be the composition of K(rx) (resp. K(TX)) with the isomorphism X : K (PG(fc)) ~ Chk(G) (resp. dim : K (P(k)) ~ Z). Given an object F of PG{X) we will write XX,T (resp. XGT) for XX([F]) (resp. #G([JF])). In fact we can express Xx in terms of XG. Lemma 1.7. Let F be an object ofPG(X). Then V 6 I r r f c G where F®Vy is by definition F®ox ^V where V is the free G-Ox -module corresponding to the constant G-vector bundle X x V. Proof. Xx,r = ^2 (XX,F,XV)XV v e i n * G v e i n * G = £ (Xx,r®vv,id)Xv-v e i n * G We used H ^ ( J r ) ® V v = H^(JT ® V v ) where V is a representation of G. B y Lemma 1.5, we have that i>0 = £ ( - l ) M i m f f x ( / - ® V v ) G i>0 = £ ( - l ) i d i m H G ' i ( J : - ( 8 ) y v ) i > 0 \>G — A-X,T®VW i and the result follows. • A.6 The smooth case Let G be a finite group and k be an algebraically closed field whose characteristic does not divide the order of G. Let E be a smooth projective G-curve over k. In this section, we will compute the homomorphism XE just as in [EL80] and in [Bor03]. Let P G sp(.E) and suppose F is a G-C^-module such that Fp is a free Gp-Op,p-module of finite rank. Define Xpp := Xpp G Oh*, Gp XGr : = d i m F G p G Z where Fp = k ®oE P Fp, which is naturally a fc-representation of Gp. Appendix A. Equivariant Riemann-Roch for Nodal Curves 62 De f in i t i on 1.8. If the set of points with trivial stabilizers is dense in E we call the G-action generical ly free. Suppose E is endowed with a generically free G-action. If P e E is a smooth point, then the stabilizer G p is cyclic as it has a faithful one dimensional representation on the tangent space of E at P. L e m m a 1.9. Let E be a smooth projective curve over k with a generically free G-action and let TT : E —* C be the quotient. Let F be a locally free G-0E-module of finite rank. Denote by QE the sheaf of Kahler differentials of E. For each P e E let ep = \Gp\. Then 1 1 e p _ 1 degTrfjT = — deg F - — V Y~ j- XGp n i 1 1 1 1 P€E j=0 Proof. The canonical morphism TT*TTGF —> F is injective and we have an exact sequence of G-Ofi-modules 0 -+ -K*-KGF ->F^K^0. (A.5) where 1Z is a G-C^-rnodule supported at the ramification points of TT : E —> C. Thus deg ir?F = —!— deg w*wGF deg7r = j ^ | (deg .F -deg7e ) = - L ( d e g . F - £ l ( f t p ) ) ' ' PEE where l(TZP) is the length of the CV^-module KP. Let P € E and Q = TT(P). B y localizing and completing Equation A.5 we have 0 -+ dE,P ® 6 c q F%p ^ F P ^ K P ^ 0. (A.6) Now O E , P — M M ] where u is an eigenvector for Gp, spaning a faithful representation with character XP:QE, which implies OC,Q — O g p — fc[[uep]]. We have Fp c± OE,P <8>fc Fp ~ k[[u]] <S>k Fp- The inclusion in Equation A.6 is just the homomorphism k[[u]} ®k[[ueP\] {k[[u}] ® f c FP)Gp -> k[[u}} ®fe FP coming from the inclusion (k[[u]\ <g>fc FP)Gp -> k[[u}} ®k FP by tensoring by k[[u]}. The group Gp is cyclic, hence the irreducible representations of Gp are all one di-mensional and their characters are homomorphisms Gp —> k*. We can talk about the eigenspace Vx of some character X and a representation V of Gp. The representation ^ E , P of Gp is faithful and thus powers of XptnE run over all irreducible characters of Gp. Appendix A. Equivariant Riemann-Roch for Nodal Curves 63 It follows that (k[[u]] ®k FP)Gp = 0 k[[u]]xv®k(FP)Xv Velrrk GP ep-l = 0 kM\Xi ®k {Fp)xi 3=0 ep-l = 0 k[[ue"]] • V> ®k {FP)x3 and e p - l <5 £,P ® A C Q ^ = 0 k[[u}} ® M [ U . P ] ] / c [ [ U e " ]K ® f c ( F P ) . e p - l < V? > ®k{FP). "P,Q 0  u J  ®k{FP) j where < wJ > is the ideal in k[[u]] generated by v?. It follows that. ep—1 Hi d i m ( F p ) ^ 3=0 e p - l ( A : p , n v . ; t > , ^ ) G p j=o • Now we compute X9 and X E . Proposition 1.10. Let E be a smooth projective curve over k with a generically free G-action and let it : E —• C be the quotient. Let F be a locally free G-0E-module of finite rank. For each P G E let ep = \Gp\. Then 1 1 1 1 P<EE j =0 / 1 \ 1 e p - l XE<r = W l degF + Xc,oc r^F) Xk[G] - — £ E 3 • ™%P ( x P ^ a E ) • v 1 1 7 1 1 P e £ j=o Proof. Using Lemma 1.4 and Lemma 1.5 we have XE:F = XQ^GJT. Applying the Riemann-Roch and Lemma 1.9 to irGF (which is locally free of the same rank as F by Lemma 1.3): XC,*GT = degTifjF -I- Xc,oc rk-T7 1 1 e p _ 1 = —degF + Xc,oc rkF - — E E 3 • (XpflV,XPtT)Gp 1 1 1 1 PeE j=0 Appendix A. Equivariant Riemann-Roch for Nodal Curves 64 and the first statement follows. By Lemma 1.7 XE,F — YJ X^-p^ywXy. V 6 I r r f c G We apply the first statement of the Proposition to each of the locally free G-O^-module T <g> V v to get XBf = E \W\ d e g ( ^ ® ^ ) + XcVc ® ^ V ) - T^T E E J " <4,fiv , ^ ® V v ) G p ) Xv V e l r r f c G \ 1 1 1 I P G B j = 0 ' E J = E U T7T d e ^ + xc,Oc r k f ) d k V 4 E b ' (4nv><*V R E S G P ^ V ) G P 1 * v M l / I I P C ;? „•_ n X) (Itesg i , AV ) 4 i n B A>^> G p AV= £ (Xv,lndGp^XPtQEXP^)GXv VehrkG V 6 l r r F C G • A.7 Normalization of a G-curve Let £ be a curve and n : E —> E be its normalization. Any G-action on i? induces a unique G-action on E making n into a G-equivariant morphism. L e m m a 1.11. Suppose E is a projective G-curve and 7r : E —> C is the quotient. Let rjB '• E —> E be a normalization of E. Then the induced morphism E/G —> C is a normalization of C. Proof. First note that given a smooth dense open subset C ' C C then the normalization of C is the unique smooth projective curve C (up to unique isomorphism) containing C' as a dense open subset together with a morphism r]G : C —> G whose restriction to C' is just the inclusion. Let n : E/G —* C be the induced morphism from the G-invariant morphism £" —> 15 —> G . Let E° be the smooth locus of E. Then E°/G is a smooth and dense subset of C . But this inclusion factors through E/G —> G . And as . E / G is smooth and projective, it is the unique smooth projective curve containing E°/G as a dense open subset, thus it is the normalization of G . • Appendix A. Equivariant Riemann-Roch for Nodal Curves 65 It follows that we have a commutative diagram E *E C ^ C where TJE and r\c are normalizations and TT and TT are quotients. A.8 Admissible action If / : Y —> X is a G-morphism of G-schemes, then it induces a canonical monomoprhims of stabilizers GQ —> Gj^ for every Q G Y. Definition 1.12. A G-morphism / : Y —• X is called fully faithful if for every point Q £ V the canonical monomorphism GQ —> G J ( Q ) is an isomorphism, in which case XQJ'T = xf(Q),r a n d {XQ,fT,XQ,fc)GQ = ( Xf(Q),F, Xf(Q),c)Gf(Qy Definition 1.13. A G-action on a curve E is called admissible if the normalization rj : E —> E is a fully faithful G-morphism. Lemma 1.14. Let G be a finite group and k be an algebraically closed field whose char-acteristic does not divide the order of G. Suppose E is a nodal curve with an admissible G-action defined over k. Then the quotient C = E/G is a nodal curve, and the projection TT : E —> G sends nodes to nodes. Moreover if E° and C° are the smooth loci of respectively E and C, then \ c\ c°\  = w\ £  |Gp|-1 1 P€E\E° Proof. Let P G E be a node and let Q\ and Q2 be the two points in E above P. Since 77 is a dominant morphism of algebraic varieties, we have an injective G-morphism OE —> V*®E of G-O^-sheaves. Localizing at P and taking the completion we get an exact sequence 0 -> k[\u, v]]/{uv) t k[[u)) © k[[v]] ^ k ^ O (A.7) where OE,P — k[[u,v]]/(uv), 0EQ ~ k[[u]] and O E Q 2 — k[[v]], the homomorphism <f> is given by p(u,v) (p(u,0),p(0,vj) and S by (p(u),q'(v)) H-> (p(0) - (7(0)). Since Gp fixes Q i and Q2, we have a Gp-action on k[[u]] and fc[[v]] making Equation A.7 Gp-equivariant, where Gp acts trivially on k. Since Gp is finite, there are positive integers m and n such that fc[[um]] = fc[[u]]Gp and fc[[t;n]] = fc[[w]]Gp. Since taking invariants preserves exactness ( |Gp| is invertible in k), we have that (k[[u, v]}/(uvY)Gp is the kernel of the restriction of S to k[[uk]} © k[[v% It follows that {k[[u,v]]/(uv))Gp ~ k[[um, vn]}/(umvn), which shows that 7r(P) is nodal since Cc,7r(P) — ( 7 I" GC ,B) i r(p) — ^ B , P -It follows that E\E° is the set of points lying above C\C° and the number of points above Q G G \ G° is T ^ - T for any P G £ \ S ° above Q. • Appendix A. Equivariant Riemann-Roch for Nodal Curves 66 A.9 Balanced action Let £ be a possibly singular G-curve over k. Suppose the G-action on E is admissible. Let Qi be the points on E lying above P G E. Then for any locally free G-O^.-module T of finte rank, XQ^JT are characters of the same group GQ{ = Gp as 77 is fully faithful. Def in i t ion 1.15. A n admissible G-action on a nodal curve E is ba lanced at a node P G E if XQLTQ- = XQ2 Q . A G-action is balanced if it is balanced at all nodes of E. A. 10 Equivariant Riemann-Roch for nodal curves T h e o r e m 1.16. Let G be a finite group and k be an algebraically closed field whose characteristic does not divide the order of G. Let E be a nodal projective curve over k with a generically free, admissible and balanced G-action, and let n : E —> G = E/G be the quotient. Let E° be the smooth locus of E and let r\ : E —> E be the normalization. Let T be a locally free G-Os-module of finite rank. For each P G E let ep = \Gp\. Then 1 1 1 1 PEE° j=o / 1 \ 1 e p - l * E ' T = ( TGI D 6 G ? 7 * - F + X c ' ° c r k JJ X"M ~ IG7 E E 3 • I ndg p (Xp^) • M l / I I P e E o j = 0 Proof. Note that the first statement follows easily from the second. Given a locally free G -0£-modu le T of finite rank, the canonical morphism T —> ri^rfT is injective. Indeed the normalization n : E —> E is a dominant morphism of algebraic varieties hence OE —> V*®E is injective. Tensoring with T we get an injective morphism T —• r)*0E ®oE ? and by the projection formula we have r]*0E ®oE J~ — V*1!*^- Thus we have an exact sequence of G-Og-modules 0 -> T -+ n+rfT -^K-+0 where 1Z is a G-Og-module supported at the singular locus of E. We have a long exact sequence of ^-representations of G 0 - HE(F) — K%{rt*T) > H° (K) -- ElE(F) - KE(ri*f) HE(n) 0 where we used a G-equivariant isomorphism ^(77,77*F) ~ YP^rfF) which follows from j] being affine G-morphism by a Grothendieck spectral sequence applied to the functor FE°V* '• qCohG(E) —• qCohG(/c). Hence we have the following relation between characters XE,F — XErj,jr — XE,TZ- (A.8) Appendix A. Equivariant Riemann-Roch for Nodal Curves 67 The curve E is smooth and we can apply Proposition 1.10 XE,V* T={W\ + XE/G,OE/G r k ^) Xk[G\ QGE 3=0 By Lemma 1.11 and Lemma 1.14, we have XE/G,Oe/c = Xc,o6 = (*c,oc + \C \ C°\) (A.9) = Xc,oc + E e p -1 1 PeE\E° Let P G E\E° and Q\ and Q2 be the two points in E above P. Since the G-action on E is balanced, XQ2>Q- = X^ n _ and it follows that e Q l - l e<32-l E i • I ^ G Q i ( < * W ^ , n J + E J • I n d c Q 2 (^.^4^) j = 0 j=0 ep —1 ep —1 = E J • I n d c p (^4^) + E J • I n d c p (xpAxl^y) 3=1 j=i ep—1 ep—1 By Lemma 1.6, we have e p - l E J • I ndg p (Xps*Lns) + E (<* - • I n C (^4^) 3=1 3=1 e p - l ep - E I n d G p (XpfxL^-eP • E I ndg p (xP^QxU_) (A.10) j=i = ep • ^  E I n d G p (*^4i,n J - I n d G p ^ j (A.11) = eP • I E I n d G p (*P^<*v) - I n d g p Afp^ \ V { E l r r f c Gp = ep • (Xp,H[d)" *fc[G] - I n d G p XPtr) = eP • (rk(^) • X K [ G ] - I n d g p ^ ) where we used that every irreducible character of G p = GQ1 is of the form XQIQ_ for some 0 < j < ep — 1 as the G-action on E is generically free. Thus using Equation A.9 to Appendix A. Equivariant Riemann-Roch for Nodal Curves 68 Equation A.11 and the fact that rj is an isomorphism over E°, we have XEtfT = ( | G J d e S ^ + Xc,oc R K ^ ) X m (A.12) j e Q - l - ] G | E £ j - I n d g Q ( * Q , ^ 4 , n J + |G7 E ep • rk(^) • ^ f c [ G ] ' ' PeE\E° E E i • l ^ o Q QeE\r)-^(E°) 3=0 ( | G ] d e g v * : F + Xc,oc r k ^ *k[G] + IGT E e P • Indg„ # P i Jr. ' ' PeE\E° And finally the second statement of Theorem 1.16 follows from Equation A.8, Equa-tion A.12 and XE,n = xn°E(Ti) = TQ\ E E P ' I n d G p *-P,^- ( A - 1 3 ) ' ' PEE\E° To show this, note that since TZ is a torsion sheaf H]j(7?.) = 0, and P€E\E° P£E\E° But for Q € £ \ £ ° we have 0 F p = I n d g Q F 0 . PeGQ And since Indg p Fp = Indg Q FQ for P e G • ( J , we have I n d g Q ^ = ^ £ I n d g p ^ 1 1 P6G-Q and Equation A . 13 follows. • 69 Appendix B Galois Covers B . l Some Conventions and Notations Let X be a topological space. The fundamental groupoid TTI(X) is the groupoid whose objects are points of X and whose arrows are homotopy classes of paths whose source is the ending point and the target is the starting point. The composition of arrows in ni{X) is given by the following rules. Given two paths 7 and 6 such that the source of [7] is the target of [8] we define [y][5] as the homotopy class of the path 7 o 5(t) = 7(2*) for 0 < t < \ and S(2t - 1) for \ < t < 1. Given x and y G X, denote by wi(X,x,y) the set of arrows whose target is x and whose source is y, by TTI(X,X, —) the set of arrows whose target is x, and by TTI(X,X) the group TTI(X,X,X). Given a G ni(X,x,y), denote by ca : n\(X,y) —> TV\{X,X) the homomorphism (3 i-> a / ? a _ 1 . Suppose p : X —> X is a covering. Let x G X and let x = p(x). Given a path 7 starting at x in X, denote by % the lift of 7 to the unique path in X starting at x. Given a = [7] G TTI(X,x,y) denote by x • a the element 7s(l) of E. This defines a right action of the fundamental groupoid ni(X) on X with anchor map p. Note that morphisms of covering spaces of X are 7ri(X)-equivariant. B.2 Pointed Principal C7-Bundles Let G be a finite group. Let p : E —> C be a principal G-bundle. Note that E is a disjoint union of coverings of C. Hence the fundamental groupoid TTI(C) acts on E with anchor map p. Since G acts on E by isomorphisms of covering spaces we have the following. L e m m a 2.1. The left G-action and the right wi(C)-action on E commute, that is for x G E, g G G and a G TT\ (C) such that the target of a is p(x) we have g • (x • a) = (g • x) • a. It follows that we have a well defined action A : G x TT1(C)opp x c E ^ E (g,a,x) h+g-x-a. It is easy to check that A is a transitive action if and only if C is path connected. A pair {E —> C,x) consisting of a principal G-bundle E —» C and a point x G E is called a po in ted p r i n c i p a l G-bundle . A morphism of pointed principal G-bundles is a morphism of principal G-bundles sending the marked point onto the marked point. L e m m a 2.2. Pointed principal G-bundles over path connected base space have no non-trivial automorphisms. Appendix B. Galois Covers 70 Proof. Let (p : E —> C, x) be a pointed principal G-bundle with G path connected. Let 4> be an automorphism. Let y £ E, and because A is transitive there exist g G G and /? G 7Ti(G,p(5),p(y)) such that g • x • (3 = y. Then since <f> preserves x and is both G -equivariant and 7Ti(G)-equivariant, we have <fi(y) = g • <p(x) • p = y which shows that <j> is trivial as y was arbitrary. • Let x G C and x G p~l{x). Given a G TTI{C, X), define 0(E,x)(a) as the unique element of G such that x-a = Q{E,x){a) • x-It follows from general properties of group actions that 0(E,X) 1S a group homomorphism w1(C,x)^G. L e m m a 2.3. Suppose C is path connected. Then E is path connected if and only if 9(E,X) is surjective. Proof. Suppose 9(E,X) is surjective. Let y G E and set y = p(y). Let a G TTI(C,X,y). Since 6(Etx) is surjective and A is transitive, there exists (3 G 7 r i ( G , X) such that 0(E,£)(P)'x'a = V-But 9(E,X)(P) • x • a = x • Pa which implies that x and y are in the same path connected component. The converse is trivial. • B.3 BG\C Let C be a path connected, locally path connected, and locally simply connected space C. Suppose we are given a point x G G and a homomorphism 6 : 7 r i (G, X) —• G . Let u : C —> G be the universal covering space of G , and choose x' G u~1(x). Given a G 7Ti ( G , X), there is a unique isomorphism <px'(a) : C —> G such that ^(a)(a ; ' ) = x ' • a. This defines an action of Tti(C,x) on C . Let be the quotient space Ee = G x T 1 ( C x ) G of G x C by the left action of 7i"i(G,:r) given by a • (g,y') = {g9(a~1),4>xi(a)(y')). Denote by [g,y'} the equivalence class of the pair (g,y') G G x C in Eg. Let —> G be the map induced by u : C —> C. We have a G-action on Eg induced by the G-action on the left factor of G x G , making Eg —> C into a G-bundle. Clearly Eg —> C is a principal G-bundle. The quotient map G x C —> Eg is a morphism of covering spaces of G , hence it is 7Ti(G)-equivariant. It follows that the right 7Ti (G)-action on Eg is given by EgXcTT^C)^ Eg L e m m a 2.4. Let 6 : TT\(C,X) —> G 6e o homomorphism. There exists a pointed principal G-bundle pg : (Eg,xg) —> (G, a;) SMC/I i/iai 0^Eg,xg) — 0- Moreover the pair (Eg,xg) with the above property is unique up to a unique isomorphism of pointed principal G-bundles. Appendix B. Galois Covers 71 Proof. Let Eg be the bundle constructed above and let xg = [id,x'}. Let P G iri(C,x), then we have 0(Eg,ie)(P) ' x0 = xe • P = [id, x • P] = [id,4>X'{P){x')] = [9{P)1x'} = 0(P) • xg and it follows that 9{Eg:Xg)(P) = 0(P). Let p : (F, y) —> (C, a;) be a pointed principal G-bundle such that 0(F,y) — B y the property of universal coverings, there exists a unique morphism : (U, x') —> (F, y) such that p o = u. Define : (G x U, {id,x')) -> {F,y) by $"{g,y') = g • T o P r o v e that descends to a morphism <£ : (Eg,xg) —> (F, j/) we must show that $" is constant on the 7ri(G,x)-orbits. Let a G 7Ti(C,a;), (5,2/') G G x U and /3 G iri(C,x,u(y')). We have = ^ ( a _ 1 ) •$'(^(")(2/')) = gO{a-x) •&(4,x,(a)(x'-P)) = g9{a-x) •<f>'(<Px,(a)(x')-P) = ge(a~1) • &{x' • ap) = g9(a~1) • fc'fV) • ap = gOia-1) •y-aP = g-y-P = g • •P = g • &{x' Thus we have constructed $ : (Eg,xg) —> (F, y). It is clearly G-equivariant, and thus an isomorphism of pointed principal G-bundles. It is unique as pointed principal G-bundles have no nontrivial automorphisms (Lemma 2.2). • Let G ^ 1 ^ ) denote the set Hom(7ri(G,x),G) of homomorphisms from wi(C,x) to G. The group G acts on the left of G^0'^ by conjugation on the target. L e m m a 2.5. Given two homomorphisms 9\ and 62 : iri(C,x) —• G, there exists an iso-morphisms of principal G-bundles Eg1 —> Eg1 if and only if 9\ and 92 lie in the same G-orbit in G ^ 0 ^ . More precisely, suppose h G G is such that 92 = Ch ° Oi, then there exists a unique isomorphism of principal G-bundles Qgltg2(h) : Eg1 —> Eg2 such that §gue2{h){xei) h 1 • xg2. Proof. Suppose first that 92 = Cho9\ for some h G G . We know that for i = 1 or 2, Egt is the quotient of G x U by the right action of ni(C,x) given by {g, y')-a = (g9i(a), (f>xi{a~l){y')). Denote by [g,y% the equivalence class of (g,y') in Egt. We define $g1:g2(h) : Eg1 —> Eg2 Appendix B. Galois Covers 72 by sending the equivalence class [<?,y']i to [gh 1,y']2. It is well defined as ^ A W I W W , ^ ^ " 1 ) ^ ) ] ! ) = ^ i ( a ) / i - 1 , 0 x ' ( a - 1 ) ( 2 / ' ) ] 2 = {gh-1ch(e1(a)),<f>x,(a-1)(y% = {gh-1e2(a),<f>x,(a-1)(y% = {gh-\y'h = * e i , f c W ( [ f f , S / ' ] i ) -Clearly $>g1;e2(h) is an isomorphism of principal G-bundles, sending xg1 = [id,x']\ onto [h~l,x'}2 = h~l • xg2. Again $gltg2(h) is unique such isomorphism as pointed principal G-bundles have no nontrivial automorphisms. Conversely suppose that we have an isomorphism of principal G-bundles $ : E$1 —* Eg2. Then let h e G be the unique element such that ^(xg^) = hT1 • XQ2. Then 92{p) • xg2 = xg2 • P = h-*(x01)-p = h • $(x6l • P) = h-Q{di{p)-xei) = h9l{p)-${xei) = ch(6i(P)) • *<h and it follows that 62 = c/, o 9\. • Let BG\c be the groupoid whose objects are principal G-bundles over G and whose arrows are isomorphisms of principal G-bundles. We summarize this section in the follow-ing. P r o p o s i t i o n 2.6. Let C be path connected, locally path connected and locally simply con-nected. Choose x e G . Let G act on G^C'X^ by conjugation on the target. Let G(G'!ri^C'x^) denote the transformation groupoid of this action (that is the groupoid whose objects are elements ofGni^c,x^ and whose arrows are pairs (g,6) where g e G and 9 € GWl(c'x) with source 6 and target cg o 9). Then we have an equivalence of groupoids 2 : G ( G 7 r i ( C ' x ) ) -> BG\C that sends a homomorphism 9 : TT\(C,X) —> G onto the principal G-bundle Eg, and sends an arrow g : # i — > 62 onto the isomorphism $e^,e2{g) '• Eg1 —> Eg2. Proof. We first check that E is a well defined functor. Let 0j e G 7 1 " 1 ^ ) for i = 1,2,3 and gi £ G for i = 1,2 such that 02 = cgi o d\ and #3 = cg2 o 92. Then by Lemma 2.5, S ( f f 2 : # 2 —* #3) 0 2 ( 5 1 : 9\ —> 92) is the unique isomorphism Eg1 —* £#3 sending a;^ o n t o gi1g21 • ^e 3, that is E(g2gi : 9\ —> #3). Thus 5 is well defined. We now show that S is fully faithfull. Let 9\ and # 2 £ GT1^C'X\ We want to show that 5 : H o mG ( G ' r i ( c ^ ) ) ( 6 , i ' 6 , 2 ) HomBG\c(Eei,Ee2) g ^$eug2{g) Appendix B. Galois Covers 73 is a bijection. It is injective as g is the unique element of G such that ^gltg2(g)(xg1) = g~~lxg2. It is surjective by Lemma 2.5. To show that E is essentially surjective, let p : E —> C be a principal G-bundle, choose x G p^1x, and note that by the Lemma 2.4, Eg{E - } is isomorphic to E. • C o r o l l a r y 2.7. / / A u t ^ - E g ) denotes the G-equivariant automorphisms of Eg overC, then there is an isomorphism : Aut°.(£#) —• Z G ( 0 ( 7 T I ( G , X))), such that for 4> £ A u t g ( £ e ) = * e ( ^ ) _ 1 -X0. Proof. Since S is a fully faithfull it induces an isomorphisms of groups H o mG ( G * i ( c , * ) ) ( M ) -> H o m B G | c ( £ ; e ) . The left side is just the centralizer ZC(9(TTI(C, X ) ) ) and the right side is the automorphism group AutQ(Eg). Let (f> G Aut^,(Eg), and let 5 G G be the element mapping onto <j>. Then by Lemma 2.5 <fi(xg) = g~lxg. • B.4 Principal G-Bundles Over Curves In what follows we assume that "Riemann surfaces" are not necessarily connected. If C is a Riemann surface, then for any (topological) covering p : E —+ C, the space E has a unique structure of a Riemann surface induced by p from C. Thus the results from the last section hold in the category of Riemann surfaces. Let C be a connected Riemann surface. Let xo,. • • ,xn G C be distinct points and let C° = C\ {x\,..., xn}. Given a finite covering p : E —> C°, there exists a unique branched covering E —> C and a map E —> E such that E E C° »- C is a pull back diagram. If E —> C° is a G-bundle then E —> C has a unique structure of a G-bundle such that the above diagram is a pull back diagram of G-bundles (Forster 8.4 and 8.5). For each 1 < i < n choose a path di in G ° U {x^} from £ 0 to x^, such that di passes through Xi only once. Given XQ G p~l(xo) we define £0 • di G E as l i m c _ i XQ • d\ where d\ is the path in G ° given by 11—> d,(ct). L e m m a 2.8. For each 1 < i < n, there exists an cti G wi(C0,xo) such that for every finite group G and every homomorphism 9 : 7 T i ( C ° , xo) —> G , the element 9(ai) fixes xg-di G Eg and acts on TXe.(iiEg by multiplication by exp(27rv/—1/fcj) for ki = \ Stabc(xg • di)\. Proof. F i x 1 < i < n. Let cj>: D2 —> G ° U {a;,} be an embedding of the closed disk D2 CC of radius 1, with image D, sending 0 to Xi such that XQ 0 <P(D2)- Let c G [0,1] be the Appendix B. Galois Covers 74 unique number such that dj(c) E dD and such that for all c < t < 1 we have di(t) E D° where D° is the interior of D. Let 6 be the loop: <5: [0, l]-+dD t ^ </>(exp(27rv /-li) • 4>~l(di(c))). Define Q j G 7Ti(C°, xo) as the homotopy class of the path d\ o (5 o I claim that Qj depends only on di. Let 0' : D2 —> C° U {ZJ} be another embedding of the closed disk D2 C C of radius 1, with image D'. Let c' G [0,1] and <5' : [0,1] -> dD' be defined as above with D replaced by D'. Assume that D' C D. Let dcc : [0,1] —> D \ {a:,} be the path given by 1>—> di(t(c' — c) + c) (the path along from di(c) to di(c')). Then it is easy to construct a homotopy between dcc o 5' o (d£ ) _ 1 and 5 in D \ {x^, thus [dio5'o{diri} = [5] and it follows that [d$°so {dir1} = [di o 4 o y o [dir' o (dir1} = [dfos'o(dfr1} which shows that a« doesn't depend on the choice of D' C D. Let G be a finite group and 9 : iri(C°,xo) —> G a homomorphism. Choose (p : D2 —» G° U {a i j} , .D, c and <5 as above. Then £ # | . D is a disjoint union of disks and there exists an embedding <p : D2 —• Eg\r) whose image is the component containing xg • di and such that D2 • Eg\o Pk D 2 D commutes, where pk is the map z >—> zk for k = \ Stabc^e)! (by Forster 5.10). Hence the element [6] G TTI(D \ {xi}, di(x)) fixes xg • di and acts on T^.^Eg by'multiplication by exp(27T\/—T/k), and so does on. • For each 1 < i < n choose a, G /KI(C°,XO) as in the Lemma. Let fTj < ai >—< oti > x • • • x < an > act on the right of Gn x G^c°'xo\ by G n x Q^(C,X0) x JJ < Q i > ^ G " X Gvi(c°>xo) i Let G act on the left of G n x G 7 ^ 0 - * 0 ) by G x G™ x <57ri(c'0>:i:o) • G 7 1 x ( j^ i lG 0 , ^ ) (g,9i,9) ' ^ (g%9~l,Cg o 9). Note that the two actions commute. Appendix B. Galois Covers 75 Proposition 2.9. Let MGCxi Xn^ be the category whose objects are tuples (p : E —> C, x\,... ,xn) where p : E —> C is a G-bundle that restricts to a principal G-bundle on C° and Xi G p~l(xi) are marked points, and the arrows are isomorphisms of G-bundles preserving the marked points. Denote by A/^ G x i x ^ the transformation groupoid of the G-action on the quotient (Gn x G7Tl(-c°,x°^)/(Yli < on >). Then we have an equivalence of categories & G : • A / ( C , x 1 , . . . , x n ) - > Mfc,Xl,...,xn) sending an object [gi,0] G (Gn x G 7 r i(G°' : ! ; o))/(rTj < cti >) onto (Eg —> C,gi-xg-di) and an arrow g € G from [gi,9] to [hi,p] onto the unique morphism of G-bundles &gtP(g) : Eg —> Ep extending $gtP(g). Proof. We start by showing that 9 G is well defined. Let (fcj) E Z™. Then giff(a^) -xg-di = gi • xg • di by Lemma 2.8. So © G is well defined on the objects. Given g G G , we need to check that QgtP(g) preserves the marked points. Note that hi = gig-1 and p = cg o 9. Thus we have ®e,cgoe(g)(9i • xg • di) = gt • §etCgoe{g){xe) • di = gig"1 • xCg0g • di — hi • Xp ' di. We now show the essential surjectivity. Let (p : E —» G, ii) be an object of MG(x\, ... ,xn and choose Xo G p~1(xo). Let E° be the restriction of E to G°. Let 6 — 9(B°,x0) a n d choose gi G G such that gi - XQ- di = Let us check that QG[gi,8] = (Eg —> C,gi • xg • di) is isomorphic to (E —> C,Xi) as pointed G-bundles. By Lemma 2.4, we have an isomorphism $ : Eg —> E° sending xg onto x$, inducing an isomorphism $ : Eg —> E. We need to check that $ preserves the marked points, which follows from the following computation $(9i • xe • di) = gt • $(xe) • di = 9i-xo-di We show that 0 G is fully faithfull as follows. Let 9 and p : TTI(C°,XO) —> G be two homomorphisms, and let gi, hi G G . Given $ : (Eg,gi • xg • di) —> (Ep, hi • xp- di), there is a unique g G G such that $(gtP)(g) is the restriction of <l to Eg, thus $(gtP)(g) = We need to check that g is indeed an arrow from [gi,9] to [hi,p]. By Lemma 2.5, we know that p = cg o9. We need to check that gtg^1 = hip(a^) for some ki G Z . We have hi-xp- di = §(gi • x0 • di) = gi-$(6,P){g)(xe)-di = gig'1 -xp-di that is h~1gig~1 G Stabc(xp-di). By Lemma 2.8, we know that StabG(xp-di) is generated by p(cti), and the result follows. • Appendix B. Galois Covers 76 Corollary 2.10. Let AutG (Eg, g± • xg • d\,... ,gn • xg • dn) denote the G-equivariant au-tomorphisms of Eg over C preserving the marked points, which is clearly a subgroup of AutG (Eg). Then the isomorphism ^g : AutG(Eg) —> ZC(9(K\(C° ,x0))) restricts to an isomorphism Ant%(Eg,gi -xg-du-.^gn-xg-dn) -» Zg(9{TT1{C° , x0)))n < 6(ai) > n... n < 9(an) > where < 9(on) > denotes the subgroup of G generated by 6(cti). Proof. A n element g e G gives an arrow from [gi, 9] to itself if and only if cg o 9 = 9 that is g € ZC(9(TTI(C°,XQ))), and ftg-1 = ft modulo Z that is 3 €< 9(oti) >. • B. 5 Inflation Suppose that i f is a subgroup of G or more generally that there is a homomorphism i f —> G. There is a functor G xH — from the category of if-spaces to the category of G-spaces which takes an if-space X and sends it to the quotient G x H X of the product G x X by the action (g,a;) • h »-> (g • h,h _ 1 a;) . The space G Xfj X has a natural G-action induced by the multiplication on G . This construction is functorial. Note that there is a natural if-morphism GxH : X —> G x# X induced by the if-morphism H x X —> G x X and the identification X ~ H XJJ X. Denote by 4 the inclusion i f —> G . For simplicity write A4H := M.¥N and A 4 G := A4£, „. v We will construct a funtor ML : MH —> A 7 ! 0 called the inflation. Given an object e := (p : E -* C,x\,... ,xn) of A4^Cxi Xny let A ^ e ) := (q : F —> C, yi,... ,yn) be the object of A - l 0 - , „ N constructed as follows: let F:=GxHE and for each i , let Vi •= G xH (±i). We use the convention that two functors of groupoids are said to commute if they are naturally isomorphic. Proposition 2.11. Suppose that 1 : H —* G is an injective homomorphism of groups. Then we have a commutative diagram of groupoids AfH ——*• j\4H A/"1 AfG — ^ - » - M G where A / 1 is the functor AfH —> AfG induced by the inclusion i f " X if 7 ri( c' 0> : ro) Qn x QTTI(C°,X0) Appendix B. Galois Covers 77 Proof. We have to construct a natural isomorphism r : 0 G ojV"' ~ A4l o QH. Let [hi, 9} £ (Hn •^H^c°'x°y)/(X[i < cti >), and let 9' = i o 9. Taking the quotient of inclusion H x C° —• G x G° by the action of 7 r i ( G ° , X o ) , we get if-equivariant morphism Eg —> which induces a G-equivariant isomorphism T{9) : G xH Eg —> Eg/ that sends G x# (xg) onto aV, and thus preserves the markings. This is the required natural isomorphism. • C o r o l l a r y 2.12. The functor : MH —• A 4 G is faithful and ifn>lit is fully faithful. Proof. By Corollary 2.10 and Proposition 2.11, the inflation AAl induces a homomorphism of groups Aut G (Eg,hi -xg • di) •—: >• Aut G (£ t o e, t-(hi) • xLOg • di) ZH(9(7T1(Co,x0))) n (6(ai)) n . . . n (9(an)) ^ Z G ( t o ^(7n(G°, x 0 ))) n (L o 0(ai)) n . . . n (t o 0(a„)> where the bottom arrow is an inclusion and hence A4L is faithfull. It is also an isomorphism if n > 1, and to the full faithfulness we only need to show that two objects in A4H are isomorphic if and only if their images in MG are isomorphic. B y Proposition 2.11 it suffices to show this property for AfH and AfH. Let [hi, 6} £ (Hn x Hv^c°'x°^>) / < on >), and suppose that for g £ G , we have [hi • g~l,cg o 9] £ (if™ x # 7 r i ( G °> x ° ) ) / ( rL j < a» >). Then since n > 1, we have cg o #(OJI) G i f , which implies that / i i • g~x £ H and it follows that g£H. • B.6 Monodromies Consider the map efc,Xl,...,Xn) •• (Gn x G^G°^))/(n <ai>)^Gn . It is invariant under the left G action, and thus induces a functor e (C ,x i , . . . , x„ ) • ^ ( C x i , . . . , ^ ) ~ ^ U where G is considered as a category associated to the set G . Given an n-tuple (i/\,..., vn) £ Gn, define Af^Xl x^(vi,... ,un) as the full subcategory of A ^ G X i x ^ with objects in the fiber of efCtXli„.tXn) above (J/I, . . . , un). Note that A ^ G X i Xn){vu • • •, vn) is the transformation groupoid of the G-action on the set {efcxi x -))~1(i/i, • • •, Vn)-Def in i t i on 2.13. Given a G-bundle p : E —> C of Riemann surfaces such that it is a principal G-bundle away from some x £ C. Given x £ p~x{x) we define the m o n o d r o m y of E —y C at x as the unique element vx of G that fixes x and acts on the tangent space TXE by multiplication by eyLp(2ir\f^l/k) where k = | Stab(j(x)|. Appendix B. Galois Covers 78 Proposition 2.14. Given an n-tuple (v\,...,vn) € Gn, the equivalence of categories in Proposition 2.9 restricts to an equivalence of categories JV (C , A ! L , . . . , * B )K •••,"n)-+ M f C t X l _ X n ) ( v U - • • , Vn) where M.GCxi^(v\,..., un) is the full subcategory of MGCxi x^ whose objects are the marked G-bundles (E —> C, x\,... ,xn) whose monodromy at Proof. Given [git 9) e (efcxi z , , ) ) - 1 ^ 1 ' • • • > ^ follows from Lemma 2.8 that the mon-odromy of xo • di is 9(cn). 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