Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Birational models of geometric invariant theory quotients Cheung, Elliot 2017

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
24-ubc_2017_may_cheung_elliot.pdf [ 316.01kB ]
Metadata
JSON: 24-1.0343974.json
JSON-LD: 24-1.0343974-ld.json
RDF/XML (Pretty): 24-1.0343974-rdf.xml
RDF/JSON: 24-1.0343974-rdf.json
Turtle: 24-1.0343974-turtle.txt
N-Triples: 24-1.0343974-rdf-ntriples.txt
Original Record: 24-1.0343974-source.json
Full Text
24-1.0343974-fulltext.txt
Citation
24-1.0343974.ris

Full Text

Birational Models of GeometricInvariant Theory QuotientsbyElliot CheungB.Sc., University of Toronto, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2017c© Elliot Cheung 2017AbstractIn this thesis, we study the problem of finding birational models of pro-jective G-varieties with tame stabilizers. This is done with linearizations,so that each birational model may be considered as a (modular) compacti-fication of an orbit space (of properly stable points). The main portion ofthe thesis is a re-working of a result in Kirwan’s paper ”Partial Desingu-larisations of Quotients of Nonsingular Varieties and their Betti Numbers”[3], written in a purely algebro-geometric language. As such, the proofs arenovel and require the Luna Slice Theorem as their primary tool.Chapter 1 is devoted to preliminary material on Geometric InvariantTheory and the Luna Slice Theorem.In Chapter 2, we present and prove a version of ”Kirwan’s procedure”.This chapter concludes with an outline of some differences between the cur-rent thesis and Kirwan’s original paper.In Chapter 3, we combine the results from Chapter 2 and a result from apaper by Reichstein and Youssin to provide another type of birational modelwith tame stabilizers (again, with respect to an original linearization).iiPrefaceThe topic of this thesis is based on the work of Kirwan (in [3]) and Reich-stein (in [6]).The content of Chapter 2 is based on [3]. However, the author does not di-rectly use any results from [3], and provides a novel variation of the originalideas. Therefore, the content of Chapter 2 is ultimately independent of [3],and is original work by the author.Chapter 3 is based on the work of Reichstein and Youssin (in [7]). The maintechnical results required for this chapter are directly borrowed from [7].However, it is an original observation that one may combine the work of [3](or Chapter 2 of this thesis), and [7] to provide a new result. This was anoriginal unpublished idea by the author’s supervisor, Zinovy Reichstein.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . v1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Geometric Invariant Theory . . . . . . . . . . . . . . . . . . . . . 4Luna Slice Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Kirwan’s Procedure and Stable Resolutions . . . . . . . . . 134 Abelianization Procedure . . . . . . . . . . . . . . . . . . . . . 22Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26ivAcknowledgementsI would like to express my first and foremost thanks to Zinovy Reichstein,who guided me through this Master’s project, and taught me everything Iknow about the subject of (Geometric) Invariant Theory. I am grateful forhis continual mathematical and financial support.I would also like to thank Kalle Karu, for agreeing to read my thesis andprovide helpful comments.Finally, I would like to thank Jim Bryan, who enthusiastically taught memuch of what I know about complex algebraic geometry.vChapter 1Introduction1.1 The purpose of this thesis is to investigate birational equivariant mod-els X˜ of smooth projective G-varieties X, such that there exists a lineariza-tion of X˜ with the property that all stabilizers of semi-stable points are finite(i.e. there are no strictly semi-stable points). If a linearization is providedfor X, the linearization of X˜ may be arranged so that X˜ → X is an iso-morphism along the properly stable locus of X. This work is heavily basedon Kirwan’s paper [3], where the author investigates such models. In thesecond chapter, we provide an alternative exposition of this work, includ-ing proofs entirely written in the algebro-geometric language instead of thelanguage of symplectic geometry. These new proofs make heavy use of theLuna Slice theorem and its various corollaries. In particular, the technicaldetails involved in Kirwan’s procedure may be interpreted and expressed interms of Luna’s stratification. In the case where X is generically free, wemay find a birational model X˜ab → X of X so that X˜ab has the propertythat all stabilizers of semi-stable points are finite, but in addition they arealso abelian. If G acts freely on Xps (given a linearization for X), thenX˜ab → X is an isomorphism along the properly stable locus of X.We assume that our base field κ is algebraically closed, and char(κ) = 0.1.2 We will prove the following theorem:Theorem (A). Let X be a G-linearized smooth projective variety. If Xps 6=∅, then there exists a sequence of blow ups:σ : XN := X˜ → XN−1 → · · · → X0 := Xsuch that XN satisfies:1. X˜ps = X˜ss. That is, X˜ contains only unstable and properly stablepoints.2. σ : X˜ → X is an isomorphism along Xps.1Chapter 1. IntroductionThe above theorem describes a sequence of blow ups performed on X sothat X˜ss consists only of points with finite stabilizers. We will call such avariety X˜ with a birational morphism σ : X˜ → X satisfying the conclusionsof the above theorem, a stable resolution of X. In fact, by [6] this mayindeed be alternatively achieved by performing a resolution of singularitieson the strictly semistable locus Xs.ss of X.1.3 If the action of G on X is generically free, then [7] constructs a se-quence of blow-ups peformed on X so that every reductive stabilizer sub-group of G is diagonalizable. We may combine this result with Theorem(A):Theorem (B). Suppose that X is a generically free G-linearized smoothprojective variety, and that σ : X˜ → X is a smooth stable resolution of X.Then, there exists a sequence of blow-ups:ρ : X˜M = X˜ab → X˜M−1 · · · → X˜such that1. X˜psab = X˜ssab. That is, X˜ab contains only unstable and properly points.Furthermore, any point x ∈ X˜psab has a finite and abelian stabilizersubgroup.Furthermore, if we have that G acts on Xps freely, then we may find asequence of blow-ups such that X˜M satisfies:2. ρ : X˜ab := X˜M → X˜ is an isomorphism along X˜ps. In particular, ρ◦σis an isomorphism along Xps.In particular, if X is a smooth G-linearized projective variety such that Xps= Xss, we may take X˜ = X in the above statement.1.4 Note that although X˜ps ∼= Xps, we do not necessarily have thatX˜psab∼= X˜ps ∼= Xps. In fact, this will only be true if we can choose theblow up centers containing only unstable points. This is possible if G actsfreely on Xps.However, the above theorem says that if X˜ has the property that X˜ containsonly properly stable and unstable points, then the same is true for X˜ab. We2Chapter 1. Introductionwill say that σ : Y 99K X is a stable model (of X) if: σ : Y → X isa birational morphism, and Y consists only properly stable and unstablepoints. In this case, we have that y ∈ σ−1(x) is properly stable if x ∈ X isproperly stable (see Proposition 3.4.1). In particular, a stable resolution ofX is a stable model of X with restricts to an isomorphism along the properlystable loci. Thus, Theorem (B) claims that if X˜ is a stable model of X,then X˜ab is one as well. If X˜ is a stable resolution of X, then X˜ab is one aswell, if G acts freely on Xps.1.5 Note that a stable resolution X˜ → X is not guaranteed to be smoothif the centers of blow-up are singular. However, one may show that whenchoosing the centers of blow up in the above theorem, the centers choosenare possibly singular only at unstable points. By performing equivariantresolution of singularities, one may always find a smooth stable resolutionX˜ → X. If this is done, then X˜ will be smooth. However, X˜ss will besmooth, provided that Xss is smooth. Then, X˜  G may be viewed as apartial desingularization of the GIT quotient X G in that the only singu-larities are finite quotient singularities. After applying the blow up sequenceρ, X˜ab G has at worse finite abelian quotient singularities.3Chapter 2PreliminariesGeometric Invariant TheoryWe begin this chapter with a brief summary of GIT. We assume that thereader has had a previous encounter with the relevant definitions.2.1 Suppose that X is a projective G-variety, and that there exists anample line bundle L on X with a G-action on the total space compatiblewith the G-action on X. Then, there is a power L⊗n such that the lin-ear system X ↪→ H0(X,L⊗n) defines an equivariant embedding of G intoP(H0(X,L⊗n)). In this context, G-action on P(H0(X,L⊗n)) is induced by alinear action on the vector space H0(X,L⊗n). We call a G-linearizable pro-jective variety X along with such an equivariant embedding, a linearizedprojective G-variety X (or G-linearized projective variety X).2.2 The linear equivariant embedding ofX into the projective space P(H0(X,L⊗n))allows one to define various notions of stability for points of X (i.e. prop-erly stable, unstable, semi-stable). This provides a decomposition of X intothe disjoint union of three G-invariant sets Xps(L), Xs.ss(L) and Xunst(L)(properly stable, strictly semi-stable and unstable). Recall that Xunst(L) isa closed subset of X. The significance of this decomposition is the follow-ing. One can show that a categorical quotient always exists for the openG-variety Xps(L) unionsqXs.ss(L) = X \Xunst(L) := Xss. This is of course theGIT quotient X LG of X. On the locus of properly stable points Xps, therestriction of pi defines a geometric quotient (thus, an orbit space). Thatis, there is a categorical quotient pi : Xps(L) → Xps(L) L G whose fibersparametrize precisely the orbits of G in Xps(L). One can show that theGIT quotient X L G is a projective variety, if X is projective. Further-more, XpsLG ⊆ XLG is an open subvariety, and hence is not necessarilyprojective. Hence, one may consider the GIT quotient X L G as a com-pactification of the orbit space Xps L G where the orbits of G in Xs.ss are4Luna Slice Theoremthe orbits which lie on the boundary2.3 The boundary of the geometric quotient Xps  G in X  G has thedeficiency that its points only have ”weak modular meaning”. What thisamounts to is that that the fibers above the points on the boundary are notin bijection with the orbits of G in Xs.ss. Instead, an orbit closure relationhold among the fibers: x and y in Xss lie on the same fiber of the quotientmap pi if and only if Gx ∩ Gy 6= ∅, where the closure is being taken inXss. Note that this relation is necessary for the map Xss → Xss  G tobe continuous. However, one may prove that a fiber pi−1(x) for x ∈ Xsscontains a unique closed orbit. Therefore, one may say that the fibers of theGIT quotient map pi are in bijection with closed orbits of Xss. Recall thatany orbit Gx in X has the property that Gx contains a closed orbit. Wecall a point semi-stable point x stable if Gv is closed in H0(X,L⊗n) for anaffine representation v of x. In the sequel, Xs will denote the set of stablepoints of X.Luna Slice TheoremIn this section, we introduce the Luna Slice theorem and its various corol-laries. These are some of the main technical tools required to formulate andprove the results presented in this thesis.2.4 The Luna slice theorem provides a local description of an orbit Gxof a point x in a G-variety X. In differential geometry, there is an analo-gous ”slice theorem”. It states that given a manifold X and a Lie group Gacting on X diffeomorphically, then for any point x ∈ M , one may find aG-invariant open neighbourhood U containing Gx, such that U is equivari-antly diffeomorphic to the homogeneous fiber space G ×Gx Nx. Here, Nxis a Gx-invariant direct sum complement to Tx(Gx) in Tx(X). In short,there is a tubular neighbourhood around Gx so that the action of G in thisneighbourhood has a simple description as a homogeneous fiber space.2.5 As one may expect, this theorem does not hold in the algebraic setting.Firstly, if Gx is not reductive, then a Gx-invariant complement of Tx(Gx) inTx(X) may not exist. Furthermore, Zariski open sets are too large to expectto find a Zariski open set U which is equivariantly isomorphic to G×Gx Nx.This is illustrated by the following example:5Luna Slice TheoremExample. Suppose that X = H0(P1,OP1(3)), and G = SL2, whose actionon X is induced by the standard representation on P1. Then, there is anx ∈ X such that there is no open U such that U is equivariantly isomorphicto G×Gx Nx.Indeed, consider x = u2v. Note that any point in U ∼= G ×Gx Nx has astabilizer which is conjugate to a subgroup of Gx (see Lemma 2.6.1). Onemay check that the stabilizer of x is trivial. However, one may also checkthat the action of G on X has a stabilizer in general position of order 3.This is because if κ is algebraically closed, any binary cubic with distinctroots may be transformed (under the action of SL2, by 3-transitivity of theaction of SL2 on P1) to a multiple of the form h = u3 +v3. Clearly, for sucha form, we have that Gh =(µ 00 µ−1), where µ3 = 1. Thus, as any twonon-empty Zariski open sets intersect, we see that such an open set U is notpossible (i.e. an open set U such that every point has trivial stabilizer).As we will see, this is resolved by the fact that G×Gx Nx is an e´tale neigh-bourhood of x containing Gx, as opposed to a Zariski neighbourhood. Thatis, there is an e´tale morphism from G ×Gx Nx → X with image containingGx. For details, see paragraph 2.9.Homogeneous Fiber Spaces2.6 Suppose that G is a reductive group, and that H is a closed reductivesubgroup of G. Then H acts on G by the formula h.g = gh−1. Under thisaction, G is a principal H-bundle over the homogeneous space G/H.If S is an affine H-variety, we can define a G variety by ”twisting” S by G(as a H-variety as above). H acts naturally on G × S component-wise bythe formula h.(g, s) = (gh−1, hs).Definition 2.6.1. A homogeneous fiber space is the quotient (G× S)H,where H acts on each factor as described above. A homogeneous fiber spaceis denoted G ×H S, and an element will be denoted by [g, s]. G acts onG×H S on the left factor in the obvious way.The affine GIT quotient (G× S) H above is in fact a geometric quotient.Note that G × S is an affine free H-variety, so that G × S is a principalH-bundle over (G× S) H.6Luna Slice TheoremLemma 2.6.1. The stabilizer of a point [g, x] in a homogeneous fiber spaceG×H S, under the action of G, is conjugate to a subgroup of the stabilizerHx ⊂ H of x.Proof. Since g.[e, x] = [e, x], we have that [e, x] and [g, x] are in the sameorbit. Hence, the stabilizers of [g, x] and [e, x] are conjugate. Therefore, toshow this result, we may simply consider stabilizers for points of the form[e, x]. If g.[e, x] = [g, x] = [e, x], we then have that (gh−1, hx) = (e, x) forsome h ∈ H. From this, we see that h ∈ Hx and g = h. Thus, g ∈ Hx.Conversely, it is clear that ghg−1 fixes [g, x] for any h ∈ Hx.It is clear that S embeds into G×HS by sending x ∈ S to [e, x]. Furthermore,as G × S is a free H-variety, we have that for a fixed x = (g, x) ∈ G × S,H×x→ G×S defned by (h, x)→ (gh−1, hx) ∈ G×S defines an embeddingof H into G×S. For a point x = (e, x), this embedding induces an inclusionof tangent spaces µx : Te(H) ↪→ T(e,x)(G× S) ∼= Te(G)⊕ Tx(S).Lemma 2.6.2. We have:1. Tx(G×H S) = (Te(G)⊕ Tx(S))/Te(H).2. Tx(G×HS) ∼= Tx(Gx)⊕Tx(S) as abstract vector spaces. If S is smoothat x, then G×H S is smooth at [e, x].3. If x is an H-fixed point, then we have that the embedding Te(H)µx↪−→T(e,x)(G×S) induces an identification Tx(G×H S) ∼= Tx(Gx)⊕Tx(S).Proof. As G×S is a principal H-bundle over (G×S)H, we have an exactsequence of tangent spaces:0→ Te(H) µx−→ T(e,x)(G× S)→ T[e,x]((G× S) H)→ 0.From this, 1. immediately follows.So we have, dimκ(Tx(Gx) ⊕ Tx(S)) = dimκ(Te(G)/Te(H) ⊕ Tx(S)) =dimκ(Te(G)) − dimκ(Te(H)) + dimκ(S) = dimκ((Te(G) ⊕ Tx(S))/Te(H)),and Tx(G×H S) ∼= Tx(Gx)⊕ Tx(S) as abstract vector spaces.Note that if x is an H-fixed point, then the induced tangent map µx :Te(H) ↪→ T(e,x)(G×S) ∼= Te(G)⊕Tx(S) maps Te(H) into Te(H)⊕{0Tx(S)}.Then, (Te(G)⊕ Tx(S))/Te(H) = Te(G)/Te(H)⊕ Tx(S) ∼= Tx(Gx)⊕ Tx(S).7Luna Slice TheoremRemark 2.6.1. In fact, since G× S is a principal H-bundle over G×H S,G×S is smooth at (e, x) if and only if G×HS is smooth at [e, x]. Therefore,G×H S is smooth at [e, x] if and only if S is smooth at x ∈ S.Existence of E´tale Slices2.7 We discuss the existence of e´tale slices in the smooth setting. In thissection, X is a smooth affine G-variety, where G is a reductive group.For a point x ∈ X, and a Gx-invariant affine subvariety S of X containingx, we may consider the homogeneous fiber space G×GxS. There is a naturalmap ψS : G×Gx S → X defined by ψS([g, s]) = gs. If pi1 : X → X G andpi2 : G×Gx S → G×Gx S G are GIT quotient maps, then the compositionpi1 ◦ψS : G×Gx S → XG is G-invariant. Therefore, there is a factorizationpi1 ◦ ψS = (ψS/G) ◦ pi2 for a unique map ψS/G : G ×Gx S  G → X  G.Thus, we have the following commutative diagram:G×Gx S X(G×Gx S) G X GψSpi2 pi1ψS/G(2.1)We say that a Gx-invariant affine subvariety S of X is an e´tale slice at xif in the above diagram, the following hold:1. x ∈ S, and ψS is e´tale (in particular, the image of this map containsx).2. ψS/G is e´tale.3. The above diagram is cartestian. Thus, we have that the induced mapG×Gx S → X ×XG ((G×Gx S) G) is an isomorphism.8Luna Slice Theorem2.8 If an e´tale slice S exists, then the image of ψS is an open set Ucontaining Gx. By Lemma 2.6.2, Tx(U) = Tx(Gx) ⊕ Tx(S). Thus, S istransverse to the orbit Gx at x. Conversely, we will see that if X is smoothat x, then there is an e´tale Gx-equivariant map φ : X → Tx(X); undersuitable hypotheses, if Tx(Gx) has a Gx-invariant complement Nx, one maytake an appropriate open set of φ−1(Nx) to be an e´tale slice at x.Lemma 2.8.1. Suppose that X is an affine variety of dimension n, whichis smooth at a point x ∈ X. Then, there is an e´tale (at x) morphismφ : X → An ∼= Tx(X) such that φ(x) = 0 ∈ An.Proof. Since x ∈ X is smooth, we may find regular functions f1, . . . , fn inmx ⊂ OX(X), such that their images df1, . . . , dfn in mx/m2x = (Tx(X))∨form a basis. Therefore, the map (f1, . . . , fn) : X → An is e´tale at x.Indeed, the induced map T0(An)∨ → Tx(X)∨ is given by the map dxi 7→ dfiand is an isomorphism.Lemma 2.8.2 (Luna). Suppose that X is an affine G-variety of dimensionn, which is smooth at a point x ∈ X. Suppose that Gx is linearly reductive.Then, there exists a morphism φ : X → An ∼= Tx(X) such that:1. φ is Gx-equivariant.2. φ is e´tale at x.3. φ(x) = 0.Proof. One only needs to see that the map φ in Lemma 2.8.1 can be madeGx-equivariant. This will depend on choosing appropriate elements f1, . . . , fnin mx which generate mx/m2x. Both mx and mx/m2x are Gx-modules, whereGx is linearly reductive. Furthermore, we clearly have that the natural mapτ : mx → mx/m2x is Gx-equivariant and surjective. By linear reductivity, wemay define a Gx-equivariant section s : mx/m2x → mx. Indeed, the ker(τ)has a Gx-invariant complement V in mx, which is equivariantly isomorphicto mx/m2x. Therefore, we may find f1, . . . , fn in V so that their images inmx/m2x form a basis (as a Gx-module).2.9 We note that a Gx-invariant subvariety S ⊂ X being an e´tale slice atx, is much stronger than only satisfying condition 1. (i.e. ψS is e´tale). Forexample, ψS/G being e´tale implies that the image of ψS is not only open9Luna Slice Theorem(recall that e´tale maps have open image), but a saturated open set. Indeed,this follows from ψS/G being an open map, and that the square (1) com-mutes. Some authors may call such a subvariety S satisfying only condition1. as a weak e´tale slice, and a subvariety S satisfying all three conditionsan e´tale slice or a strong e´tale slice. This is a significant distinction, asone needs a strong e´tale slice to assert (for example), the existence of a Lunastratification (see Theorem 2.10.1). Furthermore, the example presented inparagraph 2.5 illustrates a situation where one only has a weak e´tale sliceat x, and not an e´tale slice. The reason for this is that the orbit of x in thatexample is not closed.In characteristic 0, the only condition required for the existence of an e´taleslice at x (which may be taken to be an appropriate open subset of φ−1(Nx))is that the orbit Gx is closed in X.Theorem 2.9.1 (Matsushima). Suppose that X is an affine G-variety.Suppose that x ∈ X has a closed orbit Gx. Then, Gx is a reductive subgroupof G.Theorem 2.9.2 (Luna’s Slice Theorem). Suppose that X is an affineG-variety over a field κ which is algebraically closed and of characteristic 0.Suppose that x ∈ X is so that Gx is closed. Then, there exists an e´tale sliceat x of X. Furthermore, the image of the map ψS is a saturated open set ofX.Proof sketch of Luna’s Slice Theorem. One may consider φ−1(Nx) for someGx-invariant complement to Tx(Gx) ⊂ Tx(X) so that the natural map φ :G∗StabG(x)φ−1(Nx)→ X is e´tale whose image contains x (note that Tx(X) =Tx(Gx)⊕Nx ∼= Tx(G×Gx φ−1(Nx)), by Lemma 2.6.2.). Therefore, we mayfind a weak e´tale slice at x. To find a strong e´tale slice, we may restrict φto an appropriate open subset of G ∗StabG(x) φ−1(Nx). This is provided bythe lemma below (Luna’s fundamental lemma).Lemma 2.9.1 (Luna’s fundamental lemma). Suppose that φ : Y → Xis a G-equivariant e´tale (at y) morphism of affine G-varieties, where Y isnormal at a y ∈ Y . Suppose that φ(Gy) is closed in X, and that φ|Gy isinjective. Then there exists an open V ⊂ Y containing y, such that1. φ/G is e´tale at piY (y).2. The following square is cartesian:10Luna Slice TheoremV φ(V )V G φ(V ) GφpiY piXφ/G(2.2)3. V is saturated. That is, V = pi−1Y (V′) for some V ′ open in Y G.4. The image φ(V ) = U is a saturated open set of X.Luna Stratification2.10 Recall that the affine GIT quotient map (i.e. the categorical quo-tient) pi : X → Z := X  G maps saturated open sets of X to open setsof X  G. Recall that for any z ∈ Z, there exists a unique closed orbitcontained in the fiber pi−1(z). Let us write xz for a choice of an elementin pi−1(z) with closed orbit. By Luna’s slice theorem, we have a map ψS ,obtained from an e´tale slice, whose image is a saturated open set Uxz . Thus,pi(Uxz) is an open set in Z, containing z. Therefore, we may cover Z withopen sets of the form pi(Uxz). By quasi-compactness of Z, we may also finda finite cover {pi(Uxzi ) := Vi} covering Z. Note further that pi(Uxz) is aconstructible set (i.e. a finite union of locally closed subsets of Z). Thus,we have the theorem:Theorem 2.10.1 (Luna Stratification). There is finite collection of lo-cally closed subsets Vi of Z := X G such that ∪Vi = Z , with the followingproperties:1. Corresponding to the collection V1, . . . Vm, there is a finite list of re-ductive subgroups R1, . . . , Rm so that any stable point in pi−1(Vi) hasa stabilizer conjugate to Ri.2. If x ∈ pi−1(Vi), then Gx is conjugate to a subgroup of RiNow, Luna’s stratification provides a stratification of the quotient withfinitely many strata of the form Z〈H〉 := {z ∈ Z | Stab(xz) ∈ 〈H〉}. Fur-thermore, let us define X〈H〉 := {x ∈ X | G.x is closed and Stab(x) ∈ 〈H〉}.Thus, semi-stable points in X with closed orbit are stratified as above (i.e.11Luna Slice Theoremstratification induced by the stratification of the quotient), with strata la-belled by reductive subgroups H of G. Let us call the strata X〈H〉, isotropystrata of X. For an affine G-variety as above, let us denote the set of Vi byL(X), and the set of corresponding subgroups Ri by R(X). Note that fora fixed L(X), the subgroups Ri in R(X) are well-defined up to conjugacy.In many cases, the strata Z〈H〉 are intrinsic in that they are fixed by anyautomorphisms of Z. For details, see [4].12Chapter 3Kirwan’s Procedure andStable ResolutionsIn this chapter, we assume that X is a G-linearized projective variety, suchthat Xss is smooth.3.1 Under the assumption that X contains at least one properly stablepoint, Kirwan defines a sequence of blow-ups σi : Xi → Xi−1 starting witha linearization of X = X0, which produces a G-linearized variety X˜ = XNsuch that every semi-stable point is properly stable [3]. Furthermore, thecomposition σ of the blow-ups σi is an isomorphism along the properly stablelocus Xps of X. Such a birational model X˜ is known as a stable resolutionof X.Thus, the stable resolution X has the property that X˜ G is projectivewith an inclusion Xps  G ↪→ X˜  G. Hence, X˜  G may be consideredas a compactification of Xps  G with the property that the fibers of theboundary of XpsG in X˜ G are in bijection with orbits of G in X˜ss \Xps.3.2 The construction of a stable resolution is based on an analysis of sta-bility of points in a G-linearized Y , in terms of X, where Y and X arerelated by an equivariant morphism f : Y → X. Particularly, Kirwan in[3] works with the scenario when f is a blow-up map. In the case where fis a blowing up along a smooth center, a complete characterization of theproperly stable, semi-stable and unstable points of Y can be provided interms of the corresponding stability loci of X (see [6]).3.3 Suppose that σ : Xˆ → X is a blowing-up of X along a closed G-invariant center C. Then Xˆ is naturally G-linearized via the line bundle13Chapter 3. Kirwan’s Procedure and Stable ResolutionsLd := σ∗(Ld) ⊗ O(−E) for sufficiently large d. In the sequel, we will as-sume that all blow-ups of linearized varieties X are linearized in this way.Furthermore, we let Xaff denote the affine cone of the projective varietyX ⊆ P(H0(X,L)). That is, Xaff is a G-invariant subvariety of the vectorspace H0(X,L). For x ∈ X, we define xaff ∈ Xaff to be a lift of x in theaffine cone Xaff of X.3.4 In birationally modifying X (e.g. by a sequence of blow-ups) to arriveat a stable resolution X˜, one only has to modify the strictly semi-stable locusof X. That is, for any equivariant map Y → X, there is a linearization ofY ensuring that the properly stable and unstable loci of Y do not get anysmaller than those of X. In particular, the fibers in Y above such points donot introduce new strictly semi-stable points.Proposition 3.4.1. Suppose that X is a G-linearized projective variety.Suppose that f : Y → X is a G-equivariant map. Then for a suitablelinearization of Y ,1. If x ∈ X is unstable, then so is y ∈ f−1(x).2. If If x ∈ X is properly stable, then so is y ∈ f−1(x).Proof. See Mumford’s Geometric Invariant Theory [5]Thus, to define a stable resolution of X by a sequence of blow-ups, onechooses the respective centers according to this observation by choosingcenters intersecting strictly semi-stable points of X. In fact, we have thefollowing lemma:Lemma 3.4.1. If Xps 6= ∅, the following are equivalent:1. Xss = Xps.2. For every stable point x, StabG(xaff) is finite.Proof. ( 1 =⇒ 2): This is obvious.(2 =⇒ 1): Suppose that y ∈ Xss. Gyaff contains a closed orbit Gxaff . IfGxaff 6= Gyaff , then Gxaff ⊂ Gyaff\Gyaff . Then, Gxaff is strictly smaller thanGyaff in dimension. However, as StabG(xaff) is finite, Gxaff is of maximaldimension. Therefore, Gxaff = Gyaff = Gyaff and yaff is properly stable.14Chapter 3. Kirwan’s Procedure and Stable ResolutionsHence, it suffices to successively blow up X at stable (but not properlystable) points x in a way so that StabG(xaff) strictly decreases in size witheach blow up. This is the approach of Kirwan’s resolution.3.5 The following two paragraphs (3.5 and 3.6) will provide an overviewof the general argument presented in this section. The required proofs willfollow in paragraphs 3.7 and 3.8.Note that 2. in Lemma 3.4.1 is equivalent to StabG(xaff)0 = {1G}, ordim(StabG(xaff)0) = 0 for any stable point x. In fact, we have the followinglemma.Lemma 3.5.1. Suppose that x is a semi-stable point. Then, any con-nected subgroup R ⊆ Gx is contained in StabG(xaff)0. In particular, G0x =StabG(xaff)0.Proof. If R is a connected subgroup of Gx, then R acts on the line generatedby xaff . Suppose that φ : R→ Gm is the corresponding linear representationof R. Since x is semi-stable, the image of φ cannot be dense. Otherwise,0 ∈ Gxaff , and hence x would be unstable. Thus, by the connectedness ofR, the image of φ is precisely the identity subgroup {1} ⊂ Gm. Therefore,R acts trivially on the line Gm.xaff . Thus, R ⊆ StabG(xaff)0.Thus, 1. in Lemma 3.4.1 occurs when no connected positive dimensionalsubgroup R of G fixes a stable point x.Recall that for a stable point x, we have that Gxaff is closed Xaff . Fur-ther, StabG(xaff) is a reductive subgroup of G, and xaff will be contained in(Xssaff)〈H〉 for some reductive subgroup H in G. Luna’s stratification tells usthat there are finitely many possible connected components H0 of reductivestabilizers, such that (Xssaff)〈H〉 is not empty.Suppose that R is a connected reductive subgroup of G. Let 〈R〉 denote theconjugacy class formed by the subgroup R. We define:ZR(Xssaff) :=⋃H0∈(R)(Xssaff)H0This is the union of all Vi ∈ L(Xssaff) such that the corresponding stabi-lizer Ri has an identity component conjugate to a fixed connected reductivesubgroup R of G.15Chapter 3. Kirwan’s Procedure and Stable ResolutionsCorollary 3.5.1. x ∈ Xss is a stable point if and only if xaff ∈ ZR(Xssaff)for some connected reductive group R.Therefore, X is a linearized projective G-variety with no strictly semi-stablepoints if and only if P(ZR(Xssaff)) is empty for any connected reductive sub-group R of positive dimension. Furthermore, X contains only properlystable and unstable points if and only if we also have that P(ZR(Xssaff)) isnon-empty for R = {1G}. In this case, we have that P(ZR(Xssaff)) = Xps.3.6 In essence, Kirwan’s procedure works by blowing up X at P(ZR(Xssaff)),for connected reductive subgroups R of positive dimension. This is done sothat in the resulting blow-up space X′the projectivized isotropy strata in-tersecting the exceptional divisor are labelled by reductive subgroups H′whose identity components have dimension strictly smaller than the dimen-sion of R. As there are only finitely many isotropy strata, one may performfinitely many blow-ups (starting with X) and arrive at a X˜ with no strictlysemi-stable points. The blow-ups are defined iteratively: one begins withreductive subgroups of maximal dimension (:= dX), labelling the isotropystrata (there are finitely many). After finitely many blow-ups, one arrivesat a blow up space X′such that the maximal dimension among reductivesubgroups labelling the isotropy strata of (X′aff)ss is strictly less than dX .This is repeated until there are no positive dimensional connected subgroupsR such that Stab(x)0 = R for any x ∈ X˜.3.7 In the following paragraphs, we provide the details and proofs requiredto carry out the construction outlined in paragraphs 3.5 and 3.6. Supposethat R is a connected reductive subgroup of G, maximal in dimension withthe property that R = Stab(x)0 for some x ∈ X.Lemma 3.7.1. ZR(Xssaff) = GXRaff ∩Xssaff .Proof. Suppose that x ∈ Xss〈H〉aff , where H0 ∈ 〈R〉. Then, for some g ∈ G,Stab(x) = gHg−1. Therefore, ghg−1x = x, for any h ∈ H. In otherwords, g−1x ∈ XHaff . Thus, x ∈ GXRaff ∩ Xssaff . Conversely, suppose thatx ∈ GXRaff∩Xssaff . By definition, Stab(x) ∈ 〈H〉, for some H where H0 ∈ 〈R〉.It remains to show that Gx is closed. Since Stab(x) ∈ 〈H〉, it is of maximaldimension among stabilizer subgroups of G acting on Xaff . Thus, Gx is anorbit of minimal dimension and is hence closed in Xaff .Now, we prove that G(XRaff)ss = GXRaff ∩Xssaff is smooth in Caff := G(XRaff)ss(where the closure is taken in Xaff).16Chapter 3. Kirwan’s Procedure and Stable ResolutionsProposition 3.7.1. Suppose that y ∈ (Caff)ss. Then,1. y ∈ G(XRaff)ss. Hence, Cssaff = G(XRaff)ss. That is, taking the closure ofG(XRaff)ss in Xaff does not introduce any more semistable points.2. G(XRaff)ss is smooth.Proof. 1. As in the proof of Lemma 3.7.1, we see that y is contained inan orbit of minimal dimension. Therefore, Gy is closed. Now by theLuna Slice Theorem, there is an (Zariski) open U containing y, suchthat for any z ∈ U , Stab(z) is conjugate to a subgroup of Stab(y). Asy ∈ Cssaff = G(XRaff)ss, one may choose such a z ∈ U to be containedin G(XRaff)ss. Then, gRg−1 ⊂ Stab(z) for a suitable g ∈ G. In turn,we thus have that tRt−1 ⊂ Stab(y) for suitable t ∈ G. Therefore,y ∈ t.XR and y ∈ G(XRaff)ss as required. Since y is an arbitrarysemi-stable point of GXRaff , we have that (GXRaff)ss = G(XRaff)ss.2. It suffices to prove thatG(XRaff)ss is smooth for x ∈ (XRaff)ss ⊂ G(XRaff)ss.We have seen that such an x has a closed orbit, and so we apply theLuna Slice Theorem to both the G action on Xaff and the N := NG(R)action on XRaff . This provides us with e´tale slices S and S′ respectively.Then since both Xaff and XRaff are smooth at x, we have the tangentspace decompositions,Tx(Xaff) = Tx(Gx)⊕ Tx(S)Tx(XRaff) = Tx(Nx)⊕ Tx(S′)We have e´tale maps (defined at x), ΨS : Xssaff → G ×Gx S and ΨS′ :(XRaff)ss → N ×Gx S′, respectively. Furthermore, by smoothness at x,we have e´tale maps φS : S → Tx(S) and φS′ : S′ → Tx(S′). Thus, Xssaffand (XR)ss are e´tale equivalent (at x) to G×Gx Tx(S) andN ×Gx Tx(S′), respectively.Now, we have that Tx(XRaff) = Tx(Xaff)R = Tx(Gx)R ⊕ Tx(S)R =Tx(Nx)⊕Tx(S′). On the other hand, Tx(Gx)R = Tx(GxR). p ∈ (Gx)Riff p = gx for some g ∈ G and r.p = p for all r ∈ R. Therefore,(Gx)R = Nx. So, Tx(S)R ∼= Tx(S′) as Gx-modules.Consider now the following diagram:17Chapter 3. Kirwan’s Procedure and Stable ResolutionsN ×Gx S′ (XR)ssN ×Gx Tx(S′)G×Gx Tx(S) G×Gx S XssΨS′φS′iφS ΨSwhere ΨS′ , ΨS are e´tale at x, and φS , φS′ are e´tale. ΨS′ , φS′ areN -equivariant, and ΨS , φS are G-equivariant. The inclusion i is in-duced by the inclusion of N -representations Tx(S′) ↪→ Tx(S), and theinclusion N ↪→ G. Hence, i is R-equivariant.We would like to conclude that G(XRaff)ss is smooth at x. We havei : N ×Gx Tx(S′) ↪→ G ×Gx Tx(S). G acts naturally on the imagei(N ×Gx Tx(S′)), so that G.i(N ×Gx Tx(S′)) = G×Gx Tx(S′). Also, Racts trivially on i(N ×Gx Tx(S′)). As R is connected and φS is e´tale,we have φ−1S (i(N ×Gx Tx(S′))) = (G×Gx S)R. Indeed, recall that thefiber of an e´tale map over a point is set-theoretically finite. Thus bythe connectedness of R, the preimages of R-fixed points are also R-fixed points. Also, the image of (G×Gx S)R under the e´tale map ΨS is(Xssaff)R. Finally, as the maps on the bottom row in the diagram are G-equivariant, we have φ−1S (G×Gx Tx(S′)) = φ−1S (G.i(N ×Gx Tx(S′))) =G.(G×Gx S)R, and that ΨS(G.(G×Gx S)R) = G.(XRaff)ss.Since x is contained in the image of ΨS , we have that G×Gx Tx(S′) ise´tale equivalent to G.(XRaff)ss at x. Finally, as G×Gx Tx(S′) is smooth,we have that G.(XRaff)ss is smooth at x.The above proof shows the following:Corollary 3.7.1. G×N (XRaff)ss → G(XRaff)ss is e´tale. In particular, Tx(G(XRaff)ss) =Tx(Gx)⊕ Tx(XR)Proof. For x ∈ (XRaff)ss, we have seen in the proof of 2. in Proposition 3.7.1,that there is an e´tale map G.((G×Gx S)R)→ G(XRaff)ss with x contained inthe image.Then, we have an e´tale map G.(G ×Gx S)R) → G ×Gx T (S′). We havee´tale maps G ×N (XRaff)ss ← G ×N (N ×Gx S′) → G ×N (N ×Gx T (S′)) ∼=18Chapter 3. Kirwan’s Procedure and Stable ResolutionsG×Gx T (S′). It is easy to check that the above e´tale maps factors the nau-tral map G×N (XRaff)ss → G(XRaff)ss.Thus, G×N (XRaff)ss → G(XRaff)ss is e´tale.Therefore, see that ZR(Xssaff) is smooth, and thus so is P(ZR(Xssaff)) = G(XR)ss.We have the following:Lemma 3.7.2. We have:1. (G(XR)ss)aff = Caff2. P(Caff) = P(ZR(Xssaff)) = G(XR)ss, which we will denote as C ⊂ X.Proof. Note that G(XRaff)ss is Gm-invariant, hence so is Caff . Therefore,Caff is defined by a homogeneous ideal in Xaff and corresponds to a closedprojective subvariety in X. Therefore, the result follows from the projectiveNullstellensatz.By this lemma, along with Proposition 3.7.1, we see that G(XR)ss\G(XR)ssconsists of only unstable points.3.8 In this paragraph, we complete the construction of Kirwan’s stableresolution.Theorem 3.8.1. In the notation above, Suppose that σ : X ′ → X is theblow-up of X along the center P(ZR(Xssaff)) = G(XR)ss in X. Then, nosubgroup of G conjugate to R stabilizes any semi-stable point of X ′.Proof. Suppose that y is a semi-stable point of X ′, which is fixed by a sub-group of the form g−1Rg. Then, for all r ∈ R, (g−1rg)y = y, which impliesthat r(gy) = gy. Hence, gy is a semi-stable point of X ′ fixed by R. Then,σ(gy) = gσ(y) = gx is also fixed by R. Hence, gx := x′ ∈ XR and issemi-stable in X. Thus, there exists a homogeneous G-invariant polynomialf ∈ A(Xaff)G such that f(x′) 6= 0. That is, Xf is an affine open subvarietyof X which contains x′. We have that Tx′(X) = Tx′(G(XR)ss) ⊕ Nx′ =Tx′(Xf ) = Tx′(G(XRf )ss)⊕Nx′ . Where Nx′ is a G-invariant complement toTx′(G(XR)ss). We have seen that Tx′(G(XRf )ss) = Tx′(Gx′) ⊕ Tx′((Xssf )R)(corollary 3.7.1), where Tx′((Xssf )R) = Tx′((Xssf ))R = Tx′(X′)R. Therefore,19Chapter 3. Kirwan’s Procedure and Stable Resolutionsthe action of R on Nx′ contains no fixed points.However, as since x′ is a smooth point of G(XR)ss, the fiber σ−1(x′) is R-equivariantly isomorphic to P(Nx′). Now, gy ∈ σ−1(x′) is an R-fixed pointin P(Nx′). By Lemma 3.5.1, there exists an R-fixed point in Nx′ , which is acontradiction.Now, suppose that for a positive dimensional connected reductive sub-group R, x ∈ P(ZR(Xssaff)), and that x′ is a semi-stable point of X ′ withσ(x′) = x. Clearly, StabG(x′) ⊂ Gx. Since x′ is semi-stable, Proposition3.7.1 implies that StabG(x′)0 ( G0x = R. Since StabG(x′)0 is a connectedproper subgroup of G0x, we have that dim(StabG(x′)0)  dim(G0x). There-fore, for x ∈ X, we in general have that dim(StabG(x′)0) ≤ dim(G0x); equal-ity may only occur when either x is not contained in the center of the blowup, or if 0 ≤ dim(G0x) < dim(R).3.9 Therefore, to conclude the construction of the stable resolution, wemay perform the following algorithm:1. Given a linearized projective G-variety X with Xss smooth, considerthe set of Luna strata L(Xssaff), and the set of corresponding reductivesubgroups R(Xssaff). These sets are finite in cardinality, so we maydefine dX = maxRi∈R(Xssaff)(dim(Ri)). Furthermore, there are onlyfinitely many Ri ∈ R(Xssaff) with dim(Ri) = dX .2. For a reductive subgroup R < G such that (Xssaff)R 6= ∅ and dim(R) =dX , we must have that R is conjugate to a subgroup inR(Xssaff) by max-imality. Consider ZR(Xssaff) as defined in paragraph 3.5. P(ZR(Xssaff)) ⊂Xss is smooth by Proposition 3.7.1. C = P(ZR(Xssaff)) = G(XR0)ss (seeLemma 3.7.2) is such that C \ P(ZR(Xssaff)) consists only of unstablepoints.3. Consider the blow up X ′ → X centered at C. By 2., (X ′)ss is smooth.By paragraph 3.8, R((X ′)ssaff) consists of subgroups whose dimensionsare less than or equal to the dimension of those in R(Xssaff). Further-more, the number of subgroups with dim(R) = dX is strictly smaller.Indeed, no conjugate of R is contained in R((X ′)ssaff). By 1., there areonly finitely many reductive stabilizer subgroups R of Xssaff such thatdim(R) = dX . Iterate steps 1-3 until there are no longer any suchsubgroups.20Chapter 3. Kirwan’s Procedure and Stable Resolutions4. When step 3 is completed, we will have a linearized projective G-variety X ′ such that (X ′)ss is smooth, and dX′  dX . Repeat steps1-4 until we have a linearized projective G-variety X˜ such that dX˜ = 0.By Lemma 3.4.1, X˜ → X is a stable resolution of X.That is, the above procedure provides a proof of Theorem (A) stated in theintroduction (chapter 1).3.10 In this next paragraph, we conclude the section with a few remarks.1. In [3], Kirwan proves that blowing up X, in the way described inparagraph 3.6 under the condition that the center of the blow-up issmooth. However, C = P(ZR(Xssaff)) may be singular. Thus, Kirwanfirst performs a resolution of singularities on C before blowing up alongit. Fortunately, one does not need to know the explicit form of the res-olution of singularities of C to continue with Kirwan’s procedure. Thisis due to the fact that P(ZR(Xssaff)) is smooth. Thus, the singularitiesof C lie strictly on the boundary of C, which consists only of unstablepoints.2. We may take Y to be the resulting blow-up space obtained by re-solving the singularities of C in X. The linearization of Y may betaken to be the one described in paragraph 3.3. The result is that onehas a sequence of blow-ups Xk (:= Y ) → Xk−1 → .. → X0 (:= X)such that the center of each blow-up Xi → Xi−1 is contained in theunstable locus of Xi−1. Therefore, the composition σ of the aboveblow-ups induces an isomorphism σ : Y ss∼−→ Xss. Furthermore,as each blow-up is equivariant, σ also induces an isomorphism alongP(ZR(Y ssaff )) and P(ZR(Xssaff)). Then, Kirwan considers blowing up Yalong P(ZR(Y ssaff )) ∼= P(ZR(Xssaff)), where now P(ZR(Y ssaff )) is smooth inY .3. Therefore, if X is smooth, and one performs blow-ups along smoothcenters C, then one may construct a stable resolution X˜ → X suchthat X˜ is smooth (so that not just X˜ss is guaranteed to be smooth).21Chapter 4Abelianization ProcedureIn the last chapter, we provided a proof of Theorem (A) as presented inthe introduction (see paragraph 3.9). In this chapter, we present a proof ofTheorem (B).4.1 An important consequence of Proposition 3.4.1 is that once one hasobtained a stable resolution X˜ → X, then the composition X˜ ′ → X˜ → Xis also a stable resolution of X (for a suitable linearization of X˜ ′) for anyG-equivariant birational morphism X˜ ′ → X˜. Even more, if Y → X˜ is anyG-equivariant map, then Proposition 3.4.1 says that Y also has the propertythat, for a suitable linearization of Y , all points of Y are either properly sta-ble, or unstable. Thus, if X˜ → X is a stable model of X, then one cannotlose the property of being a stable model by resolving X˜ further.4.2 In [7] Reichstein and Youssin define a type of resolution of generi-cally free G-variety X: it is a sequence of blow-ups pi : Xnpin−→ Xn−1 pin−1−−−→. . . X1pi1−→ X so that Xn is in standard form with respect to a certan divisorD of Xn. The significance of this is that there are choices of D so that thestabilizers in Xn will be ”tame” stabilizers.Definition 4.2.1 (G-variety in standard form [7]). A generically freeG-variety X is said to be in standard form with respect to a divisor D if:1. X is smooth, and D is a normal crossing divisor on X.2. The action of G on X \D is free3. For each irreducible component Di of D, for any g ∈ G, we either havegDi = Di, of gDi ∩Di = ∅22Chapter 4. Abelianization Procedure4.3 One of the main theorems in [7] is the following theorem. We state ithere for reference (Theorem 3.2):Theorem 4.3.1. Let X be a smooth G-variety and Y ( X be a closed G-invariant subvariety such that the action of G on X \Y is free. Then, thereis a sequence of equivariant blow-ups:pi : Xn → Xn−1 · · · → Xwith smooth G-invariant centers Ci ⊂ Xi such that Xn is in standard formwith respect to the the divisor D := En∪pi−1(Y ), where En is the exceptionaldivisor of pi.4.4 Let X be a smooth linearized generically free G-variety. Then, wemay apply Theorem 4.3.1 and obtain a Xn → X such that Xn is in standardform with respect to some divisor D. We show in this paragraph that for asuitable linearization of Xn, all stable points have abelian stabilizers. Thus,we define Xab := Xn, where Xab implicitly depends on the divisor D. Xabis linearized following Proposition 3.4.1 (as usual).Corollary. If x ∈ Xssab is stable, then Gx is an abelian subgroup of G.Proof. Note that since x is stable, Gx is a reductive subgroup of G. We havethat the action of G on Xab \D is free, so all such points have trivial stabi-lizers. It suffices to consider x ∈ Xssab ∩ D. Suppose that x is contained incomponents D1, . . . , Dk components of D. By property 3. in the definitionof a G-variety in standard form, we have that each Di is Gx-invariant, andso is W = D1 ∩ · · · ∩Dk. Then, we have the tangent space decompositionsTx(Xab) = Tx(Di)⊕Vi, where Vi is Gx-invariant. Hence, Vi is 1-dimensional,and Gx acts on Vi by a character χi.Furthermore, we have then that Tx(Xab) = Tx(W ) ⊕ (⊕mi=1 Vi) by thenormal crossing property 1. in Definition 4.2.1. This is a Gx-invariant de-composition, and Gx acts on V :=⊕ki=1 Vi by the characters (χ1, . . . , χk).That is, we have a homomorphism χ : Gx → Gkm. If χ has trivial kernel,then Stabx(G) is necessarily abelian.Suppose that ker(χ) is non-trivial. Then, x ∈ (Xssab)ker(χ). In fact, as xis semi-stable, we may find a G-invariant affine open subset (Xab)f so thatx ∈ (Xab)ker(χ)f ⊂ Xssab. As the action of G on Xab \ D is free, we actually23Chapter 4. Abelianization Procedurehave that (Xab)ker(χ)f is contained in D. As (Xab)f is smooth and affine,(Xab)ker(χ)f is smooth. Thus, (Xab)ker(χ)f is entirely contained in a singlecomponent Di for some i.We have that Tx((Xab)ker(χ)f ) = Tx((Xab)f )ker(χ) ⊂ Tx(Di). However, bydefinition Vi ⊂ Tx((Xab)f )ker(χ). As Vi and Tx(Di) are direct sum comple-ments of each other, we have a contradiction. Therefore, ker(χ) = {1Gx},and Gx is finite and abelian.4.5 Let us illustrate the above theorem (and corollary) when one has astable resolution σ : X˜ → X, where X is a generically free G-linearizedprojective variety with Xps 6= ∅. Note then that X˜ is also a generically freeG-variety, and we of course have that X˜ss = X˜ps (containing points withfinite stabilizer subgroups). By paragraph 3.10, we can arrange X˜ to besmooth by constructing a stable resolution X˜ → X by blowing up alongsmooth centers. Alternatively, one may equivariantly resolve the singular-ities of any stable resolution X˜ → X to arrive at a smooth X˜sm that is astable resolution of X (see the remark in paragraph 4.1). Therefore, let usassume that we have constructed a stable resolution X˜ → X such that X˜ issmooth. Applying Theorem 4.3.1 to X˜, we obtain a smooth X˜ab in standardform with respect to some divisor D, with an equivariant birational mor-phism X˜ab → X˜. X˜ab contains only properly stable and unstable points, sothat X˜ab → X is a stable model of X. Furthermore, every x ∈ X˜psab is sothat Gx is a finite abelian subgroup of G.4.6 In Theorem 4.3.1, the centers of blow up contain only points of Y andits preimages (see the proof in [7], Theorem 3.2). If G acts freely on Xps,then we may take Y = X \Xps. The result of this choice of Y will be thatρ : Xab → X is an isomorphism along Xps. Therefore, we see that if G actsfreely on Xps, then we may find a (smooth) stable resolution X˜ and X˜ab asabove, such that X˜ab → X is an isomorphism along Xps. Thus, we have inthis case that X˜ab → X is a still a stable resolution if X˜ → X is a stableresolution. Note that if G acts freely on Xps, then it acts freely on X˜ps.Remark 4.6.1. Note that in the statement of Theorem 4.3.1, X is requiredto be a generically free G-variety. It is necessary to impose a condition on thestabilizer in general position of X, as Xab → X is a birational map. Indeed,if X has a non-abelian stabilizer in general position, then such a birational24Chapter 4. Abelianization Proceduremap is impossible. Xab is a G-variety with an open set containing pointswith abelian stabilizer, and thus cannot have a non-abelian stabilizer ingeneral position. An example of such an X is the space of cubic curves onP2, with the natural action of PGL3.Remark 4.6.2. A stronger form of the above corollary is contained in [7] asTheorem 4.1. The authors prove that if X is in standard form with respectto a divisor D, then Gx is isomorphic to a semidirect product of a unipotentgroup, and a diagonalizable group. In particular, If Gx is reductive, thenit is a abelian. Note that the statement of this result does not involve anylinearizations. Therefore, the claim that stable points of X have abelianstabilizers is independent of a choice of linearization of X.Remark 4.6.3. In the case, where G is a finite group, the above corollarywas independently proved by Batyrev [1] and Borisov-Gunnels [2].25Bibliography[1] V. V. Batyrev. Canonical abelianization of finite group actions. ArXivMathematics e-prints, Sept. 2000.[2] L. A. Borisov and P. E. Gunnells. Wonderful blowups associated to groupactions. Selecta Math. (N.S.), 8(3):373–379, 2002.[3] F. C. Kirwan. Partial desingularisations of quotients of nonsingular va-rieties and their betti numbers. Ann. Of Math. (2), 122(1):45–85, 1985.[4] J. Kuttler and Z. Reichstein. Is the luna stratification intrinsic? Ann.Inst. Fourier, 58(2):689–721, 2008.[5] D. Mumford, J. Fogarty, and F. Kirwan. Geometric Invariant Theory,3rd Edition. Springer-Verlag, Berlin, 1994.[6] Z. Reichstein. Stability and equivariant maps. Invent. Math., 96(2):349–383, 1989.[7] Z. Reichstein and B. Youssin. Essential dimensions of algebraic groupsand a resolution theorem for G-varieties. Canad. J. Math., 52(5):1018–1056, 2000.26

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.24.1-0343974/manifest

Comment

Related Items