aa r X i v : . [ m a t h . AG ] D ec Quotients by non-reductive algebraic group actions
Frances KirwanMathematical Institute, Oxford OX1 3BJ, [email protected] 22, 2018
This paper is dedicated to Peter Newstead, from whose Tata Institute Lecture Notes [Ne] I learnt aboutGIT and moduli spaces some decades ago, with much appreciation for all his help and support over the yearssince then.
Geometric invariant theory (GIT) was developed in the 1960s by Mumford in order to construct quotients ofreductive group actions on algebraic varieties and hence to construct and study a number of moduli spaces,including, for example, moduli spaces of bundles over a nonsingular projective curve [MFK, Ne, Ne2]. Modulispaces often arise naturally as quotients of varieties by algebraic group actions, but the groups involved arenot always reductive. For example, in the case of moduli spaces of hypersurfaces (or, more generally, completeintersections) in toric varieties (or, more generally, spherical varieties), the group actions which arise naturallyare actions of the automorphism groups of the varieties [Co, CK]. These automorphism groups are not ingeneral reductive, and when they are not reductive we cannot use classical GIT to construct (projectivecompletions of) such moduli spaces as quotients for these actions.In [DK] (following earlier work including [Fa, Fa2, GP, GP2, Wi] and references therein) a study was madeof ways in which GIT might be generalised to non-reductive group actions; some more recent developmentsand applications can be found in [AD, AD2]. Since every affine algebraic group H has a unipotent radical U E H such that H/U is reductive, [DK] concentrates on unipotent actions. It is shown that when aunipotent group U acts linearly (with respect to an ample line bundle L ) on a complex projective variety X , then X has invariant open subsets X s ⊆ X ss , consisting of the ‘stable’ and ‘semistable’ points forthe action, such that X s has a geometric quotient X s /U and X ss has a canonical ‘enveloping quotient’ X ss → X//U which restricts to X s → X s /U where X s /U is an open subset of X//U . (When it is necessaryto distinguish between stability and semistability for different group actions on X we shall denote X s and X ss by X s,U and X ss,U .) However, in contrast to the reductive case, the natural morphism X ss → X//U isnot necessarily surjective; indeed its image is not necessarily a subvariety of
X//U , so we do not in generalobtain a categorical quotient of X ss . Moreover X//U is in general only quasi-projective, not projective,though when the ring of invariants ˆ O L ( X ) U = L k ≥ H ( X, L ⊗ k ) U is finitely generated as a C -algebra then X//U is the projective variety Proj( ˆ O L ( X ) U ).In order to construct a projective completion of the enveloping quotient X//U when the ring of invariantsˆ O L ( X ) U is not finitely generated, and to understand its geometry, it is convenient to transfer the problem ofconstructing a quotient for the U -action to the construction of a quotient for an action of a reductive group G which contains U as a subgroup, by finding a ‘reductive envelope’. This is a projective completion G × U X of the quasi-projective variety G × U X (that is, the quotient of G × X by the free action of U acting diagonallyon the left on X and by right multiplication on G ), with a linear G -action on G × U X which restricts tothe induced G -action on G × U X , such that ‘sufficiently many’ U -invariants on X extend to G -invariants on1 × U X . If (as is always possible) we choose the linearisation on G × U X to be ample (or more generally tobe ‘fine’), then the classical GIT quotient G × U X//G is a (not necessarily canonical) projective completion
X//U of X//U , and hence also of its open subset X s /U if X s = ∅ , and we have X ¯ s ⊆ X s ⊆ X ss ⊆ X ¯ ss where X ¯ s (respectively X ¯ ss ) denotes the open subset of X consisting of points of X which are stable(respectively semistable) for the G -action on G × U X under the inclusion X ֒ → G × U X ֒ → G × U X. In principle, at least, we can apply the methods of classical GIT to study the geometry of this projectivecompletion
X//U in terms of the geometry of the G -action on G × U X ; for example, techniques from sym-plectic geometry can be used to study the topology of GIT quotients of complex projective varieties bycomplex reductive group actions [AB, JK, JKKW, Ki, Ki2, Ki3, Ki4].Given a linear H -action on X where H is an affine algebraic group with unipotent radical U , if an amplereductive envelope G × U X is chosen in a sufficiently canonical way that the GIT quotient G × U X//G = X//U inherits an induced linear action of the reductive group R = H/U , then(
X//U ) //R is a projective completion of the geometric quotient X s,H /H = ( X s,H /U ) /R where X s,H is the inverse image in X s,U of the open set of points in X s,U /U ⊆ X//U ⊆ X//U which arestable for the action of R = H/U . Moreover we can study the geometry of (
X//U ) //R in terms of that of X//U using classical GIT and symplectic geometry.The aim of this paper is to discuss, for suitable actions on projective varieties X of a non-reductive affinealgebraic group H with unipotent radical U , how to choose a reductive group G ≥ U and reductive envelopes G × U X . In particular we will study the family of examples given by moduli spaces of hypersurfaces in theweighted projective plane P (1 , , H of P (1 , ,
2) (cf. [Fa3]). This automorphism group is a semidirect product H = R ⋉ U where R ∼ = GL (2; C )is reductive and U ∼ = ( C + ) is the unipotent radical of H , acting on P (1 , ,
2) as[ x : y : z ] [ x : y : z + λx + µxy + νy ]for λ, µ, ν ∈ C .For simplicity we will work over C throughout. The layout of the paper is as follows. § § § C + ) r . Finally § P (1 , , In the preface to the first edition of [MFK], Mumford states that his goal is “to construct moduli schemesfor various types of algebraic objects” and that this problem “appears to be, in essence, a special and highlynon-trivial case” of the problem of constructing orbit spaces for algebraic group actions. More precisely,when a family X of objects with parameter space S has the local universal property (that is, any otherfamily is locally equivalent to the pullback of X along a morphism to S ) for a given moduli problem, and a2roup acts on S such that objects parametrised by points in S are equivalent if and only if the points lie inthe same orbit, then the construction of a coarse moduli space is equivalent to the construction of an orbitspace for the action (cf. [Ne, Proposition 2.13]). Here, as in [Ne], by an orbit space we mean a G -invariantmorphism φ : S → M such that every other G -invariant morphism ψ : S → M factors uniquely through φ and φ − ( m ) is a single G -orbit for each m ∈ M .Of course such orbit spaces do not in general exist, in particular because of the jump phenomenon: theremay be orbits contained in the closures of other orbits, which means that the set of all orbits cannot beendowed naturally with the structure of a variety. This is the situation with which Mumford’s geometricinvariant theory [MFK] attempts to deal, when the group acting is reductive, telling us (in suitable circum-stances) both how to throw out certain (unstable) orbits in order to be able to construct an orbit space, andhow to construct a projective completion of this orbit space. Mumford’s GIT is reviewed briefly next; formore details see [Ne2] in these proceedings, or [Do, MFK, Ne, PV]. Example 2.1.
Let G = SL (2; C ) act on ( P ) in the standard way. Then { ( x , x , x , x ) ∈ ( P ) : x = x = x = x } is a single orbit which is contained in the closure of every other orbit. On the other hand, the open subset of( P ) where x , x , x , x are distinct has an orbit space which, using the cross-ratio, can be identified with P − { , , ∞} . Let X be a complex projective variety and let G be a complex reductive group acting on X . To applygeometric invariant theory we require a linearisation of the action; that is, a line bundle L on X and a liftof the action of G to L . Usually L is assumed to be ample, and then we lose little generality in supposingthat for some projective embedding X ⊆ P n the action of G on X extends to an action on P n given by arepresentation ρ : G → GL ( n + 1) , and taking for L the hyperplane line bundle on P n . We have an induced action of G on the homogeneouscoordinate ring ˆ O L ( X ) = M k ≥ H ( X, L ⊗ k )of X . The subring ˆ O L ( X ) G consisting of the elements of ˆ O L ( X ) left invariant by G is a finitely generatedgraded complex algebra because G is reductive [MFK], and so we can define the GIT quotient X//G to bethe variety Proj( ˆ O L ( X ) G ). The inclusion of ˆ O L ( X ) G in ˆ O L ( X ) defines a rational map q from X to X//G ,but because there may be points of X ⊆ P n where every G -invariant polynomial vanishes this map willnot in general be well-defined everywhere on X . The set X ss of semistable points in X is the set of those x ∈ X for which there exists some f ∈ ˆ O L ( X ) G not vanishing at x . Then the rational map q restricts toa surjective G -invariant morphism from the open subset X ss of X to the quotient variety X//G . However q : X ss → X//G is still not in general an orbit space: when x and y are semistable points of X we have q ( x ) = q ( y ) if and only if the closures O G ( x ) and O G ( y ) of the G -orbits of x and y meet in X ss .A stable point of X (‘properly stable’ in the terminology of [MFK]) is a point x of X ss with a neigh-bourhood in X ss such that every G -orbit meeting this neighbourhood is closed in X ss , and is of maximaldimension equal to the dimension of G . If U is any G -invariant open subset of the set X s of stable pointsof X , then q ( U ) is an open subset of X//G and the restriction q | U : U → q ( U ) of q to U is an orbit spacefor the action of G on U , so we will write U/G for q ( U ). In particular there is an orbit space X s /G for theaction of G on X s , and X//G is a projective completion of this orbit space.3 s ⊆ X ss ⊆ X open open y y X s /G ⊆ X//G = X ss / ∼ open (1) X s , X ss , and X//G are unaltered if the line bundle L is replaced by L ⊗ k for any k > G , so it is convenient to allow fractional linearisations.Recall that a categorical quotient of a variety X under an action of G is a G -invariant morphism φ : X → Y from X to a variety Y such that any other G -invariant morphism ˜ φ : X → ˜ Y factors as ˜ φ = χ ◦ φ for a unique morphism χ : Y → ˜ Y [Ne, Chapter 2, § φ : X → Y such that each fibre φ − ( y ) is a single G -orbit, and a geometric quotient is an orbitspace φ : X → Y which is an affine morphism such that(i) if U is an open affine subset of Y then φ ∗ : O ( U ) → O ( φ − ( U ))induces an isomorphism of O ( U ) onto O ( φ − ( U )) G , and(ii) if W and W are disjoint closed G -invariant subvarieties of X then their images φ ( W ) and φ ( W )in Y are disjoint closed subvarieties of Y .If U is any G -invariant open subset of the set X s = X s ( L ) of stable points of X , then q ( U ) is an open subsetof X//G and the restriction q | U : U → q ( U ) of q to U is a geometric quotient for the action of G on U . Inparticular X s /G = q ( X s ) is a geometric quotient for the action of G on X s , while q : X ss → X//G is acategorical quotient of X ss under the action of G .The subsets X ss and X s of X are characterised by the following properties (see Chapter 2 of [MFK] or[Ne]). Proposition 2.2. (Hilbert-Mumford criteria) (i) A point x ∈ X is semistable (respectively stable) for theaction of G on X if and only if for every g ∈ G the point gx is semistable (respectively stable) for the actionof a fixed maximal torus of G .(ii) A point x ∈ X with homogeneous coordinates [ x : . . . : x n ] in some coordinate system on P n is semistable(respectively stable) for the action of a maximal torus of G acting diagonally on P n with weights α , . . . , α n if and only if the convex hull Conv { α i : x i = 0 } contains (respectively contains in its interior). In [MFK] the definitions of X s and X ss are extended as follows to allow L to be not ample and X notprojective. However it is not necessarily the case that U ss = U ∩ X ss or that U s = U ∩ X s when U is a G -invariant open subset of X . Definition 2.3.
Let X be a quasi-projective complex variety with an action of a complex reductive group G and linearisation L on X . Then y ∈ X is semistable for this linear action if there exists some m ≥ f ∈ H ( X, L ⊗ m ) G not vanishing at y such that the open subset X f := { x ∈ X | f ( x ) = 0 } is affine, and y is stable if also the action of G on X f is closed with all stabilisers finite.When X is projective and L is ample and f ∈ H ( X, L ⊗ m ) G \ { } for some m ≥
0, then X f is affinewhen f is nonconstant, so this is equivalent to the previous definition.4 emark 2.4. The reason for introducing the requirement that X f must be affine in Definition 2.3 aboveis to ensure that X ss has a categorical quotient X ss → X//G , which restricts to a geometric quotient X s → X s /G (see [MFK] Theorem 1.10); the quotient X//G is quasi-projective, but need not be projectiveeven when X is projective, if L is not ample. However when X is projective we can define ‘naively stable’and ‘naively semistable’ points by omitting the condition that X f should be affine. More precisely, let I = S m> H ( X, L ⊗ m ) G and for f ∈ I let X f be the G -invariant open subset of X where f does not vanish.Then a point x ∈ X is naively semistable if there exists some f ∈ I which does not vanish at x , so that theset of naively semistable points is X nss = [ f ∈ I X f , whereas X ss = S f ∈ I ss X f where I ss = { f ∈ I | X f is affine } . The set of naively stable points of X is X ns = [ f ∈ I ns X f where I ns = { f ∈ I | the action of G on X f is closed with all stabilisers finite } while X s = S f ∈ I s X f where I s = { f ∈ I ss | the action of G on X f is closed with all stabilisers finite } . If ˆ O L ( X ) = M k ≥ H ( X, L ⊗ k )is finitely generated as a complex algebra, then so is its ring of invariants ˆ O L ( X ) G because G is reductive,and then X//G = Proj( ˆ O L ( X ) G )is a projective completion of X//G and the categorical quotient X ss → X//G is the restriction of a natural G -invariant morphism X nss → X//G, which by analogy with Definition 3.3 below we will call an enveloping quotient of X nss .Throughout the remainder of § X is projective and L is ample. When X is nonsingular then X s /G has only orbifold singularities, but if X ss = X s the GIT quotient X//G is likely to have more serious singularities. However if X s = ∅ there is a canonical procedure (see [Ki2]) forconstructing a partial resolution of singularities ˜ X//G of the quotient
X//G . This involves blowing X upalong a sequence of nonsingular G -invariant subvarieties, all contained in the complement of the set X s ofstable points of X , to obtain eventually a nonsingular projective variety ˜ X with a linear G -action, liftingthe action on X , for which every semistable point of ˜ X is stable. The blow-down map π : ˜ X → X inducesa birational morphism π G : ˜ X//G → X//G which is an isomorphism over the dense open subset X s /G of X//G , and if X is nonsingular the quotient ˜ X//G has only orbifold singularities.This construction works as follows [Ki2]. Let V be any nonsingular G -invariant closed subvariety of X and let π : ˆ X → X be the blowup of X along V . The linear action of G on the ample line bundle L over X lifts to a linear action on the line bundle over ˆ X which is the pullback of L ⊗ k tensored with O ( − E ), where E is the exceptional divisor and k is a fixed positive integer. When k is large the line bundle π ∗ L ⊗ k ⊗ O ( − E )is ample on ˆ X , and this linear action satisfies the following properties:(i) if y is semistable in ˆ X then π ( y ) is semistable in X ;5ii) if π ( y ) is stable in X then y is stable in ˆ X ;(iii) if k is large enough then the sets ˆ X s and ˆ X ss of stable and semistable points of ˆ X with respect tothis linearisation are independent of k (cf. [Rei]).Now X has semistable points which are not stable if and only if there exists a nontrivial connected reductivesubgroup of G which fixes some semistable point. If so, let r > G fixing semistable points of X , and let R ( r ) be a set of representatives of conjugacy classesin G of all connected reductive subgroups R of dimension r such that Z ssR = { x ∈ X ss : R fixes x } is nonempty. Then [ R ∈R ( r ) GZ ssR is a disjoint union of nonsingular closed subvarieties of X ss , and GZ ssR ∼ = G × N R Z ssR where N R is the normaliser of R in G .By Hironaka’s theorem we can resolve the singularities of the closure of S R ∈R ( r ) GZ ssR in X by performinga sequence of blow-ups along nonsingular G -invariant closed subvarieties of X − X ss . We then blow up alongthe proper transform of the closure of S R ∈R ( r ) GZ ssR to get a nonsingular projective variety ˆ X . The linearaction of G on X lifts to an action on this blow-up ˆ X which can be linearised using suitable ample linebundles as above, and it is shown in [Ki2] that the set ˆ X ss of semistable points of ˆ X with respect to any ofthese suitable linearisations of the lifted action is the complement in the inverse image of X ss of the propertransform of the subset φ − φ [ R ∈R ( r ) GZ ssR of X ss , where φ : X ss → X//G is the canonical map. Moreover no point of ˆ X ss is fixed by a reductivesubgroup of G of dimension at least r , and a point in ˆ X ss is fixed by a reductive subgroup R of G ofdimension less than r if and only if it belongs to the proper transform of the subvariety Z ssR of X ss .The same procedure can now be applied to ˆ X to obtain ˆ X such that no reductive subgroup of G ofdimension at least r − X ss . After repeating enough times we obtain ˜ X satisfying ˜ X ss = ˜ X s .If we are only interested in ˜ X ss and the partial resolution ˜ X//G of X//G , rather than in ˜ X itself, then thereis no need in this procedure to resolve the singularities of the closure of S R ∈R ( r ) GZ ssR in X . Instead we cansimply blow X ss up along S R ∈R ( r ) GZ ssR (or equivalently along each GZ ssR in turn) and let ˆ X ss be the setof semistable points in the result, and then repeat the process.Thus the geometric quotient X s /G has two natural compactifications X//G and ^ X//G = ˜
X//G , whichfit into a diagram X s /G ֒ → ^ X//G = ˜ X ss /G ← ˜ X ss = ˜ X s ֒ → ˜ X || ↓ ↓ ↓ X s /G ֒ → X//G ← X ss ֒ → X. The GIT quotient
X//G depends not just on the action of G on X but also on the choice of linearisation L of the action; that is, the choice of the line bundle L and the lift of the action to L . This should of course bereflected in the notation; to avoid ambiguity we will sometimes add appropriate decorations, as in X// L G and also X s, L and X ss, L . There is thus a natural question: how does the GIT quotient X// L G vary asthe linearisation L varies? This has been studied by Brion and Procesi [BP] and Goresky and MacPherson6GM] (in the abelian case) and by Thaddeus [Th], Dolgachev and Hu [DH] and Ressayre [Res] for generalreductive groups G .A very simple case is when the line bundle L is fixed, but the lift τ : G × L → L of the action of G on X to an action on L varies. The only possible such variation is to replace τ with( g, ℓ ) χ ( g ) τ ( g, ℓ )where χ : G → C ∗ is a character of G . The Hilbert-Mumford criteria (Proposition 2.2 above) can be used tosee how X s and X ss are affected by such a variation. Example 2.5.
Consider the linear action of C ∗ on X = P r + s , with respect to the hyperplane line bundle L on X , where the linearisation L + is given by the representation t diag( t , . . . , t , t, t − , . . . , t − )of C ∗ in GL ( r + s + 1; C ) in which t occurs with multiplicity r and t − occurs s times. For this linearisationwe have X ss, L + = X s, L + = { [ x : . . . : x r + s ] ∈ P r + s : x , . . . , x r are not all 0 and x r +1 , . . . , x r + s are not all 0 } and X// L + C ∗ is isomorphic to the product of a weighted projective space P (3 , . . . , ,
1) of dimension r and theprojective space P s − . The same action of C ∗ on X has other linearisations with respect to the hyperplaneline bundle; let L and L − denote the linearisations given by multiplying the representation above by thecharacters χ ( t ) = t − and χ ( t ) = t − . Then L − is given by the representation t diag( t, . . . , t, t − , t − , . . . , t − )of C ∗ in GL ( r + s + 1; C ) and for this linearisation X ss, L − = X s, L − = { [ x : . . . : x r + s ] ∈ P r + s : x , . . . , x r − are not all 0 and x r , . . . , x r + s are not all 0 } while X// L − C ∗ is isomorphic to the product of the projective space P r − and a weighted projective space P (1 , , . . . ,
3) of dimension s . Finally for the linearisation L given by the representation t diag( t , . . . , t , , t − , . . . , t − )of C ∗ in GL ( r + s + 1; C ) we have X ss, L = X s, L (semistability does not imply stability) and the quotient X// L C ∗ has more serious singularities than theorbifolds X// L ± C ∗ . It can be identified with the result of collapsing [0 : . . . : 0 : 1] × P s − to a point in P (3 , . . . , , × P s − , and also with the result of collapsing P r − × [1 : 0 : . . . : 0] to a point in P r − × P (1 , , . . . , Remark 2.6.
The general case when the line bundle L is allowed to vary, as well as the lift of the G -actionfrom X to the line bundle, can be reduced to the case above by a trick due to Thaddeus [Th]. Suppose thata given action of G on X lifts to ample line bundles L , . . . , L m giving linearisations L , . . . , L m over X , andconsider the projective variety Y = P ( L ⊕ · · · ⊕ L m ) . Then the induced action of G on Y has a natural linearisation, and the complex torus T = C m +1 also acts on Y , commuting with the action of G , with a natural linearisation which can be modified using any character χ of T . Taking χ to be the j th projection χ j : T → C ∗ and using the fact that the GIT quotient operationswith respect to G and T commute, we find that X// L j G ∼ = ( Y //G ) // χ j T and hence the general question of variation of GIT quotients with linearisations reduces to the special casewhen the variation is by multiplication by a character of the group. The conclusion [DH, Res, Th] is that,7oughly speaking, the space of all possible ample fractional linearisations of a given G -action on a projectivevariety X is divided into finitely many polyhedral chambers within which the GIT quotient is constant.Moreover, when a wall between two chambers is crossed, the quotient undergoes a transformation whichtypically can be thought of as a blow-up followed by a blow-down. If L + and L − represent fractionallinearisations in the interiors of two adjoining chambers, and L represents a fractional linearisation in theinterior of the wall between them, then we have inclusions X s, L ⊆ X s, L + ∩ X s, L − and X ss, L + ∪ X ss, L − ⊆ X ss, L inducing morphisms X// L ± G → X// L G which are isomorphisms over X s, L /G . In addition, under mild conditions there are sheaves of ideals on thetwo quotients X// L ± G whose blow-ups are both isomorphic to a component of the fibred product of the twoquotients over the quotient X// L G on the wall. Remark 2.7. If φ : X → Y is a categorical quotient for a G -action on a variety X then its restriction toa G -invariant open subset of X is not necessarily a categorical quotient for the action of G on U . In thesituation above, as in Example 2.5, we have X ss, L + ⊆ X ss, L but the restriction of the categorical quotient X ss, L → X// L G to X ss, L + is not a categorical quotient for the G action on X ss, L + . Translation actions appear all over geometry, so it is not surprising that there are many cases of moduliproblems which involve non-reductive group actions, where Mumford’s GIT does not apply. One exampleis that of hypersurfaces in a toric variety Y . The case we shall consider in detail in this paper is when Y isthe weighted projective plane P (1 , ,
2) (cf. [Fa3]), with homogeneous coordinates x, y, z (that is, Y is thequotient of C \ { } by the action of C ∗ with weights 1,1 and 2, and x, y and z are coordinates on C \ { } ).Let H be the automorphism group of Y = P (1 , , C ∗ of a semidirect product ofthe unipotent group U = ( C + ) acting on Y via[ x : y : z ] [ x : y : z + λx + µxy + νy ] for ( λ, µ, ν ) ∈ ( C + ) and the reductive group GL (2; C ) × GL (1; C ) acting on the ( x, y ) coordinates and the z coordinate. H actslinearly on the projective space X d of weighted degree d polynomials in x, y, z . Example 3.1.
When d = 4, a basis for the weighted degree d polynomials is { x , x y, x y , xy , y , x z, xyz, y z, z } , and with respect to this basis, the U -action is given by λ λ µ λ λµ ν µ λ λν + µ ν µ µν ν ν λ µ ν . The tautological family H ( d ) parametrised by X d of hypersurfaces in Y has the following two properties:(i) the hypersurfaces H ( d ) s and H ( d ) t parametrised by weighted degree d polynomials s and t are isomorphicas hypersurfaces in Y if and only if s and t lie in the same orbit of the natural action of H ∼ = U ⋊ GL (2; C )on X d , and 8ii) (local universal property) any family of hypersurfaces in Y is locally equivalent to the pullback of H ( d ) along a morphism to X d .This means that the construction of a (coarse) moduli space of weighted degree d hypersurfaces in Y isequivalent to constructing an orbit space for the action of H on X d ([Ne] Proposition 2.13).Now let H be any affine algebraic group, with unipotent radical U , acting linearly on a complex projectivevariety X with respect to an ample line bundle L . Of course the most immediate difficulty when trying togeneralise Mumford’s GIT to a non-reductive situation is that the ring of invariantsˆ O L ( X ) H = M k ≥ H ( X, L ⊗ k ) H is not necessarily finitely generated as a graded complex algebra, so that Proj( ˆ O L ( X ) H ) is not well-definedas a projective variety. However Proj( ˆ O L ( X ) H ) does make sense as a scheme, and the inclusion of ˆ O L ( X ) H in ˆ O L ( X ) gives us a rational map of schemes q from X to Proj( ˆ O L ( X ) H ), whose image is a constructiblesubset of Proj( ˆ O L ( X ) H ) (that is, a finite union of locally closed subschemes).The action on X of the unipotent radical U of H is studied in [DK] (building on earlier work such as[Fa, Fa2, GP, GP2, Wi]), where the following definitions are made and results proved. Definition 3.2.
Let I = S m> H ( X, L ⊗ m ) U and for f ∈ I let X f be the U -invariant affine open subsetof X where f does not vanish, with O ( X f ) its coordinate ring. A point x ∈ X is naively semistable if therational map q from X to Proj( ˆ O L ( X ) U ) is well-defined at x ; that is, if there exists some f ∈ I which doesnot vanish at x . The set of naively semistable points is X nss = S f ∈ I X f . The finitely generated semistableset of X is X ss,fg = [ f ∈ I fg X f where I fg = { f ∈ I | O ( X f ) U is finitely generated } . The set of naively stable points of X is X ns = [ f ∈ I ns X f where I ns = { f ∈ I | O ( X f ) U is finitely generated, and q : X f −→ Spec( O ( X f ) U ) is a geometric quotient } . The set of locally trivial stable points is X lts = [ f ∈ I lts X f where I lts = { f ∈ I | O ( X f ) U is finitely generated, and q : X f −→ Spec( O ( X f ) U ) is a locally trivial geometric quotient } . Definition 3.3.
Let q : X ss,fg → Proj( ˆ O L ( X ) U ) be the natural morphism of schemes. The envelopedquotient of X ss,fg is q : X ss,fg → q ( X ss,fg ), where q ( X ss,fg ) is a dense constructible subset of the envelopingquotient X//U = [ f ∈ I ss,fg Spec( O ( X f ) U )of X ss,fg . Note that q ( X ss,fg ) is not necessarily a subvariety of X//U , as is demonstrated by the examplestudied in [DK] § U = C + acting on X = P n via the n th symmetric product of its standard representationon C when n is even. 9 roposition 3.4. ([DK] 4.2.9 and 4.2.10). The enveloping quotient
X//U is a quasi-projective variety withan ample line bundle L H → X//U which pulls back to a positive tensor power of L under the natural map q : X ss,fg → X//U . If ˆ O L ( X ) U is finitely generated then X//U is the projective variety
Proj( ˆ O L ( X ) U ) . Now suppose that G is a complex reductive group with U as a closed subgroup. Let G × U X denote thequotient of G × X by the free action of U defined by u ( g, x ) = ( gu − , ux ) for u ∈ U , which is a quasi-projectivevariety by [PV] Theorem 4.19. There is an induced G -action on G × U X given by left multiplication of G on itself. If the action of U on X extends to an action of G there is an isomorphism of G -varieties G × U X ∼ = ( G/U ) × X given by [ g, x ] ( gU, gx ) . (2)When U acts linearly on X with respect to a very ample line bundle L inducing an embedding of X in P n , and G is a subgroup of SL ( n + 1; C ), then we get a very ample G -linearisation (which by abuse ofnotation we will also denote by L ) on G × U X as follows: G × U X ֒ → G × U P n ∼ = ( G/U ) × P n , by taking the trivial bundle on the quasi-affine variety G/U . If we choose a G -equivariant embedding of G/U in an affine space A m with a linear G -action we get a G -equivariant embedding of G × U X in A m × P n ⊂ P m × P n ⊂ P nm + m + n and the G -invariants on G × U X are given by M m ≥ H ( G × U X, L ⊗ m ) G ∼ = M m ≥ H ( X, L ⊗ m ) U = ˆ O L ( X ) U . (3) Definition 3.5.
The sets of
Mumford stable points and
Mumford semistable points in X are X ms = i − (( G × U X ) s ) and X mss = i − (( G × U X ) ss ) where i : X → G × U X is the inclusion given by x [ e, x ]for e the identity element of G . Here ( G × U X ) s and ( G × U X ) ss are defined as in Definition 2.3 for theinduced linear action of G on the quasi-projective variety G × U X .In fact it follows from Theorem 3.10 below that X ms and X mss are equal and are independent of thechoice of G . Definition 3.6. A finite separating set of invariants for the linear action of U on X is a collection ofinvariant sections { f , . . . , f n } of positive tensor powers of L such that, if x, y are any two points of X then f ( x ) = f ( y ) for all invariant sections f of L ⊗ k and all k > f i ( x ) = f i ( y ) ∀ i = 1 , . . . , n. If G is any reductive group containing U , a finite separating set S of invariant sections of positive tensorpowers of L is a finite fully separating set of invariants for the linear U -action on X if(i) for every x ∈ X ms there exists f ∈ S with associated G -invariant F over G × U X (under theisomorphism (3)) such that x ∈ ( G × U X ) F and ( G × U X ) F is affine; and(ii) for every x ∈ X ss,fg there exists f ∈ S such that x ∈ X f and S is a generating set for O ( X f ) U .This definition is in fact independent of the choice of G (see [DK] Remark 5.2.3). Definition 3.7.
Let X be a quasi-projective variety with a linear U -action with respect to an ample linebundle L on X , and let G be a complex reductive group containing U as a closed subgroup. A G -equivariantprojective completion G × U X of G × U X , together with a G -linearisation with respect to a line bundle L which restricts to the given U -linearisation on X , is a reductive envelope of the linear U -action on X ifevery U -invariant f in some finite fully separating set of invariants S for the U -action on X extends to a G -invariant section of a tensor power of L over G × U X .10f moreover there exists such an S for which every f ∈ S extends to a G -invariant section F over G × U X such that ( G × U X ) F is affine, then ( G × U X, L ′ ) is a fine reductive envelope , and if L is ample (in whichcase ( G × U X ) F is always affine) it is an ample reductive envelope .If every f ∈ S extends to a G -invariant F over G × U X which vanishes on each codimension 1 componentof the boundary of G × U X in G × U X , then a reductive envelope for the linear U -action on X is called a strong reductive envelope.It will be useful to add an extra definition which does not appear in [DK]. Definition 3.8.
In the notation of Definitions 3.6 and 3.7 above, a reductive envelope is called stably fine iffor every x ∈ X ms there exists a U -invariant f which extends to a G -invariant section F over G × U X suchthat x ∈ ( G × U X ) F and both ( G × U X ) F and ( G × U X ) F are affine. Definition 3.9.
Let X be a projective variety with a linear U -action and a reductive envelope G × U X .The set of completely stable points of X with respect to the reductive envelope is X s = ( j ◦ i ) − ( G × U X s )and the set of completely semistable points is X ss = ( j ◦ i ) − ( G × U X ss ) , where i : X ֒ → G × U X and j : G × U X ֒ → G × U X are the inclusions, and G × U X s and G × U X ss are thestable and semistable sets for the linear G -action on G × U X . Following Remark 2.4 we also define X nss = ( j ◦ i ) − ( G × U X nss );then X nss = X ss when the reductive envelope is ample, but not in general otherwise. Theorem 3.10. ([DK] 5.3.1 and 5.3.5).
Let X be a normal projective variety with a linear U -action, for U a connected unipotent group, and let ( G × U X, L ) be any fine reductive envelope. Then X s ⊆ X lts = X ms = X mss ⊆ X ns ⊆ X ss,fg ⊆ X ss = X nss . The stable sets X s , X lts = X ms = X mss and X ns admit quasi-projective geometric quotients, given byrestrictions of the quotient map q = π ◦ j ◦ i where π : ( G × U X ) ss → G × U X//G is the classical GIT quotient map for the reductive envelope and i, j are as in Definition 3.9. The quotientmap q restricted to the open subvariety X ss,fg is an enveloped quotient with q : X ss,fg → X//U an envelopingquotient. Moreover
X//U is an open subvariety of G × U X//G and there is an ample line bundle L U on X//U which pulls back to a tensor power L ⊗ k of the line bundle L for some k > and extends to an ampleline bundle on G × U X//G .If furthermore G × U X is normal and provides a fine strong reductive envelope for the linear U -actionon X , then X s = X lts and X ss,fg = X nss . Definition 3.11. ([DK] 5.3.7). Let X be a projective variety equipped with a linear U -action. A point x ∈ X is called stable for the linear U -action if x ∈ X lts and semistable if x ∈ X ss,fg , so from now on wewill write X s (or X s,U ) for X lts and X ss (or X ss,U ) for X ss,fg .Thus in the situation of Theorem 3.10 we have a diagram of quasi-projective varieties X s ⊆ X s ⊆ X ns ⊆ X ss ⊆ X ss = X nss ↓ ↓ ↓ ↓ ↓ X s /U ⊆ X s /U ⊆ X ns /U ⊆ X//U ⊆ G × U X//G where all the inclusions are open and all the vertical morphisms are restrictions of π : ( G × U X ) ss → G × U X//G , and each except the last is a restriction of the map of schemes q : X nss → Proj( ˆ O L ( X ) U ))associated to the inclusion ˆ O L ( X ) U ⊆ ˆ O L ( X ). In particular we have11 s ⊆ X ss ⊆ X ↓ ↓ X s /U ⊆ X//U (4)which looks very similar to the situation for reductive actions (see diagram (1) above), with the majordifferences that(i)
X//U is not always projective,and (even if the ring of invariants ˆ O L ( X ) U is finitely generated and X//U = Proj( ˆ O L ( X ) U ) is projective)(ii) the morphism X ss → X//U is not in general surjective.
Remark 3.12.
The proofs of [DK] Theorem 5.3.1 and Theorem 5.3.5 show that if X is a normal projectivevariety with a linear U -action and ( G × U X, L ) is any stably fine reductive envelope in the sense of Definition3.8, then X ¯ s ⊆ X s ⊆ X ns ⊆ X ss and if furthermore G × U X is normal and provides a reductive envelope which is both strong and stablyfine, then X ¯ s = X s . Indeed, this is still true even if ( G × U X, L ) is not a reductive envelope at all, providedthat it satisfies all the conditions except for omitting (ii) in Definition 3.6.There always exists an ample, and hence fine, but not necessarily strong, reductive envelope for anylinear U -action on a projective variety X , at least if we replace the line bundle L with a suitable positivetensor power of itself, by [DK] Proposition 5.2.8. By Theorem 3.10 above a choice of fine reductive envelope G × U X provides a projective completion X//U = G × U X//G of the enveloping quotient
X//U . This projective completion in general depends on the choice of reductiveenvelope, but when ˆ O L ( X ) U is finitely generated then X//U = Proj( ˆ O L ( X ) U ) is itself projective, whichimplies that X//U = G × U X//G for any fine reductive envelope G × U X .The proof of [DK] Theorem 5.3.1 also gives us the following result for any reductive envelope, notnecessarily fine or strong. Proposition 3.13.
Let X be a normal projective variety with a linear U -action, for U a connected unipotentgroup, and let ( G × U X, L ) be any reductive envelope. Then X s ⊆ X s ⊆ X ss ⊆ X nss , and if the graded algebra L k ≥ H ( G × U X, L ⊗ k ) is finitely generated then the projective completion G × U X//G = Proj( M k ≥ H ( G × U X, L ⊗ k )) of G × U X//G (cf. Remark 2.4) is a projective completion of
X//U with a commutative diagram X s ⊆ X s ⊆ X ss ⊆ X nss ↓ ↓ ↓ ↓ X s /U ⊆ X s /U ⊆ X//U ⊆ X//U = G × U X//G. (5)
Let H be a connected affine algebraic group over C . Then H has a unipotent radical U , which is a normalsubgroup of H with reductive quotient group R = H/U . We can hope to quotient first by the action of U , and then by the induced action of the reductive group H/U , provided that the unipotent quotient issufficiently canonical to inherit an induced linear action of the reductive group R . Moreover U has canonicalseries of normal subgroups { } = U ≤ U ≤ · · · ≤ U s = U such that each successive subquotient is12somorphic to ( C + ) r for some r (for example the descending central series of U ), so we can hope to quotientsuccessively by unipotent groups of the form ( C + ) r , and then finally by the reductive group R . Thereforewe will concentrate on the case when U ∼ = ( C + ) r for some r ; of course this is the situation in our exampleconcerning hypersurfaces in the weighted projective plane P (1 , , H is the automorphism group of P (1 , ,
2) and U is its unipotent radical.More generally, let us assume first that U is a unipotent group with a one-parameter group of automor-phisms λ : C ∗ → Aut( U ) such that the weights of the induced C ∗ action on the Lie algebra u of U are allnonzero. When U = ( C + ) r we can take λ to be the inclusion of the central C ∗ in Aut( U ) ∼ = GL ( r ; C ). Thenwe can form the semidirect product ˆ U = C ∗ ⋉ U given by C ∗ × U with group multiplication( z , u ) . ( z , u ) = ( z z , ( λ ( z − )( u )) u ) .U meets the centre of ˆ U trivially, so we have an inclusion U ֒ → ˆ U → Aut( ˆ U ) → GL (Lie ˆ U ) = GL ( C ⊕ u )where ˆ U maps to its group of inner automorphisms. Thus U is isomorphic to a closed subgroup of thereductive group G = SL ( C ⊕ u ).In particular when U = ( C + ) r we have U ≤ G = SL ( r + 1; C ), and then G/U ∼ = { α ∈ ( C r ) ∗ ⊗ C r +1 | α : C r → C r +1 is injective } with the natural G -action gα = g ◦ α . Since the injective linear maps from C r to C r +1 form an open subsetin the affine space ( C r ) ∗ ⊗ C r +1 whose complement has codimension two, it follows that U = ( C + ) r is aGrosshans subgroup of G = SL ( r + 1; C ) and O ( G ) U ∼ = O ( G/U ) ∼ = O (( C r ) ∗ ⊗ C r +1 )is finitely generated [Gr]. ( C + ) r which extend to SL ( r + 1; C ) Let X be a normal projective variety with a linear action of U = ( C + ) r with respect to an ample line bundle L . Suppose first that the linear action of U = ( C + ) r on X extends to a linear action of G = SL ( r + 1; C ),giving us an identification of G -spaces G × U X ∼ = ( G/U ) × X as at (2) via [ g, x ] ( gU, gx ). Then (as in the Borel transfer theorem [Do, Lemma 4.1])ˆ O L ( X ) U ∼ = ˆ O L ( G × U X ) G ∼ = [ O ( G/U ) ⊗ ˆ O L ( X )] G is finitely generated [Gr2] and we have a reductive envelope G × U X = P r ( r +1) × X, where P r ( r +1) = P ( C ⊕ (( C r ) ∗ ⊗ C r +1 )), with G × U X//G ∼ = X//U = Proj( ˆ O L ( X ) U ) . More precisely, if we choose for our linearisation on G × U X the line bundle L ( N ) = O P r ( r +1) ( N ) ⊗ L with N > X ¯ s = X s and X ¯ ss = X ss . (6)13 emark 4.1. Even if X is nonsingular, this quotient X//U = ( P r ( r +1) × X ) //G may have serious singularities if there are semistable points which are not stable. However provided that X s = ∅ we can construct a partial desingularisation ^ X//U ( G ) = ( ^P r ( r +1) × X ) //G as in § P r ( r +1) × X up successively along G -invariant closed subvarieties, all disjoint from( P r ( r +1) × X ) s and hence from X s = X s , to get a linear G -action on the resulting blow-up ^P r ( r +1) × X forwhich all semistable points are stable. This construction is determined by the linear G -action, and if X isnonsingular the resulting quotient is an orbifold. Since X ¯ s = X s and the morphism ^ X//U ( G ) = ( ^P r ( r +1) × X ) //G → ( P r ( r +1) × X ) //G = X//U is an isomorphism over ( P r ( r +1) × X ) s , it follows that ^ X//U ( G ) → X//U is an isomorphism over X s /U , andhence we have two compactifications of the geometric quotient X s /U : X s /U ⊆ ^ X//U ( G ) || ↓ X s /U ⊆ X//U where ^ X//U ( G ) is an orbifold. U = ( C + ) r is the unipotent radical of a parabolic subgroup P = U ⋊ GL ( r ; C ) (7)in SL ( r + 1; C ) with Levi subgroup GL ( r ; C ) embedded in SL ( r + 1; C ) as g (cid:18) g
00 det g − (cid:19) . We have G × U X ∼ = G × P ( P × U X )where P/U ∼ = GL ( r ; C ) and G/P ∼ = P r is projective. If P × U X is a P -equivariant projective completion of P × U X then G × P ( P × U X ) is a projective completion of G × U X . When the action of U on X extendsto a G -action as above, we can choose P × U X to be the closure of P × U X in G × U X = G/U × X = P r ( r +1) × X ;that is, P × U X = P r × X ⊆ G × U X where P r = P ( C ⊕ (( C r ) ∗ ⊗ C r )). There is then a birational morphism G × P ( P × U X ) → G × U X given by [ g, y ] gy which is an isomorphism over G × U X . The resulting pullback ˆ L = ˆ L ( N ) to G × P ( P × U X ) of O P r ( r +1) ( N ) ⊗ L is isomorphic to the induced line bundle G × P ( O P r ( N ) ⊗ L )on G × P ( P × U X ), where the P -action on O P r ( N ) ⊗ L is the restriction of the G -action on O P r ( r +1) ( N ) ⊗ L .If we regard G × P ( P × U X ) as a subvariety in the obvious way of G × P ( G × U X ) = G × P ( P r ( r +1) × X )14 = ( G/P ) × P r ( r +1) × X ∼ = P r × P r ( r +1) × X then the birational morphism G × P ( P × U X ) → G × U X ∼ = P r × X given by [ g, y ] gy extends to the projection P r × P r ( r +1) × X → P r ( r +1) × X and so ˆ L ( N ) is the restriction to G × P ( P × U X ) of O P r ( r +1) ( N ) ⊗ L . Thus this line bundle ˆ L = ˆ L ( N ) is notample, but its tensor product ˆ L ǫ = ˆ L ( N ) ǫ with the pullback via the morphism G × P ( P × U X ) → G/P ∼ = P r , of the fractional line bundle O P r ( ǫ ), where ǫ ∈ Q ∩ (0 , ∞ ), provides an ample fractional linearisation for theaction of G on G × P ( P × U X ) with, when ǫ is sufficiently small, an induced surjective birational morphism \ X//U = df G × P ( P × U X ) // ˆ L ǫ G → G × U X//G = X//U (8)(cf. [Ki2, Rei]) which is an isomorphism over( G × U X ¯ s ) /G ∼ = X ¯ s /U = X s /U. Note that ˆ L ǫ can be thought of as the bundle G × P ( O P r ( N ) ⊗ L ) on G × P ( P × U X ), where now the P -action on O P r ( N ) ⊗ L is no longer the restriction of the G -action on O P r ( r +1) ( N ) ⊗ L but has been twistedby ǫ times the character of P which restricts to the determinant on GL ( r ; C ). Remark 4.2.
It follows from variation of GIT [Res] for the G -action on G × P ( P × U X ) that \ X//U = \ X//U
N,ǫ is independent of N and ǫ , provided that N is sufficiently large and ǫ > N . Remark 4.3.
When ǫ > G × P ( P × U X ) equipped with the induced amplefractional linearisation on ˆ L ǫ is not in general a reductive envelope for the U -action on X , though it satisfiesall the remaining conditions when (ii) is omitted from Definition 3.6 (cf. Remark 3.12). If we use thelinearisation on ˆ L = ˆ L instead, then we do obtain a reductive envelope, but it is not ample; nonetheless theconditions of Proposition 3.13 are satisfied and we have G × P ( P × U X ) // ˆ L G = G × U X//G = X//U.
Example 4.4.
Let U = C + act linearly on a projective space P n . Then we can choose coordinates so that1 ∈ Lie( C + ) = C has Jordan normal form with blocks · · ·
00 0 1 0 · · · · · · · · · · · · of sizes k +1 , . . . , k s +1 where P sj =1 ( k j +1) = n +1 . Then the C + action extends to an action of G = SL (2; C )via the identifications C + ∼ = { (cid:18) a (cid:19) : a ∈ C } ≤ G and C n +1 ∼ = s M j =1 Sym k j ( C )where Sym k ( C ) is the k th symmetric power of the standard representation C of G = SL (2; C ). Moreover G/ C + ∼ = C \ { } ⊆ C ⊆ P = G/ C + P n // C + ∼ = Proj( C [ x , . . . , x n ] C + ) ∼ = ( P × P n ) //G with respect to the linearisation O P ( N ) ⊗ O P n (1) on P × P n for N a sufficiently large positive integer.When G = SL (2; C ) acts on P we have ( P ) ss,G = C (and ( P ) s,G = ∅ ), so since N is large we have( P × P n ) ss,G ⊆ C × P n = ( G × C + P n ) ⊔ ( { } × P n )and if semistability implies stability then P n // C + = ( P n ) s,U / C + ⊔ ( { } × P n ) //SL (2; C ) . In this example the parabolic subgroup P of G = SL (2; C ) is its Borel subgroup B = { (cid:18) a b a − (cid:19) : a ∈ C ∗ , b ∈ C } with B/ C + = C ∗ = P and B × C + P n = P × P n , while G × B B/ C + = G × B P is the blow-up of P at the origin 0 ∈ C ⊆ P . Similarly G × B ( B × C + P n ) isthe blow-up of G × C + P n ∼ = P × P n along { } × P n , and its quotient \ X//U is the blow-up of P n // C + alongits ‘boundary’ P n //SL (2; C ) ∼ = ( { } × P n ) //SL (2; C ) ⊆ ( P × P n ) //SL (2; C ) = P n // C + . Let us continue to assume that U = ( C + ) r acts linearly on X and that the action extends to G = SL ( r + 1; C ). Notice that there are surjections P × U X ss,P,ǫ → \ X//U → X//U (9)where P × U X ss,P,ǫ is the intersection of P × U X with the G -semistable set in G × P P × U X with respectto the linearisation ˆ L ǫ , and y , y ∈ P × U X ss,P,ǫ map to the same point in \ X//U if and only if the closuresof their P -orbits P y and P y meet in P × U X ss,P,ǫ .Consider the linear action of the Levi subgroup GL ( r ; C ) ≤ P on P × U X = P r × X . It follows fromthe Hilbert-Mumford criteria (Proposition 2.2 above) that P × U X ss,P,ǫ ⊆ P × U X ss,GL ( r ; C ) ,ǫ ⊆ P × U X ss,SL ( r ; C ) (10)where P × U X ss,GL ( r ; C ) ,ǫ and P × U X ss,SL ( r ; C ) (independent of ǫ ) denote the GL ( r ; C ) and SL ( r ; C )-semistablesets of P × U X after twisting the linearisation by ǫ times the character det of GL ( r ; C ); this character is ofcourse trivial on SL ( r ; C ).It is not hard to check that if the action of GL ( r ; C ) on P/U = P r is linearised with respect to O P r (1)by twisting by the fractional character det then P/U ss,GL ( r ; C ) , / = P/U s,GL ( r ; C ) , / = GL ( r ; C ) ⊆ ( C r ) ∗ ⊗ C r ⊆ P r . (11)Thus, if instead of choosing ǫ close to 0 we choose ǫ to be approximately N/
2, where N is the sufficientlylarge positive integer chosen above, then we see from the Hilbert-Mumford criteria (Proposition 2.2) that P × U X ss,GL ( r ; C ) ,ǫ = ( P r × X ) ss,GL ( r ; C ) ,ǫ = GL ( r ; C ) × X and so quotienting we get P × U X// ˆ L ( N ) N/ GL ( r ; C ) = X.
16 GIT quotient of a nonsingular complex projective variety Y by a linear action of GL ( r ; C ) can alwaysbe constructed by first quotienting by SL ( r ; C ) and then quotienting by the induced linear action of C ∗ = GL ( r ; C ) /SL ( r ; C ): we have Y //GL ( r ; C ) = ( Y //SL ( r ; C )) // C ∗ . Therefore if we set X = P × U X// ˆ L ( N ) SL ( r ; C ) = ( P r × X ) // ˆ L ( N ) SL ( r ; C ) (12)for N > X is a projective variety with a linear action of C ∗ which we can twist by ǫ times the standard character of C ∗ , such that when ǫ = N/ X // N/ C ∗ ∼ = X (13)while for ǫ > X // ǫ C ∗ onto \ X//U , and hence onto
X//U . More precisely, the inclusion( ˆ O ˆ L ǫ ( G × P ( P × U X ))) G = ( ˆ O ˆ L ǫ ( P × U X )) P ⊆ ( ˆ O ˆ L ǫ ( P × U X )) GL ( r ; C ) induces a rational map X // ǫ C ∗ = P × U X// ˆ L ǫ GL ( r ; C ) − − → G × P P × U X// ˆ L ǫ G = \ X//U (14)whose composition with the surjection P × U X ss,GL ( r ; C ) ,ǫ → X // ǫ C ∗ induced by the inclusion ( ˆ O ˆ L ǫ ( P × U X )) GL ( r ; C ) ⊆ ( ˆ O ˆ L ǫ ( P × U X )) SL ( r ; C ) is the rational map P × U X ss,GL ( r ; C ) ,ǫ − − → G × P P × U X// ˆ L ǫ G = \ X//U which restricts to a surjection P × U X ss,P,ǫ → \ X//U.
Hence the restriction of X // ǫ C ∗ − − → \ X//U to its domain of definition is surjective.
Definition 4.5.
Let ( X // ǫ C ∗ ) ˆ ss denote the open subset of X // ǫ C ∗ which is the domain of definition of therational map (14) from X // ǫ C ∗ to \ X//U , where as above X is the projective variety X = ( P × U X ) //SL ( r ; C )with the induced linear C ∗ -action, and 0 < ǫ <<
1. Let ( X // ǫ C ∗ ) ˆ s be the open subset P × U X s,P,ǫ /GL ( r ; C )of P × U X s,GL ( r ; C ) ,ǫ /GL ( r ; C ) = ( P × U X s,GL ( r ; C ) ,ǫ /SL ( r ; C )) / C ∗ = X s,ǫ / C ∗ ⊆ X // ǫ C ∗ . Let X ˆ ss,ǫ = π − (( X // ǫ C ∗ ) ˆ ss ) and X ˆ s,ǫ = π − (( X // ǫ C ∗ ) ˆ s ) where π : X ss,ǫ → X // ǫ C ∗ is the quotient map, sothat ( X // ǫ C ∗ ) ˆ s = X ˆ s,ǫ / C ∗ . Then we have
Proposition 4.6. If ǫ > is sufficiently small, the rational map from X // ǫ C ∗ to \ X//U induced by theinclusion of ( ˆ O ˆ L ǫ ( P × U X )) P in ( ˆ O ˆ L ǫ ( P × U X )) GL ( r ; C ) restricts to surjective morphisms ( X // ǫ C ∗ ) ˆ ss → \ X//U → X//U and ( X // ǫ C ∗ ) ˆ s → X s /U. emark 4.7. Using the theory of variation of GIT [DH, Res, Th], as described in Remark 2.6, we canrelate the quotient X // ǫ C ∗ which appears in Proposition 4.6 above to X // N/ C ∗ ∼ = X via a sequence of flipswhich occur as walls are crossed between the linearisations corresponding to ǫ and to N/
2. Thus we have adiagram ( X // ǫ C ∗ ) ˆ s ⊆ ( X // ǫ C ∗ ) ˆ ss ⊆ X // ǫ C ∗ ← − → X = X // N/ C ∗ ↓ ↓ flips X s /U ⊆ \ X//U || ↓ X s /U ⊆ X//U where the vertical maps are all surjective , in contrast to (4), and the inclusions are all open.Note also that by variation of GIT if 0 < ǫ << X // ǫ C ∗ → X // C ∗ . When ǫ = 0 the inclusion( ˆ O ˆ L ( G × P ( P × U X ))) G = ( ˆ O ˆ L ( P × U X )) P ⊆ ( ˆ O ˆ L ( P × U X )) GL ( r ; C ) induces a rational map X // C ∗ = P × U X// ˆ L GL ( r ; C ) − − → G × P P × U X// ˆ L G = X//U (15)whose composition with the surjective morphism X // ǫ C ∗ → X // C ∗ is the composition of (14) with the surjective morphism \ X//U → X//U . Thus the restriction of the rationalmap (15) from X // C ∗ to X//U to its domain of definition is surjective.
Remark 4.8.
Note that the GIT quotient P r //SL ( r ; C ) is isomorphic to P , but we do not have ( P r ) ss =( P r ) s for this action of SL ( r ; C ). It is therefore convenient to replace the compactification P r of GL ( r ; C ) byits wonderful compactification f P r given by blowing up P r = { [ z : ( z ij ) ri,j =1 ] } along the (proper transformsof the) subvarieties defined by z = 0 and rank( z ij ) ≤ ℓ for ℓ = 0 , , . . . , r and by rank( z ij ) ≤ ℓ for ℓ = 0 , , . . . , r − SL ( r ; C ) on f P r , linearised with respect to a small perturbation O g P r (1) of the pullback of O P r (1), satisfies f P r ss = f P r s and f P r //SL ( r ; C ) ∼ = P . Thus if we replace P × U X = P r × X with ^ P × U X = f P r × X and define ^ X//U = G × P ( ^ P × U X ) // ˆ L ǫ G and e X = ^ P × U X// ˆ L ( N ) SL ( r ; C ) = ( f P r × X ) // ˆ L ( N ) SL ( r ; C ) (16)for N >>
0, then all the properties of X given above still hold for e X , and in addition e X fibres over P as e X = ( f P r × X ) // ˆ L ( N ) SL ( r ; C ) = ( f P r ss × X ) /SL ( r ; C ) → f P r ss /SL ( r ; C ) = P with fibres isomorphic to the quotient of X by the finite centre of SL ( r ; C ). If X is nonsingular then it turnsout that e X and e X // ǫ C ∗ (for 0 < ǫ <<
1) and ^ X//U are orbifolds, so that ^ X//U is a projective completion of X s /U which is a partial desingularisation of X//U (cf. Remark 4.1).18 .2 General ( C + ) r actions Of course the constructions described in § U = ( C + ) r on X extends to anaction of G = SL ( C ⊕ u ), which is a rather special situation when the ring of invariants ˆ O L ( X ) U is alwaysfinitely generated. Moreover at least a priori these constructions may depend on the choice of this extension,although G × U X//G = X//U = Proj( ˆ O L ( X ) U ) depends only on the linearisation of the U -action on X .So next we need to consider what happens if the linear U -action on X does not extend to a linear actionof G . Suppose that we can associate to the linear U -action on X a normal projective variety Y containing X , with an action of G = SL ( C ⊕ u ) and a G -linearisation on a line bundle L Y , which restricts to thegiven linearisation of the U -action on X and is such that every U -invariant in a finite fully separating set of U -invariants on X extends to a U -invariant on Y . Then we can embed X in the G -variety P r ( r +1) × Y as { ι } × X where ι ∈ ( C r ) ∗ ⊗ C r +1 ⊆ P r ( r +1) is the standard embedding of C r in C r +1 , and the closure of GX ∼ = G × U X in P r ( r +1) × Y will provide us with a reductive envelope G × U X . Therefore we will nextconsider how, given any linearised U -action on X , we can choose a G -variety Y with these properties. Wewill find that for any sufficiently divisible positive integer m we can choose such a variety Y m in a canonicalway, depending only on m and the linear action of U on X , giving us a reductive envelope G × U X m .Let S be any finite fully separating set of invariants (in the sense of Definition 3.6) on X . By replacingthe elements of S with suitable powers of themselves, we can assume that S ⊆ H ( X, L ⊗ m ) U for some m > L ⊗ m is very ample. Then X ⊆ P ( H ( X, L ⊗ m ) ∗ ) and every σ ∈ S extends to a U -invariant sectionof O (1) on P ( H ( X, L ⊗ m ) ∗ ).Now consider the linear action of U on V m = H ( X, L ⊗ m ) ∗ , and let P be the parabolic subgroup of G = SL ( r + 1; C ) with unipotent radical U , as at (7) above. Since P is a semi-direct product P = U ⋊ GL ( r ; C )we have P × U V m ∼ = GL ( r ; C ) × V m with the P -action on GL ( r ; C ) × V m given for ( h, v ) ∈ GL ( r ; C ) × V m by p. ( h, v ) = ( gh, ( h − uh ) .v )where p = gu with g ∈ GL ( r ; C ) and u ∈ U , and h − uh acts on v ∈ V m via the given U -action. Of course GL ( r ; C ) × V m is an affine variety with O ( GL ( r ; C ) × V m ) ∼ = C [ h ij , (det h ) − , v k ]where det h is the determinant of the r × r matrix ( h ij ) ri,j =1 and ( v k ) are coordinates on V m . Let φ V m : C r = Lie U → Lie( GL ( V m )) (17)be the infinitesimal action of U on V m and let U V m be its image in Lie( GL ( V m )). Since U is unipotent wehave V m ⊇ U V m ( V m ) ⊇ ( U V m ) ( V m ) ⊇ · · · ⊇ ( U V m ) dim V m ( V m ) = 0where ( U V m ) j ( V m ) = { u u · · · u j ( v ) : u , . . . , u j ∈ U V m , v ∈ V m } = { φ V m (˜ u ) φ V m (˜ u ) · · · φ V m (˜ u j )( v ) : ˜ u , . . . , ˜ u j ∈ Lie
U, v ∈ V m } . For 0 ≤ j ≤ dim V m − j,m be the complex vector space consisting of all polynomial functions θ : ( C r ) j × (( C r ) ∗ ⊗ C r ) dim V m − → C ( u , . . . , u j , h , . . . , h dim V m − ) θ ( u , . . . , u j , h , . . . , h dim V m − )19hich are simultaneously homogeneous of degree 1 in the coordinates of each u i ∈ C r separately, for 1 ≤ i ≤ j ,and homogeneous of total degree r (dim V m − − ( r − j in the coordinates of all the h k ∈ ( C r ) ∗ ⊗ C r together, for 1 ≤ k ≤ dim V m −
1. Let W m = dim V m − M j =0 Θ j,m ⊗ ( U V m ) j ( V m ) . (18)Then we can embed V m linearly into W m via v ψ ( v ) = dim V m − X j =0 ψ j ( v ) (19)where ψ j ( v ) : ( C r ) j × (( C r ) ∗ ⊗ C r ) (dim V m − → ( U V m ) j ( V m ) for 0 ≤ j ≤ dim V m − v ∈ V m sends( u , . . . , u j , h , . . . , h dim V m − ) to the product ofdet( h ) det( h ) · · · det( h dim V m − j − )with φ V m ( h dim V m − u ) · · · φ V m ( h dim V m − j u j )( v ) . In particular ψ ( v ) : (( C r ) ∗ ⊗ C r ) (dim V m − ) → ( U V m ) ( V m ) = V m sends ( h , . . . , h dim V m − ) to(det( h ) det( h ) · · · det( h dim V m − )) v ∈ V m . This embedding of V m into W m is U -equivariant with respect to the linear U -action on W m for which theinfinitesimal action of u ∈ Lie U = C r is given by u ( dim V m − X j =0 α j ) = dim V m − X j =0 u · α j +1 (20)with u · α j +1 : ( C r ) j × (( C r ) ∗ ⊗ C r ) dim V m − → ( U V m ) j ( V m ) defined to be 0 for j = dim V m and for any0 ≤ j < dim V m defined by u · α j +1 ( u , . . . , u j , h , . . . , h dim V m − )= α j +1 ( u , . . . , u j , adj( h dim V m − − j )( u ) , h , . . . , h dim V m − ) . Here adj( h ) is the adjoint matrix of h (so that h adj( h ) = adj( h ) h is det( h ) times the identity matrix ι ),which is homogeneous of degree r − h . Moreover this linear action of U on W m extendsto a linear action of P = U ⋊ GL ( r ; C ) on W m where g ∈ GL ( r ; C ) acts as g ( dim V m − X j =0 α j ) = dim V m − X j =0 (det g ) j g · α j (21)with g · α j : ( C r ) j × (( C r ) ∗ ⊗ C r ) dim V m − → ( U V m ) j ( V m ) defined for any 0 ≤ j < dim V m and any α j :( C r ) j × (( C r ) ∗ ⊗ C r ) dim V m − → ( U V m ) j ( V m ) by g · α j ( u , . . . , u j , h , . . . , h dim V m − )= α j ( gu , . . . , gu j , gh g − , . . . , gh dim V m − g − ) . Since the U -action on P ( W m ) extends to a linear P -action, we can construct the projective variety Y m = G × P P ( W m )and we can equip Y m with the line bundles L ǫY m = G × P O P ( W m ) (1) for ǫ ∈ Q where the P -action on O P ( W m ) (1) is induced from the given P -action on W m twisted by ǫ times the pullback to P of the characterdet of P/U ∼ = GL ( r ; C ). Equivalently L ǫY m is the tensor product of L Y m with the pullback via G × P P ( W m ) → G/P ∼ = P r of O P r ( ǫ ). 20 emark 4.9. The action of P on W m extends to a linear action of G = SL ( r + 1; C ) on the vector space W m ⊇ W m , where W m = H ( G × P P ( W m ⊗ ( M λ ∈ ∆ C λ )); G × P O (1))for a sufficiently large finite set ∆ of weights of P , with C λ denoting a copy of C on which P acts with weight λ . Thus G × P P ( W m ) ⊆ G × P P ( W m ) ∼ = G/P × P ( W m ) ∼ = P r × P ( W m ), and with respect to this identificationwe have L Y m = O P ( W m ) (1) | Y m . Hence if ǫ > L ǫY m is the restriction of the line bundle O P r ( ǫ ) ⊗ O P ( W m ) (1) on P r × P ( W m ),and so L ǫY m is ample.This gives us for every sufficiently divisible positive integer m a U -equivariant embedding X ⊆ P ( H ( X, L ⊗ m ) ∗ ) = P ( V m ) ⊆ P ( W m )in a projective variety with a linear P -action, such that every σ in the finite fully separating set of invariants S extends to a U -invariant linear functional on V m , and hence extends to a U -invariant (in fact P -invariant)linear functional on W m defined by P dim V m j =0 α j σ ( α ( ι, . . . , ι )). As each σ ∈ S extends to a P -invariantlinear functional on W m , it extends to a G -invariant section of L Y m . Thus from § Proposition 4.10.
Let X be embedded in Y m = G × P P ( W m ) as above, for a sufficiently divisible positiveinteger m , and let ι ∈ ( C r ) ∗ ⊗ C r +1 ⊆ P r ( r +1) be the standard embedding of C r in C r +1 . If N is sufficientlylarge (depending on m ), then the linear action of U = ( C + ) r on X has a reductive envelope given by theclosure G × U X m of G × U X embedded in P r ( r +1) × Y m as G ( { ι } × X ) , equipped with the restriction of the G -linearisation on O P r ( r +1) ( N ) ⊗ L Y m . Note that by Remark 4.9 the line bundle L Y m is not in general ample (although L ǫY m is ample for any ǫ > X s and X nss are defined using this reductive envelope we have Corollary 4.11. X s ⊆ X s ⊆ X ss ⊆ X nss . If moreover the ring of invariants ˆ O L ( X ) U is finitely generated and m is sufficiently divisible thatˆ O L ⊗ m ( X ) U is generated by H ( X, L ⊗ m ) U , then for N >> ρ m : M k ≥ H ( G × U X m , ( O P r ( r +1) ( N ) ⊗ L Y m ) ⊗ k ) G → ˆ O L ⊗ m ( X ) U is an isomorphism and X//U = Proj ˆ O L ( X ) U is the canonical projective completion of G × U X m //G (seeProposition 3.13 again). Even when the ring of invariants ˆ O L ( X ) U is not finitely generated, if m is sufficientlydivisible that H ( X, L ⊗ m ) contains a finite fully separating set of invariants, then for any multiple m ′ = k ′ m of m the subalgebra ˆ O m ′ L ( X ) U of ˆ O L ( X ) U generated by H ( X, L ⊗ m ′ ) U is finitely generated and provides a projective completion X//U m = Proj ˆ O m ′ L ( X ) U of X//U , while the restriction of ρ m to the subalgebra of M k ≥ H ( G × U X m , ( O P r ( r +1) ( N ) ⊗ L Y m ) ⊗ k ) G generated by H ( G × U X m , ( O P r ( r +1) ( N ) ⊗ L Y m ) ⊗ k ′ ) G gives an isomorphism onto ˆ O m ′ L ( X ) U .21y analogy with the construction of \ X//U and ^ X//U in § \ G × U X m and ^ G × U X m of G × U X = G ( { ι } × X ) in G × P ( P r × Y m ) and G × P ( f P r × Y m ) respectively. Since Y m = G × P P ( W m ) we have \ G × U X m ∼ = G × P ( P × U X m ) ∼ = G × U X m and ^ G × U X m ∼ = G × P ( ^ P × U X m )where P × U X m and ^ P × U X m are the closures of P × U X = P ( { ι } × X ) in P r × P ( W m ) and f P r × P ( W m )respectively. Definition 4.12.
Let ˆ L ǫ = ˆ L ( N ) ǫ be the tensor product of the pullback via \ G × U X m ∼ = G × P ( P × U X m ) → G/P ∼ = P r of O P r ( ǫ ) with the line bundle G × P ˆ L ( N ) on G × P ( P × U X m ) whereˆ L ( N ) = O P r ( N ) ⊗ O P ( W m ) (1) | P × U X m ;equivalently ˆ L ( N ) ǫ = G × P ˆ L ( N ) where the action of P on ˆ L ( N ) is twisted by ǫ times the character det of P/U ∼ = GL ( r ; C ). Similarly let ˜ L ǫ = ˜ L ( N ) ǫ be the tensor product of the pullback via ^ G × U X m ∼ = G × P ( ^ P × U X m ) → G/P ∼ = P r of O P r ( ǫ ) with the line bundle G × P ˜ L ( N ) on G × P ( ^ P × U X m ), where ˜ L ( N ) = O g P r ( N ) ⊗ O P ( W m ) (1) | ^ P × U X m and as in Remark 4.8 O g P r (1) is a small perturbation of the pullback of O P r (1) along f P r → P r such thatthe SL ( r ; C )-action lifts to O g P r (1) and satisfies ( f P r ) ss = ( f P r ) s and f P r //SL ( r ; C ) ∼ = P . Note that theline bundles ˆ L ( N ) ǫ on \ G × U X m ∼ = G × P ( P × U X m ) and O P r ( r +1) ( N ) ⊗ L ǫY m on \ G × U X m ∼ = G × P ( P × U X m ) ∼ = G × U X m are both G -invariant and both restrict to O P r ( N ) ⊗ O P ( W m ) (1) on P × U X m with the same P -action, sothey are isomorphic to each other.The line bundles ˆ L ǫ = ˆ L ( N ) ǫ and ˜ L ǫ = ˜ L ( N ) ǫ on G × U X m and ^ G × U X m are ample for ǫ >
0, and the G -actions on G × U X m and ^ G × U X m lift to linear actions on ˆ L ǫ and ˜ L ǫ . Definition 4.13.
For a positive integer N sufficiently large (depending on m ) let X m = P × U X m // ˆ L ( N ) SL ( r ; C ) and ˜ X m = ^ P × U X m // ˜ L ( N ) SL ( r ; C )and for ǫ > m and N ) let \ X//U m = G × U X m // ˆ L ( N ) ǫ G and ^ X//U m = ^ G × U X m // ˜ L ( N ) ǫ G. Remark 4.14.
The line bundles ˆ L ( N ) and ˜ L ( N ) are ample on P × U X m and ^ P × U X m , and for ǫ > L ( N ) ǫ and ˜ L ( N ) ǫ are ample on G × U X m and ^ G × U X m . Thus it follows from variation of GIT[Res] for the SL ( r ; C )-actions on P × U X m and ^ P × U X m and the G = SL ( r + 1; C )-actions on G × U X m and ^ G × U X m that X m and ˜ X m and \ X//U m and ^ X//U m are independent of N and ǫ , provided that m isfixed and N is sufficiently large, depending on m , and ǫ is sufficiently small, depending on N and ǫ .22 emark 4.15. Recall that ˆ L ( N ) ǫ can be identified with the line bundle G × P ˆ L ( N ) on G × P ( P × U X m )when the action of P on ˆ L ( N ) is twisted by ǫ times the character det of P/U ∼ = GL ( r ; C ). The characterdet extends to a section of the line bundle O P r ( r ) over the projective completion P r of GL ( r ; C ) on which P acts by multiplication by the character det, and thus to a section of the line bundle G × P O P r over G × P ( P × U X m ) when the action of P on O P r is twisted by this character. Tensoring with this sectiongives us an injection H ( G × U X m , ( ˆ L ( N )0 ) ⊗ k ) G → H ( G × U X m , ( ˆ L ( N + rǫ ) ǫ ) ⊗ k ) G (22)where ǫ = 1 /k , whose composition with the injection given by the restriction map ρ ǫm : M k ≥ H ( G × U X m , ( ˆ L ( N + rǫ ) ǫ ) ⊗ k ) G → ˆ O L ⊗ m ( X ) U is the restriction map ρ m = ρ m . If the ring of invariants ˆ O L ( X ) U is finitely generated and m is sufficientlydivisible that ρ m is an isomorphism, it follows that (22) is an isomorphism and thus that( G × U X m ) ss,ǫ,G ⊆ ( G × U X m ) ss, ,G and that the inclusion of ( G × U X m ) ss,ǫ,G in ( G × U X m ) ss, ,G induces a birational surjective morphism \ X//U m → X//U = Proj( ˆ O L ( X ) U ) . Even when ˆ O L ( X ) U is not finitely generated, if m and k are sufficiently divisible that H ( X, L ⊗ m ) U containsa fully separating set of invariants and O (ˆ L ( N + rǫ ) ǫ ) ⊗ k ( G × U X m ) is generated by H ( G × U X m , ( ˆ L ( N + rǫ ) ǫ ) ⊗ k ) G ,then we have a birational surjective morphism \ X//U m → Proj( ˆ O m ′ L ( X ) U ) = X//U m ′ where m ′ = km . The same is also true when \ X//U m is replaced with ^ X//U m .As in Proposition 4.6 and Remark 4.8 we obtain Theorem 4.16. If m is a sufficiently divisible positive integer and N >> then X m and e X m are projectivevarieties with linear actions of C ∗ which we can twist by ǫ times the standard character of C ∗ , such thatwhen ǫ = N/ we have X m // N/ C ∗ ∼ = X and e X m // N/ C ∗ ∼ = X, while if ǫ > is sufficiently small then the rational maps from X m // ǫ C ∗ to \ X//U m and from e X m // ǫ C ∗ to ^ X//U m induced by the inclusions of ( ˆ O ˆ L ǫ ( P × U X m )) P in ( ˆ O ˆ L ǫ ( P × U X m )) GL ( r ; C ) and ( ˆ O ˆ L ǫ ( ^ P × U X m )) P in ( ˆ O ˆ L ǫ ( ^ P × U X m )) GL ( r ; C ) restrict to surjective morphisms ( X m // ǫ C ∗ ) ˆ ss → \ X//U m and ( e X m // ǫ C ∗ ) ˆ ss → ^ X//U m where ( X m // ǫ C ∗ ) ˆ ss and ( e X m // ǫ C ∗ ) ˆ ss are open subsets of X m // ǫ C ∗ and e X m // ǫ C ∗ defined as in Definition4.5. Thus when m is sufficiently divisible we have the following diagrams (cf. Remark 4.7):( X m // ǫ C ∗ ) ˆ ss ⊆ X m // ǫ C ∗ ← − → X = X m // N/ C ∗ ↓ flips \ X//U m (23)and ( e X m // ǫ C ∗ ) ˆ ss ⊆ e X m // ǫ C ∗ ← − → X = e X m // N/ C ∗ ↓ flips ^ X//U m (24)23here the vertical maps are surjective and the inclusions are open. When the ring of invariants ˆ O L ( X ) U isfinitely generated these can be extended by Remark 4.15 to( X m // ǫ C ∗ ) ˆ s ⊆ ( X m // ǫ C ∗ ) ˆ ss ⊆ X m // ǫ C ∗ ← − → X = X m // N/ C ∗ ↓ flips ↓ \ X//U m ↓ X s /U ⊆ X//U (25)(and a similar diagram involving ^ X//U m ), where ( X m // ǫ C ∗ ) ˆ s is the inverse image of X s /U in ( X m // ǫ C ∗ ) ˆ ss .In general for sufficiently divisible m and k we have( X m // ǫ C ∗ ) ˆ s ⊆ ( X m // ǫ C ∗ ) ˆ ss ⊆ X m // ǫ C ∗ ← − → X = X m // N/ C ∗ ↓ flips ↓ \ X//U m ↓ X s /U ⊆ X//U ⊆ X//U m ′ (26)where m ′ = km . Given a linear action of U = ( C + ) r on a projective variety X , we have embedded X in a normal projectivevariety Y m such that the linear action of U on X extends to a linear action of G = SL ( r + 1; C ) ≥ U on Y m and (if m is sufficiently divisible) every U -invariant in a finite fully separating set of U -invariants on X extends to a U -invariant on Y m . We have then constructed the GIT quotients \ X//U m = G × P ( P × U X m ) // ˜ L ( N ) ǫ G, ^ X//U m = G × P ( ^ P × U X m ) // ˜ L ( N ) ǫ G and X m = P × U X m // ˆ L ( N ) SL ( r ; C ) and ˜ X m = ^ P × U X m // ˜ L ( N ) SL ( r ; C )where P × U X m and ^ P × U X m are the closures of P × U X in the projective completions P r × Y m and f P r × Y m of P × U Y m ∼ = P/U × Y m . One difficulty with using this construction in practice is that it isnot easy to tell how divisible m has to be for there to be a finite fully separating set of U -invariants on X extending to U -invariants on Y m . However the construction does have the following nice property, whichenables us to study the families \ X//U m and ^ X//U m by embedding X in other projective varieties Y . Proposition 4.17.
Let Y be a projective variety with a very ample line bundle L and a linear action of U = ( C + ) r on L , and let X be a U -invariant projective subvariety of Y with the inherited linear action of U . Then the inclusion of X in Y induces inclusions of projective varieties \ X//U m ⊆ [ Y //U m and ^ X//U m ⊆ ] Y //U m for all m > , as well as X m ⊆ Y m and ˜ X m ⊆ ˜ Y m when X m and ˜ X m are defined as in Definition 4.13 and Y m and ˜ Y m are defined similarly with Y replacing X . Proof : The construction of \ X//U m and ^ X//U m starts by embedding X into P ( V Xm ) where V Xm = H ( X, L ⊗ m ) ∗ , and then embedding P ( V Xm ) into P ( W Xm ) where W Xm = dim V Xm − M j =0 Θ Xj,m ⊗ ( U V Xm ) j ( V Xm )24nd U V Xm represents the infinitesimal action of u = Lie ( U ) = C r on V Xm . Here Θ Xj,m is the space of complexvalued polynomial functions on { ( u , . . . , u j , h , . . . , h dim V Xm − ) ∈ ( C r ) j × (( C r ) ∗ ⊗ C r ) dim V Xm − } which are homogeneous of degree 1 in the coordinates of each u ∈ C r separately, and homogeneous of totaldegree r (dim V Xm − − ( r − j in the coordinates of all the h k ∈ ( C r ) ∗ ⊗ C r together. The surjection H ( Y, L ⊗ m ) → H ( X, L ⊗ m ) given by restriction gives us an inclusion of V Xm into V Ym and of ( U V Xm ) j ( V Xm )into ( U V Ym ) j ( V Ym ) for all j ≥
0. We then get a mapΘ
Xj,m ⊗ ( U V Xm ) j ( V Xm ) → Θ Yj,m ⊗ ( U V Ym ) j ( V Ym ) α α Y where α Y ( u , . . . , u j , h , . . . h dim V Ym − ) is given bydet( h ) . . . det( h dim V Ym − dim V Xm )times α ( u , . . . , u j , h dim V Ym − dim V Xm +1 , . . . , h dim V Ym − ) . This gives us a commutative diagram of U -equivariant embeddings X → P ( V Xm ) → P ( W Xm ) ↓ ↓ ↓ Y → P ( V Ym ) → P ( W Ym )where the righthand vertical map sends dim V Xm − X j =0 α j dim V Xm − X j =0 α Yj and is equivariant with respect to the linear P -actions on W Xk and W Yk . We thus get a commutative diagramof embeddings X → P ( V Xm ) → P ( W Xm ) → G × P P ( W Xm ) ↓ ↓ ↓ ↓ Y → P ( V Ym ) → P ( W Ym ) → G × P P ( W Ym ) . ^ X//U m is the GIT quotient ^ G × U X m // ˜ L ( N ) ǫ G where ^ G × U X m is the closure of G ( { ι } × X ) ∼ = G × U X in G × P ( f P r × ( G × P P ( W Xm )), and ] Y //U m is constructed in the same way. Since the line bundle ˜ L ( N ) ǫ is amplefor ǫ >
0, this gives us an inclusion of ^ X//U m into ] Y //U m , and the other inclusions follow similarly.Suppose now that U ∼ = ( C + ) r is a normal subgroup of an algebraic group H acting linearly on X withrespect to the line bundle L . This linear action induces an action of H on V m = H ( X, L ⊗ m ) ∗ for each m >
0, and thus H acts by conjugation on GL ( V m ) and its Lie algebra. H also acts by conjugation on U and C r = Lie U as U is a normal subgroup of H , and the Lie algebra homomorphism φ V m : Lie U → Lie( GL ( V m ))defined at (17) is H -equivariant with respect to these actions. Hence the action of H on V m preserves thesubspaces ( U V m ) j ( V m ) of V m , and from this action and the action of H on C r = Lie U we get induced actionsof H on W m . If h ∈ H and p = gu ∈ P = GL ( r ; C ) ⋉ U with g ∈ GL ( r ; C ) and u ∈ U , then the actions of H and P are related by h ( pw ) = h ( guw ) = (( ψ ( h ) gψ ( h ) − )( huh − ))( hw ) = Ψ h ( p )( hw )for any w in W m , where ψ : H → GL ( r ; C ) is the group homomorphism defining the action of H onLie U = C r by conjugation andΨ h ( p ) = ( ψ ( h ) gψ ( h ) − )( huh − ) ∈ GL ( r ; C ) ⋉ U = P. P ⋊ H on W m . (Note that the action of U as a subgroup of P defined at (20) is different from the action of U as a subgroup of H defined above.) In fact the subgroup P ⋊ U of P ⋊ H is a direct product P × U , since U ∼ = ( C + ) r acts trivially on itself by conjugation, so if p ∈ P and h ∈ U ≤ H then ψ ( h ) is the identity element of GL ( r ; C ) and Ψ h ( p ) = p . H also acts on P r and on f P r via the homomorphism ψ : H → GL ( r ; C ), giving us an action of P ⋊ H on f P r × P ( W m ). Since X is H -invariant it follows that the closure ^ P × U X m of P × U X = P ( { ι } × X ) in f P r × Y m is also H -invariant. Since H normalises SL ( r ; C ) and commutes with the central C ∗ subgroup of GL ( r ; C ), we get an induced linear action of H × C ∗ on˜ X m = ^ P × U X m // L ( N ) N/ SL ( r ; C ) , preserving the open subset ( ˜ X m // ǫ C ∗ ) ˆ ss , and of H/U on ^ X//U m = ^ P × U X m // L ( N ) ǫ P. Remark 4.18.
Suppose that ( C + ) r = U E H and H acts linearly on X as above. Suppose also that H contains a one-parameter subgroup λ : C ∗ → H whose weights for the induced (conjugation) action on u = C r are all strictly positive. Then the subgroup ˆ U of H generated by λ ( C ∗ ) and U is a semidirectproduct ˆ U ∼ = U ⋊ C ∗ . Moreover this C ∗ acts on C r ( r +1) ⊆ P r ( r +1) with all weights strictly positive, and we have GL ( r ; C ) = ψ ( λ ( C ∗ )) SL ( r ; C ) with ψ ( λ ( C ∗ )) ∩ SL ( r ; C ) finite. Recall from Remark 4.8 that if P/U = f P r then P/U ss,SL ( r ; C ) = P/U s,SL ( r ; C ) and P/U //SL ( r ; C ) ∼ = P with the induced action of λ ( C ∗ ) on P a positive power of the standard action on C ∗ on P . Thus usingvariation of GIT (as in Remark 2.7) there are rational numbers δ − < δ + such that the induced action of λ ( C ∗ ) on P = P/U //SL ( r ; C ) twisted by δ times the standard character of C ∗ satisfies( P ) ss,δ = ( P ) s,δ = (cid:26) C ∗ if δ ∈ ( δ − , δ + ) ∅ if δ [ δ − , δ + ]and hence P/U ss,GL ( r ; C ) ,δ = P/U s,GL ( r ; C ) ,δ = (cid:26) GL ( r ; C ) = P/U if δ ∈ ( δ − , δ + ) ∅ if δ [ δ − , δ + ] . It follows that if the linearisation of the H -action is twisted by δ times the standard character of λ ( C ∗ ) for δ = δ − , δ + (which is possible to arrange if, for example, λ ( C ∗ ) centralises H/U ), then (for sufficiently large N ) all the points of ˜ X m = ^ P × U X m // L ( N ) SL ( r ; C ) which are semistable for the induced action of λ ( C ∗ ) arecontained in the image of P × U X ∼ = GL ( r ; C ) × X . Hence the inverse image of ( ^ X//U m ) ss,λ ( C ∗ ) under thesurjection ( ˜ X m // ǫ C ∗ ) ˆ ss → ^ X//U m in Theorem 4.16 is an open subset of ˜ X m // ǫ C ∗ which is unaffected by the flips ˜ X m // ǫ C ∗ ← − → ˜ X m // N/ C ∗ = X and thus can be identified canonically with an open subset X ss, ˆ U of X . Similarly the inverse image of( ^ X//U m ) s,λ ( C ∗ ) under the restriction of this surjection to ( X m // ǫ C ∗ ) ˆ s is an open subset X s, ˆ U of X ss, ˆ U which,like X ss, ˆ U , is independent of m when m is sufficiently divisible. Indeed it turns out that x ∈ X ss, ˆ U (respec-tively x ∈ X s, ˆ U ) if and only if x is semistable (respectively stable) for every conjugate of λ : C ∗ → ˆ U in ˆ U ,or equivalently for every one-parameter subgroup ˆ λ : C ∗ → ˆ U of ˆ U . If the linear U -action on X extends to G then the same is true when ^ X//U m is replaced with ^ X//U defined as in § H/U is reductive and is centralised by λ ( C ∗ ), then ifthe linearisation of the H -action is twisted by a suitable character of H/U the induced GIT quotients ^ X//U m // ( H/U )26or m sufficiently divisible are independent of m (and are isomorphic to ^ X//U // ( H/U ) when the linear U -action on X extends to G ). In fact the proof of [DK] Theorem 5.3.18 shows that in this situation, withthe linearisation of the H -action suitably twisted, the ring of invariants ˆ O L ( X ) H is finitely generated, withassociated projective variety X//H = Proj( ˆ O L ( X ) H ) , and we have X//H ∼ = ^ X//U m // ( H/U ) . Indeed, it turns out that in this situation, with the linearisation of the H -action suitably twisted, we haveessentially the same situation as for classical GIT for reductive group actions: there is a diagram X s,H /H ⊆ X ss,H ⊆ X ↓ ↓ X s,H /H ⊆ X//H (27)where the vertical maps are surjective and the inclusions are open, and in addition the Hilbert-Mumford cri-teria for stability and semistability hold as in the reductive case (Proposition 2.3 above), and two semistableorbits in X represent the same point of X//H if and only if their closures meet in X ss,H . We will see an ex-ample of this phenomenon for hypersurfaces in P (1 , ,
2) in § § ^^ X//U m // ( H/U )of the GIT quotient of ^ X//U m by the action of the reductive group H/U is independent of m and providesa partial desingularisation ^ X//H of X//H . Remark 4.19.
In practice it is not difficult to find the range of characters of λ ( C ∗ ) with which the lineari-sation can be twisted in order to achieve the nice situation described in Remark 4.18. The picture describedin Remark 4.18 is valid for all δ ∈ Q \ { δ − , δ + } , and if δ [ δ − , δ + ] then X ss, ˆ U,δ = ∅ and hence X// δ H = ∅ .If we attempt to use the Hilbert-Mumford criteria to calculate X ss, ˆ U,δ for all δ ∈ Q , we will find finitelymany rational numbers a < a < . . . < a q such that, when calculated according to the Hilbert-Mumfordcriteria, X ss, ˆ U,δ is empty for δ < a and for δ > a q , and is nonempty but constant for δ ∈ ( a j − , a j ) when j = 1 , . . . , q . Then we must have δ − ≤ a and δ + ≥ a q , and moreover X ss, ˆ U,δ and X ss,H,δ are as predicted bythe Hilbert-Mumford criteria for any δ = a , a q . Thus X// δ H = ∅ if δ [ a , a q ] and the situation describedin Remark 4.18 holds for every δ ∈ ( a , a q ). P (1 , , Recall from § d in the weighted projectiveplane P (1 , ,
2) is essentially equivalent to constructing a quotient for the action of H = ( C + ) ⋊ GL (2; C )on (an open subset of) the projective space X d of weighted degree d polynomials in the three weightedhomogeneous coordinates x, y, z on P (1 , , H is the automorphism group of P (1 , , α, β, γ ) ∈ U = ( C + ) acts on P (1 , ,
2) via[ x : y : z ] [ x : y : z + αx + βxy + γy ]and g ∈ GL (2; C ) acts in the standard fashion on ( x, y ) ∈ C and as scalar multiplication by (det g ) − on z .Thus g ∈ GL (2; C ) acts by conjugation on U as the standard action of GL (2; C ) on Sym ( C ) ∼ = C twistedby the character det. Remark 5.1.
Notice that the central one-parameter subgroup λ : C ∗ → GL (2; C ) of GL (2; C ) satisfiesthe conditions of Remark 4.18 above: the weights of its action (by conjugation) on u = C are all strictlypositive, as they are all equal to 4. 27e wish to study the action of H on the projective space X d = P ( C ( d ) [ x, y, z ])where C ( d ) [ x, y, z ] is the linear subspace of the polynomial ring C [ x, y, z ] consisting of polynomials p of theform p ( x, y, z ) = X i, j, k ≥ i + j + 2 k = d a ijk x i y j z k for some a ijk ∈ C [Co, CK]. Here h ∈ H acts as p h · p with h · p ( x, y, z ) = p ( h − x, h − y, h − z ). Thisrepresentation H → GL ( C d [ x, y, z ]) gives us a linearisation of the action of H on X d , which we can twist byany multiple ǫ ∈ Q of the character det of GL (2; C ) to get a fractional linearisation L ǫ . Remark 5.2. If m ≥ H ( X d , O X d ( m )) ∗ ∼ = C ( md ) [ x, y, z ] and the natural embedding of X d in theprojective space P ( H ( X d , O X d ( m )) ∗ ) is given by p ( x, y, z ) ( p ( x, y, z )) m . U = ( C + ) First let us consider the action of the unipotent radical U = ( C + ) of H on X d . Consider Y d = P ( C ⌈ d/ ⌉ [ X, Y, W, z ])where ⌈ d/ ⌉ denotes the least integer n ≥ d/
2, and C ⌈ d/ ⌉ [ X, Y, W, z ] is the space of homogeneous polynomialsof degree ⌈ d/ ⌉ in X, Y, W, z . By multiplying by x if d is odd, we can identify C ( d ) [ x, y, z ] with the set ofpolynomials of the form p ( x, y, z ) = X i ≥ ⌈ d/ ⌉ − d, j, k ≥ i + j + 2 k = 2 ⌈ d/ ⌉ a ijk x i y j z k . Then we can embed X d in Y d via p ˆ p where ˆ p ( X, Y, W, z ) equals the sum over i ≥ ⌈ d/ ⌉ − d and j, k ≥ i + j + 2 k = 2 ⌈ d/ ⌉ of a ijk X ( i − M ij ) / −⌈ ( m ij − M ij ) / ⌉ W M ij +2 ⌈ ( m ij − M ij ) / ⌉ Y ( j − M ij ) / −⌈ ( m ij − M ij ) / ⌉ z k for m ij = min { i, j } and M ij = max { i, j } . Thus ˆ p ( x , y , xy, z ) = p ( x, y, z ) if d is even and ˆ p ( x , y , xy, z ) = xp ( x, y, z ) if d is odd. For simplicity we will assume from now on that d is even; by Remark 5.2 this involvesvery little loss of generality.The action of U on X d extends to an action on Y d such that ( α, β, γ ) ∈ U acts via p ( X, Y, W, z ) p ( X, Y, W, z + αX + βW + γY ) . This extends to the standard action of G = SL (4; C ) on C d/ [ X, Y, W, z ]. Thus Y d //U = ( P × Y d ) //G and ^ Y d //U = ( G × P ( P × Y d )) //G where P = P ( C ⊕ (( C ) ∗ ⊗ C )) and P = P ( C ⊕ (( C ) ∗ ⊗ C )). Here the linearisation on P × Y d is O P ( N ) ⊗ O Y d (1) for N >> P × Y d is O P ( N ) ⊗ O Y d (1).The weights of the action of the standard maximal torus T c of G = SL (4; C ) on P = P ( C ⊕ (( C ) ∗ ⊗ C )) with respect to O P (1) are 0 (with multiplicity 1) and χ , χ , χ , χ (each with multiplicity 3) where χ , χ , χ , χ = − χ − χ − χ are the weights of the standard representation of SL (4; C ) on C . The weightsof the action of T c on Y d = P ( C d/ [ X, Y, W, z ]) with respect to O Y d (1) are { } ∪ { iχ + jχ + kχ + ℓχ : i, j, k, ℓ ≥ i + j + k + ℓ = d/ } .
28 point a = [ a : a : a : a : a : a : a : a : a : a : a : a : a ] ∈ P is semistable for thisaction of SL (4; C ) if and only if a = 0. Therefore if N >> a = 0 whenever ( a, y ) ∈ ( P × Y d ) ss,G for any y ∈ Y d . Moreover if a = [1 : a ij ] ∈ P and y ∈ Y d is represented by p ( X, Y, W, z ) = X i, j, k, ℓ ≥ i + j + k + ℓ = d/ b ijkℓ X i Y j W k z ℓ ∈ C d/ [ X, Y, W, z ]then by the Hilbert-Mumford criteria ( a, y ) ∈ ( P × Y d ) ss,G if and only if ( ga, gy ) ∈ ( P × Y d ) ss,T c for every g ∈ G , and ( a, y ) ∈ ( P × Y d ) ss,T c if and only if 0 lies in the convex hull of the set of weights { iχ + jχ + kχ + ℓχ : b ijkℓ = 0 } ∪ S ∪ S ∪ S ∪ S where S = (cid:26) { N χ + iχ + jχ + kχ + ℓχ : b ijkℓ = 0 } if ( a , a , a ) = 0 ∅ if ( a , a , a ) = 0and S , S , S are defined similarly. Let us write( P × Y d ) ss,G = ( P × Y d ) ss,G ⊔ ( P × Y d ) ss,G ⊔ ( P × Y d ) ss,G ⊔ ( P × Y d ) ss,G (28)where ( P × Y d ) ss,Gq = { ( a, y ) ∈ ( P × Y d ) ss,G : rank(( a ij )) = q } . Then( P × Y d ) ss,G = { [1 : 0 : . . . : 0] } × Y ss,Gd and ( P × Y d ) ss,G = G × U ( { [1 : ι ] } × Y ss, d )where ι = ,U is its stabiliser { ( g ij ∈ G : g = 1 and g = g = g = 0 } in G = SL (4; C ) and Y ss, d = { y ∈ Y d : uy ∈ Y ss,T c d for all u ∈ U } . Similarly ( P × Y d ) ss,G = G × U ( { [1 : ι ] } × Y ss, d )where ι = ,U is its stabiliser { ( g ij ∈ G : g = g = 1 and g = g = g = g = g = g = 0 } in G and Y ss, d = { y ∈ Y d : uy ∈ Y ss,T c d for all u ∈ U } for T c = { ( g ij ) ∈ T c : g = g } , while( P × Y d ) ss,G = G × U ( { [1 : ι ] } × Y ss, d )where Y ss, d = { y ∈ Y d : uy ∈ Y ss,T c d for all u ∈ U } for T c = { ( g ij ) ∈ T c : g = g = g } . 29 .2 The action of ˆ U = C ⋊ C ∗ Now let us consider the action of the subgroup ˆ U = C ∗ ⋉ U of H on X d , where C ∗ is the centre of GL (2; C ) andacts by conjugation on Lie ( U ) = C with weights all equal to 4. The action of t ∈ C ∗ on p ( x, y, z ) ∈ C d [ x, y, z ]is given by tp ( x, y, z ) = p ( tx, ty, t − z ) . This action extends to the action on C d/ [ X, Y, W, z ] given by tP ( X, Y, W, z ) = P ( t X, t Y, t W, t − z )and thus the action of ˆ U on X d extends to a linear action on Y d , which is the restriction of the GL (4; C )-actionon Y d via the embedding of ˆ U in GL (4; C ) such that t t t t
00 0 0 t − for t ∈ C ∗ . (29)If we twist this action by 2 δ times the standard character of C ∗ then we get a fractional linearisation of theaction of ˆ U on Y d which extends the fractional linearisation L δ on X d . We also get an action of C ∗ = ˆ U /U on P = G/U via t [ a : a ij ] = [ a : t a ij ] . Note that, since ( t , t , t , t ) = ( t τ , t τ , t τ , t − ( τ τ τ ) − ) if and only if t = t t t t and τ = t − t , τ = t − t , τ = t − t , C ∗ T c ∼ = ( C ∗ × T c ) / ( Z / Z ) is the maximal torus of GL (4; C ), acting on P with weights 0, 2 χ + χ + χ + χ , χ + 2 χ + χ + χ , χ + χ + 2 χ + χ and χ + χ + χ + 2 χ , and acting on Y d with fractional weights { iχ + jχ + kχ + ℓχ + ǫ χ + χ + χ − χ ) : i, j, k, ℓ ≥ i + j + k + ℓ = d } where χ , χ , χ , χ are now the weights of the standard representation of GL (4; C ) on C . Let us break up( P × Y d ) ss,GL (4; C ) ,δ as at (28) as( P × Y d ) ss,GL (4; C ) ,δ ⊔ ( P × Y d ) ss,GL (4; C ) ,δ ⊔ ( P × Y d ) ss,GL (4; C ) ,δ ⊔ ( P × Y d ) ss,GL (4; C ) ,δ (30)where ( P × Y d ) ss,GL (4; C ) ,δq equals { ( a, y ) ∈ ( P × Y d ) ss,GL (4; C ) ,δ : rank(( a ij )) = q } . We find by considering the central C ∗ in GL (4; C ) that( P × Y d ) ss,GL (4; C ) ,δ = ( P × Y d ) ss,GL (4; C ) ,δ = ( P × Y d ) ss,GL (4; C ) ,δ = ∅ unless δ = − d/
2, while( P × Y d ) ss,GL (4; C ) ,δ = ( P × Y d ) ss,GL (4; C ) ,δ ∼ = GL (4; C ) × ˆ U Y ss, ˆ U,δd where Y ss, ˆ U,δd = { y ∈ Y d : uy ∈ Y ss, C ∗ ,δd for all u ∈ U } . Similarly if δ = − d/ G × P ( f P × Y d )) ss,GL (4; C ) ,δ = ( G × P ( f P × Y d )) ss,GL (4; C ) ,δ = GL (4; C ) × ˆ U Y ss, ˆ U,δd and if m is sufficiently divisible( G × P ( ^ P × U Y dm )) ss,GL (4; C ) ,δ ∼ = GL (4; C ) × ˆ U Y ss, ˆ U,δd , while ( P × Y d ) s,GL (4; C ) ,δ ∼ = ( G × P ( f P × Y d )) s,GL (4; C ) ,δ ∼ = ( G × P ( ^ P × U Y dm )) s,GL (4; C ) ,δ ∼ = GL (4; C ) × ˆ U Y s, ˆ U,δd where Y s, ˆ U,δd = { y ∈ Y d : uy ∈ Y s, C ∗ ,δd for all u ∈ U } . Thus if δ = − d/
2, for sufficiently divisible m we have ^ Y d //U m // δ C ∗ = Y ss, ˆ U,δd / ∼ ˆ U and ( ^ Y d //U m ) s, C ∗ ,δ / C ∗ = Y s, ˆ U,δd / ˆ U , and so by Proposition 4.14 for sufficiently divisible m we have ^ X d //U m // δ C ∗ = X ss, ˆ U,δd / ∼ ˆ U and ( ^ X d //U m ) s, C ∗ ,δ / C ∗ = X s, ˆ U,δd / ˆ U where X ss, ˆ U,δd = { y ∈ X d : uy ∈ X ss, C ∗ ,δd for all u ∈ U } and X s, ˆ U,δd = { y ∈ X d : uy ∈ X s, C ∗ ,δd for all u ∈ U } and x ∼ ˆ U y if and only if ˆ U x ∩ ˆ U y ∩ X ss, ˆ U,δd = ∅ . Using this, the proof of [DK] Theorem 5.3.18 shows thatin fact, for the linearisation L δ when δ = − d/
2, the ring of invariants ˆ OL δ ( X d ) ˆ U is finitely generated, and X d // δ ˆ U = Proj( ˆ OL δ ( X d ) ˆ U ) = ^ X d //U m // δ C ∗ for sufficiently divisible m . Thus for all δ = − d/ X d // δ ˆ U = X ss, ˆ U,δd / ∼ ˆ U is a projective completion of X s, ˆ U,δd / ˆ U (cf. Remark 4.18). H Let T c ( GL (2; C )) be the standard maximal torus of GL (2; C ) = H/U . It now follows immediately that when δ = − d/
2, the ring of invariants ˆ OL δ ( X d ) H = ( ˆ OL δ ( X d ) ˆ U ) SL (2; C ) is finitely generated, and X d // δ H = Proj( ˆ OL δ ( X d ) H ) = X ss,H,δd / ∼ H (31)is a projective completion of X s,H,δd /H , where X ss,H,δd = { y ∈ X d : uy ∈ X ss,GL (2; C ) ,δd for all u ∈ U } = { y ∈ X d : hy ∈ X ss,T c ( GL (2; C )) ,δd for all h ∈ H } X s,H,δd = { y ∈ X d : uy ∈ X s,GL (2; C ) ,δd for all u ∈ U } = { y ∈ X d : hy ∈ X s,T c ( GL (2; C )) ,δd for all h ∈ H } and x ∼ H y if and only if Hx ∩ Hy ∩ X ss,H,δd = ∅ .The weights of the action on X d of T c ( GL (2; C )) = (cid:26)(cid:18) t t (cid:19) : t , t ∈ C ∗ (cid:27) with respect to the linearisation L δ are given by( t , t ) t i − k + δ t j − k + δ for integers i, j, k ≥ i + j + 2 k = d .Thus p ( x, y, z ) = P i + j +2 k = d a ijk x i y j z k ∈ C ( d ) [ x, y, z ] represents a point of X ss,T c ( GL (2; C ) ,δd (respectively apoint of X s,T c ( GL (2; C ) ,δd ) if and only if 0 lies in the convex hull (respectively 0 lies in the interior of the convexhull) in R of the subset { ( i − k + δ, j − k + δ ) : i, j, k ≥ , i + j + 2 k = d and a ijk = 0 } of the set of weights { ( i − k + δ, j − k + δ ) : i, j, k ≥ , i + j + 2 k = d } whose convex hull is the triangle in R with vertices ( d + δ, δ ), ( δ, d + δ ) and ( δ − d/ , δ − d/ δ = − d/ d + δ, δ ) and ( δ, d + δ ), and that we have X ss,H,δd = ∅ if δ ( − d/ , d/
2) (cf. Remark 4.19).Combining this with (31) gives an explicit description of X d // δ H and X s,H,δd /H whenever δ = − d/ Remark 5.3.
For small d this description can be expressed in terms of singularities of hypersurfaces (cf. thedescription in [MFK] Chapter 4 § P ). Classical GIT quotients in complex algebraic geometry are closely related to the process of reduction insymplectic geometry. Suppose that a compact, connected Lie group K with Lie algebra k acts smoothly ona symplectic manifold X and preserves the symplectic form ω . A moment map for the action of K on X is asmooth map µ : X → k ∗ which is equivariant with respect to the given action of K on X and the coadjointaction of K on k ∗ , and satisfies dµ ( x )( ξ ) .a = ω x ( ξ, a x )for all x ∈ X , ξ ∈ T x X and a ∈ k , where x a x is the vector field on X defined by the infinitesimal actionof a ∈ k . The quotient µ − (0) /K then inherits a symplectic structure and is the symplectic reduction at 0,or symplectic quotient, of X by the action of K .Now let X be a nonsingular complex projective variety embedded in complex projective space P n , and let G be a complex reductive group acting on X via a complex linear representation ρ : G → GL ( n + 1; C ). If K is a maximal compact subgroup of G , we can choose coordinates on P n so that the action of K preserves theFubini-Study form ω on P n , which restricts to a symplectic form on X . There is a moment map µ : X → k ∗ defined by µ ( x ) .a = ˆ x t ρ ∗ ( a )ˆ x πi || ˆ x || (32)for all a ∈ k , where µ ( x ) .a denotes the natural pairing between µ ( x ) ∈ k ∗ and a ∈ k , while ˆ x ∈ C n +1 − { } is a representative vector for x ∈ P n and the representation ρ : K → U ( n + 1) induces ρ ∗ : k → u ( n + 1) anddually ρ ∗ : u ( n + 1) ∗ → k ∗ . Note that we can think of µ as a map µ : X → g ∗ defined by µ ( x ) .a = re ˆ x t ρ ∗ ( a )ˆ x πi || ˆ x || ! a ∈ g = k ⊗ R C ; then µ satisfies µ ( x ) .a = 0 for all a ∈ i k .In this situation the GIT quotient X//G can be canonically identified with the symplectic quotient µ − (0) /K . More precisely [Ki], any x ∈ X is semistable if and only if the closure of its G -orbit meets µ − (0), while x is stable if and only if its G -orbit meets µ − (0) reg = { x ∈ µ − (0) | dµ ( x ) : T x X → k ∗ is surjective } , and the inclusions of µ − (0) into X ss and of µ − (0) reg into X s induce homeomorphisms µ − (0) /K → X//G and µ − (0) reg → X s /G. Thus the moment map picks out a unique K -orbit in each stable G -orbit, and also in each equivalence classof strictly semistable G -orbits, where x and y in X ss are equivalent if the closures of their G -orbits meet in X ss ; that is, if their images under the natural surjection q : X ss → X//G agree.
Remark 5.4.
It follows from the formula (32) that if we change the linearisation of the G -action of X bymultiplying by a character χ : G → C ∗ of G , then the moment map is modified by the addition of a centralconstant c χ in k ∗ , which we can identify with the restriction to k of the derivative of χ .When a non-reductive affine algebraic group H with unipotent radical U acts linearly on a projectivevariety X there are ‘moment-map-like’ descriptions of suitable projectivised quotients ^ X//U = ^ G × U X//G and the resulting quotients ( ^ X//U ) // ( H/U ), which are analogous to the description of a reductive GIT quo-tient
Y //G as a symplectic quotient µ − (0) /K , and can be obtained from the symplectic quotient descriptionof the reductive GIT quotient ^ G × U X//G (see [Ki5, Ki6] for more details). This is very closely related tothe ‘symplectic implosion’ construction of Guillemin, Jeffrey and Sjamaar [GJS].The case of the automorphism group H of P (1 , ,
2) acting on X d = P ( C ( d ) [ x, y, z ]) as above withrespect to the linearisation L ǫ for any ǫ = − d/ X d // ǫ H =( X d // ǫ ˆ U ) //SL (2; C ) where X d // ǫ ˆ U is the image of X ss, ˆ U,ǫd in Y d // ǫ ˆ U = ( P × Y d ) // ǫ GL (4; C ). There is amoment map µ U (4) : P × Y d → Lie U (4) ∗ for the action of the maximal compact subgroup U (4) of GL (4; C ) on P × Y d associated to the linearisation O P ( N ) × O Y d (1) for N >>
0, given by µ U (4) ( a, y ) = N µ P U (4) ( a ) + µ Y d U (4) ( y ) (33)for ( a, y ) ∈ P × Y d . Here µ P U (4) : P → Lie U (4) ∗ and µ Y d U (4) : Y d → Lie U (4) ∗ are the moment mapsgiven by formula (32) for the actions of U (4) on P and on Y d = P ( C d/ [ X, Y, W, z ]). We can identify Y d // ǫ ˆ U = ( P × Y d ) // ǫ GL (4; C ) with µ − U (4) ( − ǫ ) /U (4) = ( µ − SU (4) (0) ∩ µ − S ( − ǫ )) /S SU (4), where S is themaximal compact subgroup of the subgroup C ∗ of GL (4; C ) given at (29). We can use the standard invariantinner product on the Lie algebra of U (4) to identify Lie U (4) with Lie U (4) ∗ and with Lie( U (3) × U (1)) ⊕ (Lie( U (3) × U (1))) ⊥ , and thus withLie S ⊕ Lie S ( U (3) × U (1)) ⊕ (Lie( U (3) × U (1))) ⊥ where (Lie( U (3) × U (1))) ⊥ is the orthogonal complement to Lie( U (3) × U (1)) in Lie U (4), and S ( U (3) × U (1)) =( U (3) × U (1)) ∩ SU (4) so that U (3) × U (1) = S S ( U (3) × U (1)). With respect to this decomposition we can write µ U (4) = µ S ⊕ µ S ( U (3) × U (1)) ⊕ µ ⊥ where µ ⊥ is the orthogonal projection of µ U (4) onto (Lie ( U (3) × U (1))) ⊥ .We find that if ǫ = − d/ N is sufficiently large then Y d // ǫ ˆ U ∼ = µ − U (4) ( − ǫ ) /U (4) ∼ = ( µ − S ( − ǫ ) ∩ µ − ⊥ (0)) /S , and restricting to X d we get an identification X d // ǫ ˆ U ∼ = µ − U ( − ǫ ) /S µ ˆ U : X d → Lie( ˆ U ) ∗ ∼ = (Lie S ⊗ R C ) ∗ ⊕ (Lie( U (3) × U (1))) ⊥ is a ‘moment map’ for the action of ˆ U on X d (which takes into account the K¨ahler structure on X d , not just its symplectic structure), defined by µ ˆ U ( x ) .a = re ˆ x t ρ ∗ ( a )ˆ x πi || ˆ x || ! for a ∈ Lie( ˆ U ). Moreover if ǫ = − d/ X d // ǫ H = ( X d // ǫ ˆ U ) //SL (2; C ) can be identified with µ − U ( − ǫ ) ∩ µ − SU (2) (0) S SU (2) = µ − H ( − ǫ ) U (2)where the ‘moment map’ µ H : X d → Lie( H ) ∗ is defined by µ ( x ) .a = re ˆ x t ρ ∗ ( a )ˆ x πi || ˆ x || ! for a ∈ Lie( H ), and U (2) is a maximal compact subgroup of H . References [AD] A. Asok and B. Doran,
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