Fundamental group of a geometric invariant theoretic quotient
aa r X i v : . [ m a t h . AG ] O c t FUNDAMENTAL GROUP OF A GEOMETRIC INVARIANTTHEORETIC QUOTIENT
INDRANIL BISWAS, AMIT HOGADI, AND A. J. PARAMESWARAN
Abstract.
Let M be an irreducible smooth projective variety, defined over an alge-braically closed field, equipped with an action of a connected reductive affine algebraicgroup G , and let L be a G –equivariant very ample line bundle on M . Assume that theGIT quotient M//G is a nonempty set. We prove that the homomorphism of algebraicfundamental groups π ( M ) −→ π ( M//G ), induced by the rational map M M//G ,is an isomorphism.If k = C , then we show that the above rational map M M//G induces anisomorphism between the topological fundamental groups. Introduction
Let M be an irreducible smooth projective variety defined over an algebraically closedfield k of arbitrary characteristic. Fix a very ample line bundle L on M . Let G be aconnected reductive affine algebraic group over k acting algebraically on both M and L such that L is a G –equivariant line bundle. The geometric invariant theoretic quotient M//G for this action of G on ( M , L ) will be denoted by X . We assume that X isnonempty. The quotienting produces a rational morphism M X . This rationalmorphism produces a homomorphism h : π ( M ) −→ π ( X )between the algebraic fundamental groups.When k = C , the above rational morphism M X produces a homomorphism h ′ : π t ( M ) −→ π t ( X )between the topological fundamental groups.We prove the following (see Theorem 3.3 and Theorem 4.4): Theorem 1.1.
Let
G/k be a connected reductive algebraic affine group acting on a smoothconnected projective variety
M/k . (1) The above homomorphism h between the algebraic fundamental groups is an iso-morphism. (2) If k = C , the homomorphism h ′ between the topological fundamental groups is anisomorphism. Mathematics Subject Classification.
Key words and phrases.
Group action, GIT, fundamental group, reductive group.
Hui Li proved that for an Hamiltonian action of a compact group on a compact symplec-tic manifold M , the fundamental group of the symplectic quotient coincides with π ( M )[Li, p. 346, Theorems 1.2, 1.3]. A theorem of Kirwan and Kempf–Ness says that for alinear action of a complex reductive group on a smooth complex projective variety, if thestabilizer of every semistable point is finite, then the GIT quotient is homeomorphic tothe symplectic quotient [Ki, Theorem 7.5, Remark 8.14], [KN].2. Lifting an action
Let k be an algebraically closed field. Let G be a connected reductive affine algebraicgroup over k and M/k an irreducible smooth projective variety equipped with an algebraicaction θ : M × G −→ M of G . Continuing with the above notation we have the following proposition. Proposition 2.1.
Let ϕ : M ′ −→ M be an ´etale Galois covering morphism such that M ′ is connected. Then there is a unique action of G on M ′ θ ′ : M ′ × G −→ M ′ that lifts θ , meaning θ ◦ ( ϕ × Id G ) = ϕ ◦ θ ′ .Proof. Since M ′ × G is connected, for any two lifts θ , θ : M ′ × G −→ M ′ of themorphism θ , there is a deck transformation γ ∈ Gal( ϕ ) such that θ = γ ◦ θ . On theother hand, for an action θ ′ of G on M ′ , we have θ ′ ( z , e ) = z for all z ∈ M ′ , where e ∈ G denotes the identity element. Therefore, there can be at most one action θ ′ of G satisfying the condition that it lifts θ .We will now prove the existence of a lifted action. For any closed point x ∈ M , let ϕ x : G −→ M be the morphism defined by g θ ( x , g ). Consider the induced homomorphism ofalgebraic fundamental groups(2.1) ϕ x ∗ : π ( G, e ) −→ π ( M, x ) . We will show that(2.2) ϕ x ∗ = 0 . To prove (2.2), first take x to be such that its orbit θ ( x , , G ) ⊂ M is of minimaldimension among all the G –orbits in M . Since the boundary of θ ( x , G ) θ ( x , G ) \ θ ( x , G ) ⊂ M is preserved by the action of G , the condition that the dimension of θ ( x , G ) is the minimumone implies that this boundary is empty. In other words, the orbit θ ( x , G ) is a complete UNDAMENTAL GROUP OF A GIT QUOTIENT 3 subvariety of M . Let G x ⊂ G be the isotropy group-scheme for the point x (we note that G x need not be reduced). The reduced group(2.3) P := G x, red ⊂ G x ⊂ G is a parabolic subgroup of G because θ ( x , G ) is complete. Consider the natural projection(2.4) ξ : P \ G −→ G x \ G = θ ( x , G )given by the first inclusion in (2.3). Any solvable algebraic group defined over k is isomor-phic as a variety (not as an algebraic group) to a product of copies of A k and G m (recallthat k is algebraically closed). Therefore, from the Bruhat decomposition of G it followsthat the variety G is rational. Combining this with the fact that the quotient morphism G −→ P \ G is separable we conclude that P \ G is separably rationally connected. Thisimplies that the variety P \ G is simply connected [Ko, p. 75, Theorem 13].Let ν : e C −→ θ ( x , G )be a finite ´etale Galois covering. Let(2.5) ( P \ G ) × θ ( x ,G ) e C δ −→ e C y µ y P \ G ξ −→ θ ( x , G )be the pullback of the covering ν to P \ G by the morphism ξ in (2.4). Since P \ G is simplyconnected, the covering µ in (2.5) admits a section η : P \ G −→ ( P \ G ) × θ ( x ,G ) e C .
Now consider the composition δ ◦ η : P \ G −→ e C , where δ is the projection in (2.5). The image of δ ◦ η is a connected component of e C thatprojects isomorphically to θ ( x , G ). Indeed, this follows immediately from the fact that ξ in (2.4) is bijective on closed points. Hence we conclude that π ( θ ( x , G ) , x ) = 0 . Therefore, (2.2) holds for this point x .For another closed point y of M , let ϕ y : G −→ M be the morphism defined by g θ ( y , g ). Let ψ x : G −→ M × G and ψ y : G −→ M × G be the embeddings defined by g ( x , g ) and g ( y , g ) respectively. So, we have ϕ x = θ ◦ ψ x and ϕ y = θ ◦ ψ y . Therefore, ϕ x ∗ = θ ∗ ◦ ψ x ∗ and ϕ y ∗ = θ ∗ ◦ ψ y ∗ , where ϕ x ∗ isthe homomorphism in (2.1). Since M is projective, π ( M × G, ( y , e )) = π ( M, y ) × π ( G, e )[SGA1, Expos´e X, §
1, Corollaire 5.1]. The two groups π ( M × G, ( y , e )) and π ( M × G, ( x , e )) are identified up to conjugation, and such an identification takes the image I. BISWAS, A. HOGADI, AND A.J. PARAMESWARAN ψ x ∗ ( π ( G, e )) to ψ y ∗ ( π ( G, e )). These imply that (2.2) holds for the point y because itholds for the point x . Therefore, (2.2) holds for all x .Take M ′ in the statement of the proposition. Since M ′ is projective, we have π ( M ′ × G, ( x ′ , e )) = π ( M ′ , x ′ ) × π ( G, e )[SGA1, Expos´e X, §
1, Corollaire 5.1]. Consider the diagram M ′ × G M ′ y ϕ × Id G y ϕM × G θ −→ M Take any closed point x ′ ∈ ϕ − ( x ). Now consider the induced homomorphism of algebraicfundamental groups( θ ◦ ( ϕ × Id G )) ∗ : π ( M ′ × G, ( x ′ , e )) = π ( M ′ , x ′ ) × π ( G, e ) −→ π ( M, x ) . From (2.2) it follows immediately that we have( θ ◦ ( ϕ × Id G )) ∗ ( { e } × π ( G, e )) = 0 , where e ∈ π ( M ′ , x ′ ) denotes the identity element. Consequently, the image of ( θ ◦ ( ϕ × Id G )) ∗ coincides with the image of the homomorphism ϕ ∗ : π ( M ′ , x ′ ) −→ π ( M, x ) . This implies that the pulled back Galois ´etale covering( M ′ × G ) × M M ′ −→ M ′ × G is identified with the trivial covering M ′ × G × ϕ − ( x ) −→ M ′ × G . Consequently, thereis a unique morphism(2.6) θ ′ : M ′ × G −→ M ′ such that the following two conditions hold:(2.7) ϕ ◦ θ ′ = θ ◦ ( ϕ × Id G )and θ ′ ( x ′ , e ) = x ′ .In view of (2.7), the morphism θ ′ e : M ′ −→ M ′ defined by y θ ′ ( y , e ) is a lift ofthe identity map of M because θ ( z , e ) = z for all z ∈ M . Therefore, from the givencondition that θ ′ ( x ′ , e ) = x ′ it follows that θ ′ e = Id M ′ .Next consider the two morphisms a , b : M ′ × G × G −→ M ′ defined by ( z , g , g ) θ ′ ( θ ′ ( z , g ) , g ) and ( z , g , g ) θ ′ ( z , g g ) respectively.The morphism a (respectively, b ) is a lift of the morphism M × G × G −→ M definedby ( z , g , g ) θ ( θ ( z , g ) , g ) (respectively, ( z , g , g ) θ ( z , g g )). These twomorphisms from M × G × G to M coincide because θ is an action of G on M . Also, a ( z , e , e ) = z = b ( z , e , e ) UNDAMENTAL GROUP OF A GIT QUOTIENT 5 for all z ∈ M ′ . Therefore, we conclude that a = b . Consequently, the morphism θ ′ defines an action of G on M ′ . (cid:3) Let Γ = Gal( ϕ ) ⊂ Aut( M ′ ) be the Galois group for the Galois covering ϕ . Lemma 2.2.
The Galois action of Γ on M ′ commutes with the action of G on M ′ givenby θ ′ in Proposition 2.1.Proof. Take any γ ∈ Γ. The morphism γ ′′ : M ′ × G −→ M ′ , ( z , g ) γ ( θ ′ ( γ − ( z ) , g ))is an action of G on M ′ that lifts the action θ of G on M . Now from the uniqueness of θ ′ it follows that γ ′′ = θ ′ . This immediately implies that the actions of Γ and G on M ′ commute. (cid:3) Fundamental group of the quotient
Let L be a G –equivariant very ample line bundle on M . The action of any g ∈ G onany v ∈ L will be denoted by v · g . Let(3.1) X := M//G be the geometric invariant theoretic (GIT) quotient of M for the action of G on ( M , L )[Mu]. We assume that X is nonempty. This X is an irreducible normal projective variety.Let U ⊂ M be the largest Zariski open subset over which the rational map to the GITquotient M X is defined. Consider the homomorphism(3.2) π ( U, u ) −→ π ( X, x )induced by the quotient map, where u ∈ U is a point lying over a point x ∈ X . Thecodimension of the complement M \ U ⊂ M is at least two because this complement isthe indeterminacy locus of a rational morphism. Since M is smooth, this codimensioncondition implies that the homomorphism π ( U, u ) −→ π ( M, u ) , induced by the inclusion map U ֒ → M , is an isomorphism. Using this isomorphism, thehomomorphism in (3.2) produces a homomorphism(3.3) h : π ( M, u ) −→ π ( X, x ) . Take ( M ′ , ϕ ) as in Proposition 2.1. Consider the ample line bundle ϕ ∗ L on the covering M ′ and the action θ ′ of G on M ′ (constructed in Proposition 2.1). Since θ ′ is a lift ofthe action θ , the action of G on L produces an action of G on ϕ ∗ L . The action of any g ∈ G sends any v ∈ ( ϕ ∗ L ) x to the element in ( ϕ ∗ L ) θ ′ ( x,g ) that corresponds to v · g bythe natural identification L θ ( ϕ ( x ) ,g ) = ( ϕ ∗ L ) θ ′ ( x,g ) after we consider v as an element of I. BISWAS, A. HOGADI, AND A.J. PARAMESWARAN L ϕ ( x ) using the identification ( ϕ ∗ L ) x = L ϕ ( x ) . This action of G on ϕ ∗ L evidently lifts theaction θ ′ . Let(3.4) e X ′ := M ′ //G be the GIT quotient for the action of G on ( M ′ , ϕ ∗ L ).Let M ss ⊂ M be the semistable locus for the action of G ; it is an open subscheme of M . Let f M := ϕ − ( M ss ) ⊂ M ′ be the inverse image. We note that the subset f M is left invariant under the action of G on M ′ because M ss is preserved by the action of G on M and ϕ is G –equivariant.There is a finite collection of G –invariant nonzero sections { s i } Ni =1 of L such that thecollection U i := Spec A i = { z ∈ M | s i ( z ) = 0 } is an affine open cover of M ss . Since s i is G –invariant, the subset U i is preserved by theaction of G on M . The GIT quotient X = M//G is obtained by patching together theaffine open subschemes V i = Spec( A Gi ) (see [New, Ch. 3, §
3] and [New, Ch. 3, §
4] foraffine and projective GIT quotients respectively).Consider the affine open cover of f MU ′ i := ϕ − ( U i ) = Spec B i = { z ∈ f M | ϕ ∗ s i ( z ) = 0 } ⊂ f M .
Note that each U ′ i is preserved by the action of G on M ′ . We may patch together the affineopen subschemes V ′ i := Spec( B Gi ) to construct a quotient e X (see the proof of Theorem1.10 in [Mu, p. 38]). Clearly, e X ⊂ e X ′ is an open subscheme, where e X ′ is the quotient in (3.4). Proposition 3.1.
The natural morphism f : e X −→ X is an ´etale Galois covering with Galois group Γ = Gal( ϕ ) . Moreover the restriction ϕ | f M : f M −→ M ss is the pullback of f via the quotient map q : M ss −→ M//G = X .Proof. From (2.2) it follows immediately that the restriction of the covering ϕ to anyorbit θ ( x , G ) ⊂ M is trivial. In other words, the inverse image ϕ − ( θ ( x , G )) is a disjointunion of copies of θ ( x , G ). In view of Lemma 2.2, this implies that the Galois group Γ actssimply transitively on the set of connected components of ϕ − ( θ ( x , G )). In particular, forany y ∈ M ′ , the restriction of ϕ to the orbit θ ′ ( y , G ) is injective.Therefore, to prove the proposition, we may replace M by the spectrum of an integralfinite type algebra A , with quotient field K , equipped with an action of G . Similarly, thevariety f M and the action of G on it are replaced by a connected finite ´etale algebra B ,with the quotient field L , over A with Galois group Γ, and a lifting to B of the action of UNDAMENTAL GROUP OF A GIT QUOTIENT 7 G on A that commutes with the action of Γ on B . The quotients e X and X get replacedby Spec( A G ) and Spec( B G ) respectively. Since M is smooth, hence normal, and the map ϕ restricted to any closed orbit of G is injective, the following lemma completes the proofof the proposition. (cid:3) Lemma 3.2.
Suppose the G –equivariant finite ´etale map f : f M −→ M of affine varietiesis defined by an inclusion A ⊂ B of finite type k algebras such that • A is normal, and • f restricted to each closed orbit of G is an injection.Then the induced map on the quotients e X −→ X is also finite ´etale, and f M is the fiberproduct e X × X M .Proof. This can be found in [Dr] (see [Dr, Proposition 4.16] and [Dr, Proposition 4.18]).In [Dr] it is assumed that the characteristic of the field k is zero. However, the proof canbe checked to be characteristic free. For the convenience of the reader, we give a briefoutline of the proof.The actions of G on A and B extend to the quotient fields K and L respectively.The spaces of invariants for the action of the Galois group Γ on B and L are A and K respectively.Since the actions of G and Γ on B commute, the natural inclusion A G ⊂ ( B G ) Γ is anisomorphism. This implies that B G is finite over A G .We first observe that B G is the integral closure of A G in L . Indeed, if an element a ∈ L is integral over A G , then all G translates of a are also solutions of the sameequation. Therefore, the connectedness of G implies that a is fixed by G .Consequently, Spec( B G ) is a normal affine variety equipped with an action of Γ suchthat the quotient is Spec( A G ). So to show that the map e X −→ X in the lemma is ´etaleit is enough to check that this action of Γ is free on the closed points.Let x ∈ Spec( B G ) be a closed point such that there is an element γ ∈ Γ with γ · x = x .Let y be a closed point in the unique closed orbit in the fiber of the map f M −→ e X over x . Since γ commutes with G (see Lemma 2.2) we get that θ ′ ( γ · y , G ) = γ · θ ′ ( y , G )is also the unique closed orbit projecting to x . Hence, we have θ ′ ( γ · y , G ) = θ ′ ( y , G ).Now from the injectivity of the map from Γ to the permutations of the components of θ ′ ( y , G ) we conclude that γ · y = y . Consequently, we have γ = e . This proves that themorphism e X −→ X is ´etale.For the isomorphism f M = e X × X M , first note that the G –equivariant inclusion A ֒ → B factors via A ⊂ A ⊗ A G B G . Therefore, in order to prove that f M = e X × X M it is enough to prove it under theassumption that the natural G –equivariant homomorphism A ⊗ A G B G −→ B I. BISWAS, A. HOGADI, AND A.J. PARAMESWARAN is an isomorphism.By using the conclusion of the first part of the lemma that B G is finite and ´etale over A G of cardinality | Γ | , the base change A ֒ → A ⊗ A G B G is also finite and ´etale of the samecardinality. Since we started with a finite and ´etale algebra B over A we conclude that A ⊗ A G B G ֒ → B is also finite and ´etale.Now the isomorphism f M = e X × X M follows because both Spec B and Spec( A ⊗ A G B G )are finite and ´etale over Spec A of same fiber cardinality | Γ | . (cid:3) The construction in Proposition 3.1 of a covering e X starting from an ´etale covering M ′ of M is functorial (compatible with the standard operations on coverings), and defines ahomomorphism(3.5) H : π ( X, x ) −→ π ( M, u ) . Given an ´etale Galois covering φ : Y −→ X , the ´etale Galois covering of M corre-sponding to the homomorphism h in (3.3) is constructed as follows. Consider the pullbackof the covering φ to the open subset of M where the rational map M X is defined.This covering extends to M because the complement of the open subset has codimensionat least two and M is smooth.The above two constructions, namely the construction of a covering of X from a coveringof M , and the construction of a covering of M from a covering of X , are clearly inversesof each other. Therefore, for the two homomorphisms h and H in (3.3) and (3.5), we have h ◦ H = Id π ( X,x ) and H ◦ h = Id π ( M,u ) . Consequently, the following is proved: Theorem 3.3.
The homomorphism h in (3.3) is an isomorphism. Remark 3.4.
Let M = P k / { ∞} be the unique nodal curve of arithmetic genusone. The action of G m on P k defined by t · ( x , x ) = ( tx , x /t ) produces an action of G m on M . While M is not simply connected, the GIT quotient M// G m = Spec k is so.So the condition in Theorem 3.3 that M is smooth is essential.4. The topological fundamental group
In this section we assume that k = C . The topological fundamental group of any Y with base point y ∈ Y will be denoted by π t ( Y, y ). We recall that an irreduciblesmooth complex projective variety is also called a complex projective manifold.As before, fix a G –equivariant algebraic line bundle on the complex projective manifold M .Let ϕ : M ′ −→ M be a holomorphic ´etale Galois covering of M such that M ′ is connected. It should beclarified that the degree of ϕ is now allowed to be infinite. Proposition 4.1.
UNDAMENTAL GROUP OF A GIT QUOTIENT 9 (1)
There is a unique holomorphic action of G on M ′ θ ′ : M ′ × G −→ M ′ that lifts the action θ of G on M . (2) The actions of G and Γ = Gal( ϕ ) on M ′ commute.Proof. The proof of the first (respectively, second) part of the proposition is exactly iden-tical to the proof of Proposition 2.1 (respectively, Lemma 2.2). (cid:3)
The geometric invariant theory is not applicable for the action of G on M ′ because M ′ is not an algebraic variety in general. However we will construct from M ′ a covering ofthe GIT quotient X = M//G .Let M ss ⊂ M be the semistable locus for the action of G on M ; it is a Zariski opensubset preserved by the action of G . Let(4.1) β : M ss −→ X be the quotient map. This map β is surjective.Take an affine open subset U ⊂ X . The inverse image β − ( U ) ⊂ M ss is also an affine open subset. Fix a maximal compact subgroup K G ⊂ G .
There is a K G –invariant subset U K ⊂ β − ( U )such that • β − ( U ) admits a deformation retraction to U K , • the map β | U K : U K −→ U is surjective, • the quotient map U K /K G −→ U is a homeomorphism, and • the subset U K is contained in the union of all closed G –orbits satisfying the con-dition that the intersection with U K is a K –orbit.(See [Nee, p. 422, Corollary 1.4] and [Nee, p. 424, Theorem 2.1]; also stated in the firstparagraph of the introduction in [Nee, p. 419]. See also [KN].)Let(4.2) e U K := ϕ − ( U K ) ⊂ M ′ be the inverse image. As β − ( U ) is a nonempty Zariski open subset of M , and M issmooth, the homomorphism of topological fundamental groups(4.3) π t ( β − ( U )) −→ π t ( M ) , induced by the inclusion of β − ( U ) in M , is surjective. Since U K is a deformation retrac-tion of β − ( U ), from the surjectivity of the homomorphism in (4.3) it follows immediatelythat the homomorphism(4.4) π t ( U K ) −→ π t ( M )induced by the inclusion of U K in M is surjective. Note that surjectivity of the homomor-phism induced on fundamental groups induced by a map is equivalent to the conditionthat the pullback to the domain (of the map) of any connected ´etale covering of the targetspace (of the map) remains connected. Consequently, from the surjectivity of the homo-morphism in (4.4) it follows that the inverse image e U K in (4.2) is connected (recall that M ′ is connected). Lemma 4.2.
Take a point x ∈ M . Let Z := θ ( x , K G ) ⊂ M be the K G –orbit of x .Then ϕ − ( Z ) is a disjoint union of copies of Z . More precisely, the restriction of ϕ toeach connected component S of ϕ − ( Z ) is a homeomorphism from S to Z .Proof. Let ι x : Z ֒ → M be the inclusion map. To prove the lemma it suffices to showthat the homomorphism ι x ∗ : π t ( Z, x ) −→ π t ( M, x ) , induced by the inclusion ι x , is trivial.Let G x ⊂ G be the isotropy subgroup for x . The G –orbit θ ( x , G ) is identified with thequotient G/G x . Since M is projective, there is an irreducible smooth complex projectivevariety e Z containing G/G x as a Zariski open subset such that the inclusion map G/G x = θ ( x , G ) ֒ → M extends to a morphism(4.5) τ : e Z −→ M .
Since G is a rational variety, the quotient G/G x is unirational. So e Z is unirational.Since e Z is smooth, this implies that e Z is simply connected [Se, p. 483, Proposition 1].The above inclusion map ι x coincides with the composition Z = K G / ( K G ∩ G x ) ֒ → G/G x ֒ → e Z τ −→ M , where τ is the map in (4.5). Since e Z is simply connected, this implies that ι x ∗ = 0. Asnoted before, the lemma follows from this. (cid:3) Corollary 4.3.
The image in π t ( M ) of the fundamental group of any G orbit in M istrivial.Proof. We saw in the proof of Lemma 4.2 that the inclusion of the orbit θ ( x , G ) in M factors through the simply connected variety e Z . (cid:3) Define(4.6) e U := e U K /K G , UNDAMENTAL GROUP OF A GIT QUOTIENT 11 where e U K is defined in (4.2). We note that e U is connected because e U K is connected. Thenatural map ( β ◦ ϕ ) | e U K : e U K −→ U , is clearly K G –invariant. So it descends to a map(4.7) e ϕ U : e U −→ U , where e U is defined in (4.6).Since the actions of G and Γ on M ′ commute (Proposition 4.1(2)), the action of Γ on e U K descends to an action of Γ on the quotient e U in (4.6). Note that the quotient map(4.8) e U −→ e U /
Γis an ´etale Galois covering map with Galois group Γ because the covering ϕ | e U K : e U K −→ U K is K G –equivariant and the map in (4.8) is the quotient for the actions of K G .Since ( e U K / Γ) /K G = U and ( e U K / Γ) /K G = ( e U K /K G ) / Γ , we have e U /
Γ = U . Therefore, the map e ϕ U in (4.7) is an ´etale Galois covering with Galoisgroup Γ.The intersection of two affine open subsets of X is again an affine open subset. Takeanother affine open subset V ⊂ X , and define W = U T V . Construct e V (respectively, f W ), as done in (4.6), from V (respectively, W ).Identify a point z of e U with a point z ′ of e V if the K G –orbit in e U K for z and the K G –orbitin e V K for z ′ lie in the same G –orbit in M ′ . Note that the subset of e U consisting of pointswhose K G –orbit lie in θ ′ ( e V , G ) actually coincides with the subset f W ⊂ e U .Take a finite collection of affine open subsets {U i } ni =1 of X that cover X . Let e U i bethe topological space constructed as in (4.6) from U i . Let X ′ be the topological spaceconstructed by performing the above gluing on the disjoint union F ni =1 e U i . Recall thatall e U i are connected, and for any pair e U i and e U j , the gluing between them is done overconnected open subsets of e U i and e U j . Therefore, the resulting topological space X ′ isconnected.Recall that each e U i is an ´etale Galois covering of U i with Galois group Gal( ϕ ) (via themap e ϕ U in (4.7)). The intersection of e U i with e U j inside X ′ is an ´etale Galois covering of U i T U j with Galois group Gal( ϕ ). Therefore, X ′ is an ´etale Galois covering of X withGalois group Gal( ϕ ).Just as Theorem 3.3 was proved, we now have the following: Theorem 4.4.
The homomorphism between topological fundamental groups induced bythe rational map M M//G is an isomorphism.
Remark 4.5.
It is easy to construct examples to show that Theorem 1.1 is false if M is not assumed to be projective. Let G = G m act on M = G m by left translations.Then the quotient M//G coincides with Spec k . Clearly π ( G m ) −→ π (Spec k ) is not anisomorphism. 5. Acknowledgements
Niels Borne pointed out a gap in the proof of Proposition 2.1 given in an earlier version.We are very grateful to him for this. We are very grateful to Jakob Stix for some veryhelpful correspondences. We thank Michel Brion for a useful correspondence. We arevery grateful to the two referees for their comments. We thank the managing editors forpointing out [Li]. The first-named author acknowledges the support of the J. C. BoseFellowship.
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Jour. Lond. Math. Soc. (1959),481–484. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road,Bombay 400005, India
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