An Analytic Approach to the Quasi-projectivity of the Moduli Space of Higgs Bundles
aa r X i v : . [ m a t h . DG ] S e p AN ANALYTIC APPROACH TO THE QUASI-PROJECTIVITY OF THEMODULI SPACE OF HIGGS BUNDLES
YUE FANA bstract . The moduli space of Higgs bundles can be defined as a quotient of aninfinite-dimensional space. Moreover, by the Kuranishi slice method, it is equippedwith the structure of a normal complex space. In this paper, we will use analyticmethods to show that the moduli space is quasi-projective. In fact, following Hausel’smethod, we will use the symplectic cut to construct a normal and projective com-pactification of the moduli space, and hence prove the quasi-projectivity. The maindifference between this paper and Hausel’s is that the smoothness of the moduli spaceis not assumed. C ontents
1. Introduction 12. Preliminaries 62.1. Construction of the moduli space 62.2. The orbit type stratification 73. Infinite dimensional GIT quotient 113.1. S-equivalence classes and closures of orbits 113.2. π -saturated open neighborhoods 124. Descent lemmas for vector bundles 145. The Kähler metric on the moduli space 166. Projective compactification 176.1. Symplectic cuts 176.2. Projectivity 22References 251. I ntroduction Let M be a closed Riemann surface with genus ≥
2. A Higgs bundle ( E , Φ ) is apair of a holomorphic vector bundle E → M and a holomorphic 1-form taking valuesin End E . In [7], the moduli space M of Higgs bundles is constructed as a quotientspace of an infinite-dimensional space. By the Kuranishi slice method, it is shownthat M can be endowed with the structure of a normal complex space. Moreover, itis shown that M is canonically biholomorphic to the moduli space in the categoryof schemes, the one constructed by Simpson and Nitsure using Geometric InvariantTheory (GIT) (see [26] and [20]). As a consequence, M is a quasi-projective variety.It is natural to ask whether the quasi-projectivity of M can be proved directly byanalytic methods without using the biholomorphism just mentioned. The purpose ofthis paper is to give a positive answer. Therefore, the first step toward this goal is tocompactify the moduli space M . In [12], Hausel used the symplectic cut to compactifythe moduli space M when it is smooth and the underlying smooth bundle of ( E , Φ ) Date : October 1, 2020. is of rank 2. He further showed that the compactification is projective and thus thequasi-projectivity of M follows. In this paper, we will follow the same method tocompactify the moduli space M and prove the projectivity of the compactification.However, we will not impose the smoothness conditions as Hausel did. Therefore,this paper is a generalization of Hausel’s results. It should be noted that Simpsoncompactified the moduli space using algebro-geometric methods in [24]. However,the projectivity of the compactification was not proved. In a recent paper [3], deCataldo followed Simpson’s method and constructed a projective compactification ofthe moduli space. It can be shown that our compactification is isomorphic to deCataldo’s compactification.To state the results and set up the notations, we fix a smooth Hermitian vectorbundle E → M . For convenience, we assume that E is of degree 0. This condition isnot essential. Since dim C M =
1, the space A of unitary connections on E parametrizesholomorphic structures on E . Let g E → M be the bundle of the skew-Hermitianendomorphisms of E , and set C = A × Ω ( g C E ) . The configuration space of Higgsbundles (with the fixed underlying bundle E ) is defined as(1.1) B = { ( A , Φ ) ∈ C : ∂ A Φ = } .The complex gauge group G C = Aut ( E ) acts on B by(1.2) ( ∂ A , Φ ) · g = ( g − ◦ ∂ A ◦ g , g − Φ g ) , g ∈ G C , ( A , Φ ) ∈ B .To define the moduli space M , let us recall various stability conditions. A Higgs bun-dle ( E , Φ ) is semistable if µ ( F ) ≤ µ ( E ) for every Φ -invariant holomorphic subbundle F with 0 ( F ( E , where µ ( F ) means the slope of F . If the equality µ ( F ) = µ ( E ) can-not occur, then ( E , Φ ) is stable. Finally, ( E , Φ ) is polystable if it is a direct sum of stableHiggs bundles of the same slope. Consequently, there are G C -invariant subspaces B ss , B s and B ps of B consisting of semistable, stable and polystable Higgs bundles, re-spectively. Moreover, the polystability has a gauge-theoretic interpretation as follows.Recall that C is an infinite-dimensional affine hyperKähler manifold that is modeledon Ω ( g E ) ⊕ Ω ( g C E ) (see [17, §6]). If we identify Ω ( g E ) with Ω ( g C E ) , an L -metricon C is given by(1.3) g ( a , η ; a , η ) = √− π Z M tr ( a ∗ a + ηη ∗ ) , ( a , η ) ∈ Ω ( g C E ) ⊕ Ω ( g C E ) .Let I be the complex structure given by the multiplication by √− Ω I its associatedKähler form, and G the subgroup of G C consisting of unitary gauge transformations.The G -action on C is Hamiltonian with respect to the Kähler form Ω I , and the momentmap is given by(1.4) µ ( A , Φ ) = π ( F A + [ Φ , Φ ∗ ]) : C → Ω ( g E ) .Then, the Hitchin-Kobayashi correspondence states that a Higgs bundle ( A , Φ ) ∈ B ispolystable if and only if ( A , Φ ) · g satisfies Hitchin’s equation µ = g ∈ G C .A stronger version of this result states that the inclusion µ − ( ) ∩ B ֒ → B ps inducesa homeomorphism ( µ − ( ) ∩ B ) / G ∼ −→ B ps / G C , where the inverse is induced by theretraction r : B ss → µ − ( ) ∩ B defined by the Yang-Mills-Higgs flow (for more details,see [30] and [31]).By definition, the moduli space M is defined as the quotient B ps / G C equipped withthe C ∞ -topology. In [7], it is shown that M is a normal complex space. One of themain results in this paper is that M admits a compactification. HE QUASI-PROJECTIVITY OF THE MODULI SPACE OF HIGGS BUNDLES 3
Theorem A.
There is a normal compact complex space M in which the moduli space M embeds as an open dense subset. Moreover, the complement Z = M \ M is a closed complexsubspace of pure codimension 1. As a consequence, the quasi-projectivity of M follows if we can show that the com-pactification M is projective. Therefore, we need to construct an ample line bundle on M . To construct such a line bundle, we need a descent lemma for vector bundles. In[5], Drezet and Narasimhan proved a descent lemma for good quotients of algebraicvarieties. A natural analogue of good quotients in our settings is the quotient map π : B ss → B ss (cid:12) G C , where B ss (cid:12) G C is the quotient space of B ss by the S -equivalencerelation of Higgs bundles. Recall that two Higgs bundles are S -equivalent if thegraded objects of their Seshadri filtrations are isomorphic. Heuristically, we think of π : B ss → B ss (cid:12) G C as an infinite-dimensional GIT quotient and naturally expect thatits properties are similar to those of good quotients of algebraic varieties. To justifythis heuristic thinking, we will first prove in Section 3.1 that the inclusion B ps ֒ → B ss induces a homeomorphism M → B ss (cid:12) G C , and hence will routinely identify M with B ss (cid:12) G C . Then, we will show the following. Theorem B.
The quotient map π : B ss → M satisfies the following properties: (1) π identifies G C -orbits whose closures in B ss intersect. (2) Every fiber of π contains a unique G C -orbit that is closed in B ss . Moreover, a G C -orbitis closed in B ss if and only if it contains a polystable Higgs bundle. (3) O M = π ∗ O G C B ss . In other words, if U is an open subset of M , the map O M ( U ) → O B ss ( π − ( U )) G C given by f π ∗ f is a bijection. (4) π is a categorical quotient in the sense that every G C -invariant holomorphic map from B ss into a complex space factors through the quotient map π : B ss → B ss (cid:12) G C . To make sense of ( ) in Theorem B, we equip the space B with a naive structuresheaf by restricting the sheaf of I -holomorphic functions on C to B . Moreover, O M denotes the structure sheaf of the moduli space M .Generalizing the descent lemma for vector bundles in [5], we will prove the follow-ing (cf. [27, Lemma 2.13]). Theorem C.
Let E → B ss be a holomorphic G C -bundle. Suppose that the stabilizer G C ( A , Φ ) acts trivially on the fiber E ( A , Φ ) for every ( A , Φ ) ∈ µ − ( ) . Then, there is a holomorphicvector bundle E over M such that π ∗ E = E . Moreover, O ( E ) = π ∗ O ( E ) G C , where O ( E ) and O ( E ) are sheaves of holomorphic sections of E and E , respectively. Now we are able to construct an ample line bundle on M as follows. Recall that A isan infinite-dimensional Kähler manifold that is modeled on Ω ( g E ) (see [2, p.587]). In[4], Donaldson constructed a holomorphic line bundle on A together with a Hermitianmetric whose curvature is a multiple of Kähler form on A (also see [22]). Moreover,the G C -action on A lifts to this line bundle. By pulling back this line bundle to C by theprojection map C → A , we obtain an I -holomorphic line bundle L → C , and the G C -action on C lifts to L . By slightly modifying the pullback Hermitian metric, we are ableto show that the curvature of the resulting Hermitian metric h on L is − π √− Ω I .Then, the projectivity of the compactification M is shown in the following result. Theorem D. (1)
The restriction of the line bundle L → C to B ss descends to M and defines a linebundle L → M . (2) L extends to a line bundle L on M . YUE FAN (3) L is ample.Therefore, M is projective, and hence M is quasi-projective. In the proof of ( ) in Theorem D, a byproduct is that the moduli space M has a weakKähler metric. More precisely, we recall that M admits an orbit type stratification suchthat each stratum Q is a complex submanifold of M together with a Kähler form ω Q (see Section 2.2). A weak Kähler metric on M is a family of continuous stratum-wisestrictly plurisubharmonic functions ρ i : U i → R such that { U i } is an open covering of M and that ρ i − ρ j = ℜ ( f ij ) for some holomorphic function f ij ∈ O M ( U i ∩ U j ) . Here,a continuous stratum-wise strictly plurisubharmonic function is a continuous func-tion that is smooth and strictly plurisubharmonic along every stratum Q in the orbittype stratification of M . Note that stratum-wise strictly plurisubharmonic functionsare not necessarily strictly plurisubharmonic. If each ρ i can be chosen to be strictlyplurisubharmonic, then { ρ i : U i → R } defines a (strong) Kähler metric on M . (see [14]for more details on strictly plurisubharmonic functions). Finally, since √− ∂∂ ( ρ i | Q ) patches together, the Kähler metric on M restricts to Q . Then, our last result is thefollowing. Theorem E.
The moduli space M admits a weak Kähler metric whose restriction to eachstratum Q in the orbit type stratification of M is the Kähler form ω Q . In [8], it is shown that each stratum Q in the orbit type stratification of M alsoadmits a complex symplectic form such that they glue together to define a complexPoisson bracket on the structure sheaf O M of M . In fact, each stratum Q is a hyper-Kähler manifold. The result in [8] and Theorem E show that the stratum-wise definedhyperKähler structure on M can be extended to two global holomorphic objects, acomplex Poisson bracket and a weak Kähler metric. Finally, we should remark thatwe are unable to prove that the weak Kähler metric on M is strong, although it ishighly likely.Now we describe the structures of the paper and the ideas behind the proofs of themain theorems. The key tools in the proof of Theorem B are a local slice theorem forthe G C -action and the retraction r : B ss → µ − ( ) ∩ B defined by the Yang-Mills-Higgsflow. We will prove Theorem B in Section 3. To prove Theorem C, we will first provea descent lemma for analytic Hilbert quotients of complex spaces. By definition, if G is a complex Lie group, then an analytic Hilbert quotient of a holomorphic G -space X is a G -invariant surjective holomorphic map π : X → Z such that inverseimages of Stein subspaces are Stein, and O Z = π ∗ O GX . This notion is an analyticanalogue of good quotients of algebraic varieties (see [16]). The proof of Theorem C isan adaptation of Drezet and Narasimhan’s argument in [5]. Then, this result will beapplied to Kuranishi local models that are used to construct the moduli space M , sinceKuranishi local models are analytic Hilbert quotients of Kuranishi spaces. In this way,we can show that every point in B ss admits an open neighborhood that is saturatedwith respect to the quotient map π : B ss → M and in which the vector bundle E inquestion is trivial. This shows that E descends to M . These results will be proved inSection 4. After Theorem B and C are proved, we are ready to prove Theorem E and ( ) in Theorem D. By verifying the hypothesis in Theorem C for the line bundle L | B ss ,we can easily show that it defines a line bundle L → M . To show Theorem E, we maychoose an open covering { U i } of M such that L is trivial over each π − ( U i ) . Then,we choose a holomorphic section s i of L over each π − ( U i ) that is G C -equivariant andnowhere vanishing. Then, we consider the functions(1.5) u i = − π log | s i | h , HE QUASI-PROJECTIVITY OF THE MODULI SPACE OF HIGGS BUNDLES 5 where h is the Hermitian metric on L . Since it is G -invariant, its restriction to π − ( U i ) ∩ µ − ( ) defines a continuous map u i ,0 : U i → R . It will be shown that the restriction ofeach u i ,0 to a stratum Q is smooth and a Kähler potential for the Kähler form ω Q on Q .In this way, we obtain a family of continuous stratum-wise strictly plurisubharmonicfunctions u i ,0 : U i → R such that { U i } covers M . Then, the normality of M and thefact that codim x ( M \ M s ) ≥ x ∈ M \ M s show that { u i ,0 : U i → R } defines aweak Kähler metric. These results will be proved in Section 5.Then, we will prove Theorem A and the rest of the statements in Theorem D inSection 6. Following Hausel’s strategy in [12], we will use the symplectic cut to com-pactify M . Recall that M admits a holomorphic C ∗ -action. Moreover, the induced U ( ) -action is stratum-wise Hamiltonian. More precisely, the restriction of the G -invariant map(1.6) f ( A , Φ ) = − π k Φ k L : C → R to µ − ( ) defines a continuous map f : M → R . When restricted to a stratum Q , f | Q is smooth and a moment map for the induced U ( ) -action with respect to the Kählerform ω Q on Q . In this sense, f is a stratum-wise moment map on M . Then, we con-sider the direct product M × C . If we let C ∗ act on C by multiplication, M × C admits adiagonal C ∗ -action. The induced U ( ) -action is also stratum-wise Hamiltonian. Here,the stratification of M × C is given by the disjoint union of Q × C , where Q ranges inthe orbit type stratification of M . Moreover, the stratum-wise moment map on M × C is given by(1.7) e f = f − k · k .By [17, Theorem 8.1] or [30, Theorem 2.15], the Hitchin fibration h is proper, and hencethe nilpotent cone h − ( ) is compact. Therefore, we are able to choose a level c < h − ( ) ⊂ f − [ c ) . Then the symplectic cut of M at the level c is defined asthe singular symplectic quotient e f − ( c ) / U ( ) , and it should be a compactification of M . Here, the rough idea is that the subspace f − [ c ] is compact by the propernessof f (see [17, Proposition 7.1]). Moreover, if a Higgs bundle is away from f − [ c ] ,following its C ∗ -orbit, it “flows” into f − [ c ] , since the 0-limit of the C ∗ -action on aHiggs bundle always exists, and hence the limiting point is a C ∗ -fixed point and iscontained in the nilpotent cone. Therefore, the moduli space M should be “containedin” e f − ( c ) / U ( ) , which is compact because of the properness of f .To carry out this idea rigorously, we first need to equip e f − ( c ) / U ( ) with the struc-ture of a complex space. Let ( M × C ) ss be the subspace of semistable points in M × C determined by the stratum-wise moment map e f − c . More precisely, it consists ofpoints in M × C whose C ∗ -orbit closures intersect e f − ( c ) . To show that the analyticHilbert quotient of ( M × C ) ss by C ∗ exists, we run into a technical difficulty. Since weare unable to prove that the Kähler metric on M is a strong one, we cannot directlyapply the analytic GIT developed by Heinzner and Loose in [15] and must take a de-tour. To motivate the following detour, let us recall that a complex reductive Lie groupacts properly at a point if and only if its stabilizer at that point is finite, provided thata local slice theorem is available around that point. Since the C ∗ -stabilizers are finiteaway from the nilpotent cone h − ( ) , it is reasonable to expect that the C ∗ -action actsproperly away from the nilpotent cone. Hence, we consider the C ∗ -invariant opensubset W = ( M × C ) \ ( h − ( ) × { } ) . By the properness of the Hitchin fibration h ,we can show that the C ∗ -action on W is proper, and hence the analytic Hilbert quo-tient of W by C ∗ exists. Moreover, W / C ∗ is a geometric quotient. Then, we use the YUE FAN properness of h and f to show that W = ( M × C ) ss = C ∗ e f − ( c ) . It then follows thatthe inclusion e f − ( c ) ֒ → W induces a homeomorphism e f − ( c ) / U ( ) → W / C ∗ . Now,note that W can be written as a disjoint union(1.8) W = ( M \ h − ( ) × { } ) ∪ ( M × C ∗ ) .We will show that the quotient ( M × C ∗ ) / C ∗ is biholomorphic to the moduli space M , and therefore M = W / C ∗ is a compactification of M . To show the rest of thestatements in Theorem D, we pullback the line bundle L → M to M × C by the pro-jection map M × C → M to obtain a line bundle L C → M × C . By slightly modifyingthe Hermitian metric h on L C , we can easily show that the resulting Hermitian met-ric, again denoted by h , is smooth along each stratum Q × C , and the curvature is − π √− ω Q × C , where ω Q × C is the product Kähler metric on Q × C . By the descentlemma for the analytic Hilbert quotients, the restriction of L C to W induces a line bun-dle L → M such that the restriction of L to ( M × C ∗ ) / C ∗ is isomorphic to L → M .In this sense, the line bundle L → M extends to the line bundle L → M . Moreover,the Hermitian metric h on L C also induces a Hermitian metric h on L . Then, we willuse Popovici’s bigness criterion (see [21, Theorem 1.3]) to show that the restriction of L to any irreducible closed complex subspace (not reduced to a point) of M is big.Then, the ampleness of L follows from a theorem of Grauert (see [9]): a line bundleover a compact complex space is ample if its restriction to any irreducible closed com-plex subspace (not reduced to a point) admits a nontrivial holomorphic section thatvanishes somewhere on that subspace. These results will be proved in Section 6.Finally, we remark that all the complex spaces in this paper are assumed to bereduced. Moreover, if necessary, we will work with the L k -topology on C and L k + -topology on G C , where k > M doesnot depend on the choice of k . Acknowledgments . This paper is part of my Ph.D. thesis. I would like to thank myadvisor, Professor Richard Wentworth, for suggesting this problem and his generoussupport and guidance. I also thank Reyer Sjamaar, Daniel Greb, and Ruadhai Dervanfor helpful discussions. 2. P reliminaries
Construction of the moduli space.
In this section, we review the results in [7].Recall that every Higgs bundle ( A , Φ ) ∈ B defines a deformation complex(2.1) C µ C ( A , Φ ) : Ω ( g C E ) D ′′ −→ Ω ( g C E ) ⊕ Ω ( g C E ) D ′′ −→ Ω ( g C E ) ,where D ′′ = ∂ A + Φ . If the Higgs bundle ( A , Φ ) is understood, we will simply write C µ C instead of C µ C ( A , Φ ) . Proposition 2.1 ([25, §1] and [26, §10]) . The complex C µ C is an elliptic complex. Moreover,the formal L -adjoint ( D ′′ ) ∗ satisfies the Kähler identities (2.2) ( D ′′ ) ∗ = − i [ ∗ , D ′ ] , ( D ′ ) ∗ = + i [ ∗ , D ′′ ] , where D ′ = ∂ A + Φ ∗ and ∗ is the Hodge star. Let H denote the harmonic space for the cohomology H ( C µ C ) . Then, we re-view the Kuranishi slice method that is used to construct the moduli space. Fix ( A , Φ ) ∈ m − ( ) with G -stabilizer K , where m = ( µ , µ C ) and µ C ( A , Φ ) = π ∂ A Φ .As a consequence, K C is the G C -stabilizer at ( A , Φ ) . Note that since the G -action on C is proper, K is compact. Then, there are an open ball in the L -topology around 0 HE QUASI-PROJECTIVITY OF THE MODULI SPACE OF HIGGS BUNDLES 7 in H and a Kuranishi map θ : B → C . Its associated Kuranishi space Z is defined as Z = θ − ( B ) . It can be shown that Z is a closed complex subspace of B . Since B ss isopen in B , if B is sufficiently small, then we obtain a restriction θ : Z → B ss . We listsome of its properties that will be used later. Proposition 2.2. (1) θ : B → C is I-holomorphic, and θ ( ) = ( A , Φ ) . (2) The derivative of θ : B → C at is the inclusion map H ֒ → T ( A , Φ ) C . (3) θ : B → C extends to a K C -equivariant holomorphic map θ : BK C → C . (4) If B is sufficiently small, then θ preserves the stabilizers in the sense that ( K C ) x =( G C ) θ ( x ) for any x ∈ Z . (5) If x ∈ Z has a closed K C -orbit in H , then θ ( x ) is a polystable Higgs bundle. Moreover, there is a local slice theorem for the G C -action. Proposition 2.3.
If B is sufficiently small, the map Z K C × K C G C → B ss given by [ x , g ] θ ( x ) g is a homeomorphism onto an open neighborhood of ( A , Φ ) in B ss . It is shown that Z K C is a closed complex subspace of BK C . Moreover, it admits ananalytic Hilbert quotient (see [16] and [15]) π : Z K C → Z K C (cid:12) K C . In other words, π isa surjective K C -invariant map such that it is Stein in the sense that preimages of Steinsubspaces are Stein. Moreover, the structure sheaf of Z K C (cid:12) K C is given by π ∗ O K C Z K C .As a topological space, Z K C (cid:12) K C is defined as a quotient space by the equivalencerelation that x ∼ y if the closures of xK C and yK C intersect. Then, it is shown that θ induces a well-defined map ϕ : Z K C (cid:12) K C → M such that ϕ [ x ] = [ r θ ( x )] for any x ∈ Z .We also call ϕ a Kuranishi map. Here, r : B ss → m − ( ) is the retraction defined bythe negative gradient flow of the Yang-Mills-Higgs functional. By [31, Theorem 1.4], r ( A , Φ ) is isomorphic to Gr ( A , Φ ) , where Gr ( A , Φ ) is the graded object associatedwith the Seshadri filtration of ( A , Φ ) . Moreover, ϕ is a homeomorphism onto an openneighborhood of [ A , Φ ] . Finally, there is a unique structure of a normal complex spaceon B ps / G C such that ϕ is a biholomorphism onto its image.More can be said about the singularities in the moduli space. Note that H admitsa linear hyperKähler structure that is induced by the inclusion H ֒ → T ( A , Φ ) C . Let ω and ω C denote the Kähler form associated with the complex structure I and thecomplex symplectic form associated with the other complex structures, respectively.The G C -stabilizer K C acts linearly on H and preserves ω C . As a consequence, thereis a canonical complex moment map ν C : H → H ( C µ C ) given by ν C ( x ) = H [ x , x ] ,where H : Ω ( g C E ) → H ( C µ C ) is the harmonic projection. Then, ν − C ( ) , as an affinevariety, admits a GIT quotient π : ν − C ( ) → ν − C ( ) (cid:12) K C . In fact, it can be realizedas a singular hyperKähler quotient as follows. The G -stabilizer K acts linearly on H and preserves the Kähler form ω . As a consequence, the K -action on H admits aunique moment map ν such that ν ( ) =
0. Then, n = ( ν , ν C ) can be regardedas a hyperKäher moment map. By [15], the inclusion n − ( ) ֒ → ν − C ( ) induces ahomeomorphism n − ( ) / K ∼ −→ ν − C ( ) (cid:12) K C . Finally, it can be shown that Z = B ∩ ν − C ( ) , and Z K C (cid:12) K C is an open neighborhood of [ ] in ν − C ( ) (cid:12) K C . In summary, [ A , Φ ] admits an open neighborhood that is biholomorphic to an open neighborhoodof [ ] in ν − C ( ) (cid:12) K C .2.2. The orbit type stratification.
In this section, we first review the results in [8], andthen prove some technical results that will be used later.
YUE FAN
Let K be a G -stabilizer at some Higgs bundle in m − ( ) and ( K ) the conjugacy classof K in G . The subspace(2.3) m − ( ) ( K ) = { ( A , Φ ) ∈ m − ( ) : G ( A , Φ ) ∈ ( H ) } is G -invariant. The orbit type stratification of the singular hyperKähler quotient m − ( ) / G is defined as(2.4) m − ( ) / G = ∐ ( H ) components of m − ( ) ( H ) / G .Similarly, if L is a G C -stabilizer at some Higgs bundle in B ps , and ( L ) denotes theconjugacy class of L in G C , then the subspace(2.5) B ps ( L ) = { ( A , Φ ) ∈ B ps : ( G C ) ( A , Φ ) ∈ ( L ) } is G C -invariant. The orbit type stratification of M is defined as(2.6) M = ∐ ( L ) components of B ps ( L ) / G C .It can be proved that the subgroup L appearing in the orbit type stratification of M isalways equal to K C for some compact subgroup K of G . Then, we have the followingresults. Proposition 2.4. (1)
Every stratum Q in the orbit type stratification of m − ( ) / G is a locally closedsmooth manifold, and π − ( Q ) is a smooth submanifold of C such that the restriction π : π − ( Q ) → Q is a smooth submersion. Moreover, the restriction of the hyperKählerstructure from C to π − ( Q ) descends to Q. (2) Every stratum Q in the orbit type stratification of M is a locally closed complex sub-manifold of M , and π − ( Q ) is a complex submanifold of C with respect to the complexstructure I such that the restriction π : π − ( Q ) → Q is a holomorphic submersion.This decomposition is a complex Whitney stratification. (3)
If Q is a stratum determined by the orbit type ( K C ) , and [ A , Φ ] ∈ Q, then the tangentspace of Q at [ A , Φ ] can be identified with ( H ) K C . Moreover, the Hitchin-Kobayashi correspondence i : m − ( ) / G ∼ −→ B ps / G C preservesthe stratifications in the following sense. Proposition 2.5.
If Q is a stratum in the orbit type stratification of m − ( ) / G , then i ( Q ) isa stratum in the orbit type stratification of M , and the restriction i : Q → i ( Q ) is a biholomor-phism with respect to the complex structure I Q on Q coming from C and the natural complexstructure on i ( Q ) . Finally, Kuranishi maps preserve the stratifications in the following sense. Let [ A , Φ ] ∈ M such that ( A , Φ ) ∈ m − ( ) . From Section 2.1, we see that the Kuran-ishi map θ : Z → B ss induces a biholomorphism(2.7) ϕ : Z K C (cid:12) K C → U ⊂ M onto an open neighborhood U of [ A , Φ ] . Moreover, e U = Z K C (cid:12) K C is an open neigh-borhood of [ ] in ν − C ( ) (cid:12) K C . Now, we can stratify ν − C ( ) (cid:12) K C by K C -orbit types,since there is a homeomorphism ν − C ( ) ps / K C → ν − C ( ) (cid:12) K C . Here, ν − C ( ) ps is thesubspace of ν − C ( ) consisting of polystable points with respect to the K C -action. Simi-larly, we may stratify the singular hyperKähler quotient n − ( ) / K by K -orbit types. By[19], we obtain similar results for the stratifications on ν − C ( ) (cid:12) K C and n − ( ) / K by HE QUASI-PROJECTIVITY OF THE MODULI SPACE OF HIGGS BUNDLES 9 replacing M , m − ( ) / G C and i in Proposition 2.4 and 2.5 by ν − C ( ) ps / K C , n − ( ) / K and n − ( ) / K ∼ −→ ν − C ( ) ps / K C , respectively. Then, it is shown that the biholomor-phism ϕ : e U → U preserves the induced stratifications on e U and U . (Here, we mayneed to refine the induced stratifications into connected components if strata are notconnected.)Now, we start to prove some technical results that will be used later. Proposition 2.6. (1)
Every G C -stabilizer of a polystable Higgs bundle is connected. (2) There are finitely many strata in M .Proof. Let ( A , Φ ) be a polystable Higgs bundle. By definition, we may write(2.8) ( E A , Φ ) = ( E , Φ ) ⊕ m ⊕ · · · ⊕ ( E r , Φ r ) ⊕ m r , m i ≥ ( E , Φ ) , · · · , ( E r , Φ r ) are pairwise non-isomorphic stable Higgs bundles thathave the same slope as ( E A , Φ ) , and ( E A , Φ ) is the Higgs bundle determined by ( A , Φ ) .As a consequence,(2.9) ( G C ) ( A , Φ ) = r ∏ i = GL ( m i , C ) .This proves ( ) and ( ) . Here, we have used the fact that if f is a morphism betweentwo stable Higgs bundles of the same slope, then either f ≡ f is an isomorphism.Moreover, every endomorphism of a stable Higgs bundle must be a scalar. (cid:3) Proposition 2.7.
Let Q and Q be two strata in M . If Q ⊂ Q , then dim Q > dim Q .Proof. Since Kuranishi maps preserve the orbit type stratifications, this problem canbe transferred to ν − C ( ) (cid:12) K C . Write(2.10) H = F ⊕ ( H ) K C ,where F is the ω C -orthogonal complement of ( H ) K C . By definition of ν C ,(2.11) ν − C ( ) = ( ν C | F ) − ( ) × ( H ) K C so that(2.12) ν − C ( ) (cid:12) K C = ( ν C | F ) − ( ) (cid:12) K C × ( H ) K C .Therefore, it is clear that the unique stratum containing [ ] is ( H ) K C . If L is a propersubgroup of K C , then(2.13) ( ν − C ( ) (cid:12) K C ) ( L ) = (( ν C | F ) − ( ) (cid:12) K C ) ( L ) × ( H ) K C ,where the subscript ( L ) denote the orbit type stratum determined by ( L ) . As a conse-quence,(2.14) dim ( ν − C ( ) (cid:12) K C ) ( L ) = dim (( ν C | F ) − ( ) (cid:12) K C ) ( L ) + dim ( H ) K C .Now, we claim that if F = (( ν C | F ) − ( ) (cid:12) K C ) ( L ) = ∅ , then(2.15) dim (( ν C | F ) − ( ) (cid:12) K C ) ( L ) > Q . Hence, Q is asingleton, and its preimage in ( ν C | F ) − ( ) ps is a single K C -orbit xK C for some x = L -norm k · k L to the orbit xK C attainsa minimum value r >
0. Therefore, we may assume that k x k L = r . Now, we showthat if t x and t x are in the same K C -orbit for some t , t >
0, then t = t . In fact, if t x = gt x for some g ∈ K C , then t r = t k gx k L ≥ t r so that t ≥ t . Applying thesame argument to g − t x = t x , we obtain that t ≤ t . Since the K C -action is linear, tx is also polystable and has the same orbit type of x for every t ∈ (
0, 1 ] . Therefore, Q contains a subspace { [ tx ] : t ∈ (
0, 1 ] } , which is a contradiction. (cid:3) Using Proposition 2.7, we can show that closures of strata are closed complex sub-spaces of M . Proposition 2.8.
If Q is a stratum in M , then Q is a closed complex subspace of M , and dim x Q = dim Q for every x ∈ Q.Proof.
We prove by induction. Note that every stratum is pure dimensional. By Propo-sition 2.6, let d < d < · · · < d k be possible values of dimensions among all thestrata. By Proposition 2.7, every stratum Q of dimension d is closed, and hencedim x Q = dim Q for all x ∈ Q . Now, suppose that the statement is true for all thestrata of dimensions smaller than d i . Let Q be a stratum of dimension d i . Therefore, Q is a closed complex subspace of M \ ∂ Q . Write(2.16) ∂ Q = Q l ∪ · · · ∪ Q l k = Q l ∪ · · · ∪ Q l k ,where each Q l i is a stratum of dimension smaller than d i . By induction, each Q l i isa closed complex subspace and dim x Q l i = dim Q l i for all x ∈ Q l i . Hence, dim Q > dim ∂ Q . By the Remmert-Stein theorem, Q is a closed complex subspace. Now weshow that dim x Q = dim Q for every x ∈ Q to finish the proof. If x ∈ Q , then theopenness of Q in Q implies that dim x Q = dim x Q . Therefore, we may assume that x ∈ ∂ Q . Since Q is open and dense in Q , ∂ Q is nowhere dense. Hence, by [11, Lemmaof Ritt], dim x ∂ Q < dim x Q . Let S be an irreducible component of Q containing x suchthat dim S = dim x Q . If S ⊂ ∂ Q , then dim S ≤ dim x ∂ Q , which is a contradiction.Hence, S ∩ Q = ∅ . As a consequence, since Q is open in Q ,(2.17) dim x Q = dim S = dim x ( Q ∩ S ) ≤ dim Q .Now, by the upper semicontinuity of the function x dim x Q (see [11, p.94]), there isan open neighborhood U of x in Q such that dim y Q ≤ dim x Q for all y ∈ U . Since Q is open and dense in Q , we may choose y ∈ U ∩ Q . Hence,(2.18) dim Q = dim y ( U ∩ Q ) = dim y Q ≤ dim x Q .Hence, dim Q = dim x Q . (cid:3) As a corollary, we obtain a codimension estimate of M \ M s , where M s is the modulispace of stable Higgs bundles. Although this is a well-known result (see [6, TheoremII.6] and [26, Lemma 11.2]), we couldn’t find an analytic proof in the literature. Corollary 2.9. M s is open and dense in M , and codim x ( M \ M s ) ≥ g − for everyx ∈ M \ M s , where g is the genus of the Riemann surface M.Proof. The first statement follows from [29, Corollary 3.24] and Proposition 2.5. Toshow the second statement, we write(2.19) M \ M s = Q ∪ Q ∪ · · · ∪ Q k = Q ∪ · · · ∪ Q k ,where each Q i is a stratum. As a consequence,(2.20) dim x ( M \ M s ) = dim x Q j = dim Q j HE QUASI-PROJECTIVITY OF THE MODULI SPACE OF HIGGS BUNDLES 11 for some j (depending on x ). Therefore, by Proposition 2.8, we obtaincodim x ( M \ M s ) = dim x M − dim x ( M \ M s )= dim M s − dim Q j .(2.21)By [29, Corollary 3.24] again, dim M s − dim Q j ≥ g − (cid:3)
3. I nfinite dimensional
GIT quotient
S-equivalence classes and closures of orbits.
In Section 3, we will prove Theo-rem B. We start with a simple lemma in point-set topology.
Lemma 3.1.
Let Y be a first countable topological space on which a topological group G acts.Let x n be a sequence in Y such that [ x n ] converges to [ x ] for some x ∈ Y in Y / G. Then, thereexists a subsequence x n k and a sequence g k ∈ G such that x n k g k converges to x in Y.Proof. Let π : Y → Y / G be the quotient map. Since Y is first countable, we can findnested open neighborhoods(3.1) · · · ⊂ U k ⊂ U k − ⊂ · · · ⊂ U of x that form a neighborhood basis. Now, for each U k , π ( U k ) is open and contains [ x ] .Hence, there exists some [ x n k ] ∈ π ( U k ) . Therefore, there exists some g k ∈ G such that x n k g k ∈ U k . Now we claim that x n k g k converges to x . Let V be an open neighborhoodof x . Then, U j ⊂ V for some j . If k > j , then x n k g k ∈ U k ⊂ U j ⊂ V . (cid:3) Then, among other properties, we first show that π identify G C -orbits whose clo-sures in B ss intersect. Proposition 3.2.
Two semistable Higgs bundles are S-equivalent if and only if the closures oftheir G C -orbits in B ss intersect.Proof. Let r : B ss → m − ( ) be the retraction defined by the Yang-Mills-Higgs flow.Since the Yang-Mills-Higgs flow preserves G C -orbits, if ( A , Φ ) is a semistable Higgsbundle, then r ( A , Φ ) is contained in ( A , Φ ) G C . Moreover, by [31, Theorem 1.4], r ( A , Φ ) is isomorphic to Gr ( A , Φ ) . Therefore, if ( A , Φ ) and ( A , Φ ) are two semistableHiggs bundles that are S -equivalent, then r ( A , Φ ) is isomorphic to r ( A , Φ ) . There-fore,(3.2) ( A , Φ ) G C ∋ r ( A , Φ ) ∼ G C r ( A , Φ ) ∈ ( A , Φ ) G C ,where ∼ G C means the equivalence relation induced by the G C -action.Conversely, suppose that(3.3) ( B , Ψ ) ∈ ( A , Φ ) G C ∩ ( A , Φ ) G C ∩ B ss .By replacing ( B , Ψ ) by r ( B , Ψ ) , we may assume that ( B , Ψ ) is polystable. Now, r ( A , Φ ) is also polystable and contained in ( A , Φ ) G C . Since ( A , Φ ) G C contains a uniquepolystable orbit (see [7, Lemma 3.7]), r ( A , Φ ) is isomorphic to ( B , Ψ ) . Similar argu-ment shows that r ( A , Φ ) is isomorphic to ( B , Ψ ) . Since r ( A i , Φ i ) is further isomor-phic to Gr ( A i , Φ i ) for i =
1, 2. We see that ( A , Φ ) and ( A , Φ ) are S -equivalent. (cid:3) Using the local slice theorem (Proposition 2.3) for the G C -action, we are able toprove that polystable orbits in B ss are exactly closed orbits. Proposition 3.3.
A semistable Higgs bundle is polystable if and only if its G C -orbit is closedin B ss . Proof.
The same proof of [27, Proposition 2.4(ii)] works. For the sake of complete-ness, we spell out the details. Let ( A , Φ ) be a semistable Higgs bundle and r : B ss → m − ( ) be the retraction defined by the Yang-Mills-Higgs flow. Since the Yang-Mills-Higgs flow preserves the G C -orbits, if ( A , Φ ) G C is closed in B ss , then obvi-ously r ( A , Φ ) ∈ ( A , Φ ) G C . This means that ( A , Φ ) is polystable. Conversely, as-sume that ( A , Φ ) is polystable. By the Hitchin-Kobayashi correspondence, we mayassume that ( A , Φ ) lies in m − ( ) . Let ( A , Φ ) g i be a sequence converging to some ( B , Ψ ) ∈ B ss . Since ( A , Φ ) is polystable, r [( A , Φ ) g i ] is isomorphic to ( A , Φ ) . Bythe Hitchin-Kobayashi correspondence, r [( A , Φ ) g i ] ∈ ( A , Φ ) G . Moreover, by continu-ity, r [( A , Φ ) g i ] converges to r ( B , Ψ ) . Since the G -action is proper, ( A , Φ ) G is closed,and hence r ( B , Ψ ) ∈ ( A , Φ ) G . On the other hand, r ( B , Ψ ) ∈ ( B , Ψ ) G C , and hence ( A , Φ ) ∈ ( B , Ψ ) G C . By Proposition 2.2, there is an G C -invariant open neighborhood U of ( A , Φ ) such that Z K C × K C G C → U is a homeomorphism. As a consequence, ( B , Ψ ) ∈ U .Then, it suffices to show that ( A , Φ ) G C is closed in U . By the homeomorphism Z K C × K C G C → U , it suffices to prove that if [ g i ] converges to [ x , g ] , then x =
0. ByLemma 3.1, there is a subsequence g i k and a sequence h k ∈ K C such that ( · h − k , h k g i k ) converges to ( x , g ) . This immediately shows that x = (cid:3) The following result allows us to identify M with B ss (cid:12) G C . From now on, we willuse [ · ] S and [ · ] to denote S-equivalence classes and isomorphism classes, respectively. Proposition 3.4.
The inclusion B ps ֒ → B ss induces a homeomorphism (3.4) B ps / G C ∼ −→ B ss (cid:12) G C . Proof.
Let r : B ss → m − ( ) be the retraction defined by the Yang-Mills-Higgs flow.We claim that r induces the inverse of the map(3.5) j : B ps / G C ∼ −→ B ss (cid:12) G C ,where j : B ps ֒ → B ss is the inclusion. By definition of the S -equivalence, r induces awell-defined continuous map(3.6) r : B ss (cid:12) G C → B ps / G C [ A , Φ ] S [ r ( A , Φ )] .Then, if ( A , Φ ) is a polystable Higgs bundle,(3.7) rj [ A , Φ ] = r [ A , Φ ] S = [ r ( A , Φ )] .Since ( A , Φ ) is polystable, it is isomorphic to the graded object Gr ( A , Φ ) and hence to r ( A , Φ ) . Therefore, [ r ( A , Φ )] = [ A , Φ ] . Conversely, if ( A , Φ ) is semistable, then(3.8) jr [ A , Φ ] S = j [ r ( A , Φ )] = [ r ( A , Φ )] S .By definition of the S -equivalence, ( A , Φ ) is S -equivalent to r ( A , Φ ) , since r ( A , Φ ) isisomorphic to Gr ( A , Φ ) . Hence, [ r ( A , Φ )] S = [ A , Φ ] S . (cid:3) Corollary 3.5.
Every fiber of π : B ss → B ss (cid:12) G C contains a unique G C -orbit that is closed in B ss .Proof. This follows from Proposition 3.3 and 3.4. (cid:3) π -saturated open neighborhoods. To further study the quotient map π : B ss → M , we will improve the local slice theorem (Proposition 2.3) so that the open neigh-borhood in B ss provided by the theorem is not only G C -invariant but also saturatedwith respect to π . Lemma 3.6.
Let U be a G C -invariant open subset of B ss . Then the following are equivalent HE QUASI-PROJECTIVITY OF THE MODULI SPACE OF HIGGS BUNDLES 13 (1) If ( A , Φ ) ∈ U, then the closure of its G C -orbit in B ss is contained in U. (2) U is π -saturated.Proof. By Proposition 3.2, ( ) implies ( ) . To show that ( ) implies ( ) , suppose that ( B , Φ ) ∈ U and ( B ′ , Φ ′ ) ∈ B ss such that π ( B , Ψ ) = π ( B ′ , Ψ ′ ) . We need to show that ( B ′ , Ψ ′ ) ∈ U . By Corollary 3.5, there exists a polystable Higgs bundle ( B ′′ , Ψ ′′ ) suchthat(3.9) ( B ′′ , Ψ ′′ ) G C ⊂ ( B , Ψ ) G C ∩ ( B ′ , Ψ ′ ) G C ∩ B ss .By assumption ( ) , ( B ′′ , Ψ ′′ ) ∈ U . If ( B ′ , Ψ ′ ) / ∈ U , then(3.10) ( B ′ , Ψ ′ ) G C ∩ U ∩ B ss = ∅ .This is a contradiction. (cid:3) Proposition 3.7.
Let ( A , Φ ) be a polystable Higgs bundle. Then, every G C -invariant openneighborhood of ( A , Φ ) in B ss contains a π -saturated open neighborhood.Proof. Let U be an G C -invariant open neighborhood of ( A , Φ ) in B ss . Take a neighbor-hood basis V n of [ A , Φ ] S in B ss (cid:12) G C such that V n ⊂ V n − for all n ≥
1. We claim that π − ( V n ) is contained in U for some n , where π : B ss → B ss (cid:12) G C is the quotient map.Assuming the contrary, we can choose a sequence ( A n , Φ n ) such that(1) ( A n , Φ n ) / ∈ U .(2) [ A n , Φ n ] S converges to [ A , Φ ] S in M .Since U is G C -invariant, the closure of ( A n , Φ n ) G C in B ss is contained in B ss \ U . Then,since the closure of ( A n , Φ n ) G C in B ss contains a unique polystable orbit (Proposi-tion 3.5), we may assume that each ( A n , Φ n ) is polystable. As a consequence, [ A n , Φ n ] converges to [ A , Φ ] in B ps / G C . By Lemma 3.1, there is a subsequence ( A n k , Φ n k ) anda sequence g k ∈ G C such that ( A n k , Φ n k ) · g k converges to ( A , Φ ) . This is impossible,since ( A n k , Φ n k ) · g k / ∈ U . (cid:3) Theorem 3.8.
Let ( A , Φ ) ∈ m − ( ) . If B is sufficiently small, then the map (3.11) θ : Z K C × K C G C → B ss [ x , g ] θ ( x ) gis a homeomorphism onto an π -saturated open neighborhood of ( A , Φ ) in B ss .Proof. Let U be the open neighborhood of ( A , Φ ) in B ss provided by Proposition 2.2.By Proposition 3.7, let U ′ be an open neighborhood of ( A , Φ ) in B ss that is π -saturatedand contained in U . Then, there is a K C -invariant open neighborhood Q of 0 in H such that θ maps ( Z K C ∩ Q ) × K C G C into U ′ . Now, let B ′ be an open ball around 0 in H such that B ′ ⊂ B ∩ Q . Then, we see that(3.12) ( B ′ ∩ Z ) K C ⊂ ( Z ∩ Q ) K C ⊂ Z K C ∩ Q .Let Z ′ = B ′ ∩ ν − C ( ) = B ′ ∩ Z , and we have Z ′ ⊂ Z . Let U ′′ be the image of Z ′ K C × K C G C under θ . Then(3.13) θ : Z ′ K C × K C G C → U ′′ is a homeomorphism, and U ′′ is contained in U ′ .We prove that U ′′ is π -saturated. Let θ ( x ) g ∈ U ′′ for some x ∈ Z ′ and g ∈ G C . ByLemma 3.6, we need to show that the closure of θ ( x ) g G C = θ ( x ) G C in B ss is containedin U ′′ . Let g n be a sequence in G C such that θ ( x ) g n converges in B ss . Since U ′ is π -saturated, the limiting point is in U ′ , and we may assume that it is θ ( y ) h for some y ∈ Z and h ∈ G C . Since θ is a homeomorphism, we see that [ x , g n ] converges to [ y , h ] in Z K C × K C G C . By Lemma 3.1, there is a subsequence g n j ∈ G C and a sequence k j ∈ K C such that ( xk − j , k j g n j ) converges to ( y , h ) in Z K C × G C . By [28, Corollary 4.9], Z ′ K C is saturated with respect to the quotient map Z K C → Z K C (cid:12) K C . Hence y ∈ Z ′ so that [ y , h ] ∈ Z ′ K C × K C G C and θ ( y ) h ∈ U ′′ . (cid:3) Now we can obtain Kuranishi local models for B ss (cid:12) G C in the following way. Fix [ A , Φ ] S ∈ B ss (cid:12) G C such that ( A , Φ ) ∈ m − ( ) . By Theorem 3.8, the natural map(3.14) θ : Z K C × K C G C → U is a homeomorphism onto an π -saturated open neighborhood U of ( A , Φ ) . By theresults in Section 2.1, θ induces a well-defined map(3.15) ϕ : Z K C (cid:12) K C → π ( U ) ⊂ B ss (cid:12) G C [ x ] [ θ ( x )] S ,and π ( U ) is an open neighborhood of [ A , Φ ] S in B ss (cid:12) G C . By Proposition 3.4, we seethat it is a biholomorphism.Moreover, we can also describe the structure sheaf of M in the following way. Proposition 3.9.
The structure sheaf of M is equal to π ∗ O G C B . In other words, for any opensubset V in M , the natural map π ∗ : O ( V ) O ( π − V ) G C is a bijection.Proof. It suffices to prove the following. Let ( A , Φ ) ∈ m − ( ) . By Theorem 3.8 and theremark after it, we see that the natural map(3.16) θ : Z K C × K C G C → U is a homeomorphism onto an π -saturated open neighborhood of ( A , Φ ) in B ss . More-over, it induces a biholomorphic map ϕ : Z K C (cid:12) K C → π ( U ) . By the definition ofthe structure sheaf of B , we easily see that θ is actually a biholomorphism. As aconsequence, there is a chain of isomorphisms(3.17) O ( π ( U )) ϕ ∗ −→ O ( Z K C (cid:12) K C ) π ∗ −→ O ( Z K C ) K C ∼ −→ O ( Z K C × K C G C ) G C θ − −→ O ( U ) G C .Moreover, the composition is exactly π ∗ : O ( π ( U )) → O ( U ) G C . (cid:3) As a corollary, the quotient map π : B ss → B ss (cid:12) G C is a categorical quotient in thefollowing sense. Corollary 3.10.
Let Z be a complex space and g : B ss → Z a G C -invariant holomorphic map.Then, g induces a unique holomorphic map g : M → Z.Proof.
Define g [ A , Φ ] S = g ( A , Φ ) . By Proposition 3.2, it is well-defined. The holomor-phicity of f follows from Proposition 3.9. (cid:3) Proof of Theorem B.
This follows from Proposition 3.2, 3.3, 3.5, 3.9 and Corollary 3.10. (cid:3)
4. D escent lemmas for vector bundles
In this section, we will first generalize the descent lemma for vector bundles in [5,Theorem 2.3] to analytic Hilbert quotients, and then prove a similar descent lemmafor the quotient map π : B ss → B ss (cid:12) G C .Let G be a complex reductive Lie group acting holomorphically on a complex space X . Suppose that X admits an analytic Hilbert quotient. In other words, there is asurjective G -invariant holomorphic map π : X → X (cid:12) G such that(1) π is Stein in the sense that inverse images of Stein subspaces are Stein.(2) O X (cid:12) G = π ∗ O GX . In other words, for every open subset U of X (cid:12) G , the map π ∗ : O X (cid:12) G ( U ) → O X ( π − U ) G is an isomorphism. HE QUASI-PROJECTIVITY OF THE MODULI SPACE OF HIGGS BUNDLES 15
Proposition 4.1.
Let E → X be a holomorphic G-bundle over X. If G x acts trivially on thefiber E x for every x ∈ X whose G-orbit is closed, then there is a vector bundle F → X (cid:12) Gsuch that π ∗ F = E. Moreover, O ( F ) = π ∗ O ( E ) G , where O ( F ) and O ( E ) are the sheaves ofholomorphic sections of F and E, respectively.Proof. We closely follow the proof of [5, Theorem 2.3]. Fix x ∈ X such that Gx is closedin X . Choose a basis σ , · · · , σ r for E x . Then, we may consider the map(4.1) s i : G / G x → E s i ( gG x ) = g σ i .Since G x acts trivially on E x , s i is well-defined and holomorphic. Now, choose an openStein neighborhood U of π ( x ) . Since π is an analytic Hilbert quotient, π − ( U ) is anopen Stein neighborhood containing Gx . Since Gx is closed in π − ( U ) , by [13, Propo-sition 3.1.1], Gx is a closed complex subspace of π − ( U ) . Since G acts transitively on Gx , Gx is smooth. Therefore, the natural map G / G x → Gx is a biholomorphism. Asa consequence, we obtain a G -equivariant map(4.2) s i : Gx → E s i ( gx ) = g σ i ,which is a holomorphic section of E over Gx . Since Gx is a closed complex subspaceof π − ( U ) , and π − ( U ) is Stein, s i can be extended to a holomorphic section of E over π − ( U ) . By averaging over a maximal compact subgroup K of G , we may assume thateach s i is K -equivariant. Since G is the complexification of K , the argument in the proofof [23, Theorem 1.1] shows that each s i is also G -equivariant. Since G x acts triviallyon E x , s i ( x ) = σ i , and hence { s i } is linearly independent over an open neighborhood V of x in π − ( U ) . Since each s i is G -equivariant, V can be chosen to be G -invariant.By [19, Proposition 3.10], we may further assume that V = π − ( U ′ ) for some smalleropen neighborhood U ′ ⊂ U of π ( x ) in M .Now, note that every fiber of π contains a unique closed orbit ([16, §3, Corollary3]). Therefore, the argument in the above paragraph provides an open covering U i of X (cid:12) G such that E is trivial over π − ( U i ) , and the transition functions g ij : π − ( U i ) ∩ π − ( U j ) → GL n ( C ) are G -invariant. As a consequence, by the definition of thestructure sheaf of X (cid:12) G , they descend to holomorphic functions e g ij : π ( π − ( U i ) ∩ π − ( U j )) → GL n ( C ) . Since every fiber of π contains a unique closed orbit, π ( π − ( U i ) ∩ π − ( U j )) = U i ∩ U j . Then, the data { e g ij , U i } defines a holomorphic vector bundle F over X (cid:12) G . It is easy to see that π ∗ F = E and O ( F ) = π ∗ O ( E ) G . (cid:3) Proof of Theorem C.
Fix ( A , Φ ) ∈ m − ( ) . By Theorem 3.8, the map(4.3) θ : Z K C × K C G C → B ss induced by the Kuranishi map θ : Z K C → B ss for ( A , Φ ) is a G C -equivariant homeo-morphism onto a π -saturated open neighborhood of ( A , Φ ) in B ss . Then, we considerthe pullback bundle θ ∗ L on Z K C × K C G C . Clearly, θ ∗ L is the restriction of θ ∗ L to Z K C . By Proposition 2.2, if x ∈ Z has a closed K C -orbit, then θ ( x ) is polystable, and ( K C ) x = ( G C ) θ ( x ) . Therefore, ( K C ) x acts trivially on the fiber ( θ ∗ L ) x = L θ ( x ) for every x ∈ Z that has a closed K C -orbit. By Proposition 4.1, the bundle θ ∗ L descends to Z K C (cid:12) K C . By shrinking Z if necessary, we may assume that the descended bundleis trivial over Z K C (cid:12) K C . As a consequence, there is a holomorphic frame { σ i } for θ ∗ L over Z K C such that each σ i is K C -equivariant. Hence, each section σ i extends to a G C -equivariant holomorphic section of θ ∗ L over Z K C × K C G C . Transported back to B ss by θ , we obtain a local frame { σ i } for L → B ss such that each σ i is a G C -equivariantholomorphic section over a π -saturated open neighborhood of ( A , Φ ) in B ss . The restfollows from the second paragraph in the proof of Proposition 4.1. (cid:3)
5. T he K ähler metric on the moduli space In this section, we will prove Theorem E. Let us start with the construction of a linebundle on the moduli space M . By [4], there is a holomorphic Hermitian line bundle L over A such that the curvature of the Hermitian metric is precisely − π √− Ω ,where(5.1) Ω ( α , α ) = π Z X tr ( α ∧ α ) , α , α ∈ Ω ( g E ) .Moreover, the G C -action on A lifts to L , and the G -action preserves the Hermitianmetric. The vertical part of the infinitesimal action of ξ ∈ Ω ( g E ) on a smooth section s of L is given by 2 π √− h µ , ξ i s , where ξ is the vector field on A generated by ξ ,and(5.2) µ ( A ) = π F A .On the other hand, we consider the trivial line bundle Ω ( g C E ) × C over Ω ( g C E ) . AKähler potential on Ω ( g C E ) is given by(5.3) ρ ( Φ ) = π k Φ k L .Letting Ω = √− ∂∂ρ , we see that Ω + Ω = Ω I on C . Let s ( Φ ) = ( Φ , 1 ) be thecanonical section of the trivial line bundle Ω ( g C E ) × C . Setting(5.4) | s | = exp ( − πρ ) ,we obtain a Hermitian metric on the trivial line bundle Ω ( g C E ) × C such that itscurvature is − π √− Ω . Letting G C act on C trivially, we see that the G C -action on Ω ( g C E ) lifts to the trivial line bundle Ω ( g C E ) × C . Moreover, the induced G -actionpreserves the Hermitian metric, since ρ is G -invariant. Finally, the vertical part of theinfinitesimal action of ξ ∈ Ω ( g E ) on s is given by 2 π √− h µ , ξ i s , where(5.5) µ ( Φ ) = π [ Φ , Φ ∗ ] .Now, we pullback the line bundle L on A and the trivial line bundle on Ω ( g C E ) to C = A × Ω ( g C E ) , and denote the resulting line bundle still by L . We equip L with theproduct of the pullback Hermitian metrics. As a consequence, the G C -action on C liftsto L , and the G -action still preserves the resulting Hermitian metric. The curvature ofthis Hermitian metric is precisely(5.6) − π √− ( Ω + Ω ) = − π √− Ω I .Moreover, the vertical part of the infinitesimal action of ξ ∈ Ω ( g E ) on a smoothsection s of L is given by 2 π √− h µ + µ , ξ i s , and µ + µ = µ on C . The followingshows that the restriction of the line bundle L to B ss descends to the moduli space M . Proposition 5.1.
There is a line bundle L → M such that π ∗ L = L | B ss , and O ( L ) = π ∗ O ( L | B ss ) G C .Proof. We follow the proof of [27, Proposition 2.14]. By Theorem C, it suffices to provethat G C ( A , Φ ) acts on L ( A , Φ ) trivially for every ( A , Φ ) ∈ m − ( ) . If ξ ∈ Lie ( G ( A , Φ ) ) and s ∈ L ( A , Φ ) , then(5.7) ξ · s = π √− h µ , ξ i s . HE QUASI-PROJECTIVITY OF THE MODULI SPACE OF HIGGS BUNDLES 17
Therefore, if µ ( A , Φ ) =
0, then ξ · s =
0. Since G -stabilizers are connected (Propo-sition 2.6), we conclude that G ( A , Φ ) acts trivially on L ( A , Φ ) . Since ( G C ) ( A , Φ ) is thecomplexification of G ( A , Φ ) , we conclude that G C ( A , Φ ) acts trivially on L ( A , Φ ) . (cid:3) Now, we show that the moduli space M admits a weak Kähler metric. Let Q bea stratum in the orbit type stratification of M . By Proposition 2.5, i − ( Q ) is a stra-tum in the orbit type stratification of m − ( ) / G , where i : m − ( ) / G ∼ −→ B ps / G C isthe Hitchin-Kobayashi correspondence. By Proposition 2.4, i − ( Q ) is a hyperKählermanifold. Let ω i − Q be the Kähler form on i − ( Q ) induced by the Kähler form Ω I on C . Then, ( i − ) ∗ ω i − Q is a Kähler form on Q . Therefore, every stratum in M is a Kählermanifold.Let [ A , Φ ] S ∈ M . By Proposition 5.1, we may choose a G C -equivariant holomorphicsection s of L over an π -saturated open neighborhood π − ( U ) of ( A , Φ ) in B ss suchthat s vanishes nowhere in π − ( U ) , where U is an open neighborhood of [ A , Φ ] S in M . Then, we define(5.8) u = − π log | s | h ,where h is the Hermitian metric on L . Since h is preserved by the G -action, u is G -invariant. As a consequence, the restriction of u to π − ( U ) ∩ m − ( ) induces awell-defined continuous function u : U → R . Proposition 5.2.
The function u is continuous and smooth along each stratum Q. Moreover,u | Q is a Kähler potential for the Kähler form on each stratum Q in M . In particular, u is acontinuous plurisubharmonic function.Proof. By Proposition 2.4, π − ( Q ) ∩ m − ( ) → Q is a submersion. Hence, u is smoothalong Q . By construction of u and the Hermitian metric h on L ,(5.9) √− ∂∂ ( u | π − ( Q ) ∩ m − ( ) ) = Ω I | π − ( Q ) ∩ m − ( ) .Therefore, the second statement follows from the construction of the Kähler form on Q . Now we have shown that the restriction of u to M s is strictly plurisubharmonic.Since it is continuous, by the normality of M and the extension theorem of plurisub-harmonic functions (see [10]), we conclude that u is plurisubharmonic. (cid:3) Proof of Theorem E.
By Proposition 5.2, there is an open covering U i of M , and a stratum-wise strictly plurisubharmonic function ρ i : U i → R on each U i such that ρ i | M s − ρ j | M s is pluriharmonic on M s ∩ U i ∩ U j . Hence, we may write(5.10) ρ i | M s − ρ j | M s = ℜ ( f ij ) for some holomorphic function f ij : U i ∩ U j ∩ M s → C . By Corollary 2.9 and thenormality of M , f ij has a unique holomorphic extension to U i ∩ U j . Then, we have(5.11) ρ i − ρ j = ℜ ( f ij ) on U i ∩ U j .Hence, { U i , ρ i } determines a weak Kähler metric on M . (cid:3)
6. P rojective compactification
Symplectic cuts.
In this section, we will use the symplectic cut to compactify themoduli space and thus prove Theorem A. Recall that there is a holomorphic C ∗ -actionon C given by(6.1) t · ( A , Φ ) = ( A , t Φ ) , t ∈ C ∗ , ( A , Φ ) ∈ C . Clearly, B ss is C ∗ -invariant. Then, it is easy to verify that the natural map C ∗ × B ss → C ∗ × M satisfies ( ) in Theorem B, where G C acts on C ∗ trivially. Since the G C -actionand the C ∗ -action on C commute, we see that the holomorphic action C ∗ × B ss → B ss descends to a holomorphic action C ∗ × M → M . Moreover, each stratum in the orbittype stratification of M is C ∗ -invariant.Furthermore, the induced U ( ) -action on M is stratum-wise Hamiltonian. To seethis, we first note that U ( ) preserves the Kähler form Ω I on C . Then, consider thefunction f : C → R given by(6.2) f ( A , Φ ) = − π k Φ k L . Proposition 6.1.
The restriction of f to m − ( ) defines a continuous function, denoted bythe same letter f , on M that is smooth along each stratum Q of M . Moreover, the restrictionf | Q is a moment map for the U ( ) -action on Q with respect to the Kähler form on Q, the oneinduced by the Kähler form Ω I on C .Proof. It is shown in [17, p.92] that f is a moment map for the U ( ) -action on C withrespect to the Kähler form Ω I . Since f is G -invariant, its restriction to m − ( ) descendsto m − ( ) / G and hence defines a continuous function on M , which we denote by thesame letter f . Let Q be a stratum in the moduli space. By Proposition 2.4, the restric-tion of f to π − ( Q ) ∩ m − ( ) descends to a smooth function on Q which is preciselythe restriction of f : M → R to Q . Since the quotient map π − ( Q ) ∩ m − ( ) → Q is U ( ) -equivariant, we conclude that f | Q is a moment map for the U ( ) -action on Q . (cid:3) To perform the symplectic cut of M , we consider the direct product M × C and let C ∗ act on C by multiplication. Hence, C ∗ acts diagonally on M × C . Moreover, M × C admits a stratification such that each stratum Q × C is equipped with the productKähler form. The next result implies that the induced U ( ) action on M × C is alsostratum-wise Hamiltonian. Proposition 6.2.
The continuous map (6.3) e f ([ A , Φ ] S , z ) = f ([ A , Φ ] S ) − k z k is smooth along each stratum Q × C , and its restriction to Q × C is a moment map for theinduced U ( ) -action on Q × C with respect to the product Kähler form on Q × C .Proof. It is clear that e f is continuous on M × C . For each stratum Q , Proposition 6.1implies that f | Q is a smooth moment map on Q . Since U ( ) acts diagonally on Q × C ,it is easy to see that e f | Q × C is a moment map for the U ( ) -action with respect to theproduct Kähler form on Q × C . Therefore, e f is a stratum-wise moment map. (cid:3) Now we recall the definition of the Hitchin fibration. Given a Higgs bundle ( A , Φ ) ,the coefficient of λ n − i in the characteristic polynomial det ( λ + Φ ) is a holomorphic sec-tion of K iM , where n is the rank of E , i = · · · , n , and K M is the canonical bundle onthe Riemann surface M . Since these sections are clearly G C -invariant, by Theorem B,we have obtained a well-defined holomorphic map, called the Hitchin fibration,(6.4) h : M → n M i = H ( M , K iM ) .It is known that h is proper (see [30, Theorem 2.15] or [17, Theorem 8.1]). Therefore,the nilpotent cone h − ( ) is compact so that f has a lower bound on h − ( ) . We choose HE QUASI-PROJECTIVITY OF THE MODULI SPACE OF HIGGS BUNDLES 19 a constant c < h − ( ) ⊂ f − ( c , 0 ] . In other words, f − ( − ∞ , c ] does notcontain the nilpotent cone. Then, we perform the symplectic cut of M at the level c .By definition, it is the singular symplectic quotient(6.5) e f − ( c ) / U ( ) = n ([ A , Φ ] S , z ) ∈ M × C : f ([ A , Φ ] S ) − k z k = c o / U ( ) .If M × C admits a (strong) Kähler metric, then we may directly apply the analytic GITdeveloped in [15]. Since we are unable to prove this, we will have to take a detour toprove that the symplectic cut of M at the level c is a compact complex space.Let W = ( M × C ) \ ( h − ( ) × { } ) . It is clear that W is C ∗ -invariant and open. Wefirst show that the analytic Hilbert quotient W / C ∗ exists. Lemma 6.3.
The C ∗ -action on M \ h − ( ) is proper.Proof. Clearly, h − ( ) is C ∗ -invariant. Suppose that(1) x i converges to x ′ / ∈ h − ( ) .(2) t i · x i converges to y / ∈ h − ( ) .We first claim that | t i | cannot be unbounded. If not, we may assume that | t i | → ∞ andlet t ′ i = t i / | t i | . By passing to a subsequence, we may assume that t ′ i converges to t ′ ∞ ,and | t ′ ∞ | =
1. As a consequence, since x ′ / ∈ h − ( ) ,(6.6) lim i → ∞ t ′ i · x i = t ′ ∞ · x ′ / ∈ h − ( ) .On the other hand, since t i · x i converges,(6.7) lim i → ∞ h (cid:18) | t i | t i · x i (cid:19) = h is proper, by passing to a subsequence, we may assume that(6.8) 1 | t i | t i · x i = t ′ i · x i converges to an element in h − ( ) . This is a contradiction.Since t i is bounded, it has a subsequence that is convergent. We claim that such asequence cannot converge to 0. If not, suppose that t i →
0. Then,(6.9) lim i → ∞ h ( t i · x i ) = t i · x i has a subsequence converging to an element in h − ( ) . Therefore, y ∈ h − ( ) . This is a contradiction. (cid:3) Corollary 6.4.
The C ∗ -action on W is proper.Proof. Note that(6.10) W = ( M \ h − ( ) × C ) ∪ ( M × C ∗ ) .Suppose there are sequences(1) ( x i , a i ) ∈ W converges to ( x ′ , a ′ ) ∈ W (2) t i · ( x i , a i ) ∈ W converges to ( y , b ) ∈ W We need to show that t i has a subsequence that converges in C ∗ . We prove this byconsidering the following cases:(1) Suppose ( x ′ , a ′ ) ∈ M × C ∗ . Then, ( x i , a i ) ∈ M × C ∗ if i ≫
0. Hence, t i =( t i a i ) a − i converges to ba ′− . If b =
0, then ba ′− ∈ C ∗ , and we are done withthis case. If b =
0, then y / ∈ h − ( ) . Moreover, lim i → ∞ h ( t i x i ) =
0. Then, t i x i has a subsequence converging to an element in h − ( ) . Since this element hasto be y , we have shown that b = (2) Suppose that ( x ′ , a ′ ) ∈ ( M \ h − ( )) × C . Then, both ( x i , a i ) and t i · ( x i , a i ) liein ( M \ h − ( )) × C if i ≫
0. If y / ∈ h − ( ) , then Lemma 6.3 applies. Hence,we may assume that y ∈ h − ( ) and hence b =
0. If a ′ =
0, then t i = ( t i a i ) a − i converges to ba ′− , and we are done. Hence, we may assume that a ′ =
0, andtherefore t i a i → b = a i → a ′ = t i is bounded, so that t i a i converges to 0, which is a contradiction.If not, we may assume that | t i | → ∞ and let t ′ i = t i / | t i | . By passing to asubsequence, we may further assume that t ′ i converges to t ′ ∞ with | t ′ ∞ | =
1. Asa consequence, t ′ i x i converges to t ′ ∞ x ′ / ∈ h − ( ) . On the other hand,(6.12) lim i → ∞ h ( t ′ i x i ) = lim i → ∞ h (cid:18) | t i | t i x i (cid:19) = h implies that t ′ i x i contains a subsequence convergingto an element in h − ( ) , which is a contradiction. (cid:3) Corollary 6.5.
The analytic Hilbert quotient of W by C ∗ exists. Moreover, W / C ∗ is a geo-metric quotient.Proof. This follows from Corollary 6.4 and [16, §4, Corollary 2]. (cid:3)
Then, we study the relationship between the symplectic cut e f − ( c ) / U ( ) and theanalytic Hilbert quotient W / C ∗ . Let ( M × C ) ss be the semistable points in M × C determined by e f − c . In other words, ([ A , Φ ] S , z ) lies in ( M × C ) ss if and only if theclosure of its C ∗ -orbit in M × C intersects e f − ( c ) . Lemma 6.6. W = ( M × C ) ss = C ∗ · e f − ( c ) .Proof. We first show that e f − ( c ) ⊂ W . Suppose that this is not true, and we choosesome ([ A , Φ ] S , z ) ∈ e f − ( c ) such that ([ A , Φ ] S , z ) / ∈ W . In other words, [ A , Φ ] S ∈ h − ( ) and z =
0. Hence, e f ([ A , Φ ] S , z ) = f ([ A , Φ ] S ) = c . This cannot happen by the choice ofthe level c .Then, we show that ( M × C ) ss ⊂ W . If the closure of the C ∗ -orbit of a point ([ A , Φ ] S , z ) in M × C meets e f − ( c ) , then it must meet W , since W is open. Since W isalso C ∗ -invariant, W contains ([ A , Φ ] S , z ) .Finally, we show that W ⊂ ( M × C ) ss . For every ([ A , Φ ] S , z ) ∈ W , consider thefunction(6.13) q ( t ) = f ([ A , t Φ ]) − t k z k − c t > q ( t ) = t > t · ([ A , Φ ] S , z ) lies in e f − ( c ) for some t >
0. Since h ([ A , t Φ ]) → t →
0, the properness of h implies that there exists asequence t n ⊂ C ∗ such that t n → [ A , t n Φ ] converges to some [ B , Ψ ] S ∈ h − ( ) .Hence, letting n → ∞ , we see that(6.14) lim n → ∞ q ( t n ) = f ([ B , Ψ ] S ) − c > f ≤
0, we have(6.15) q ( t ) ≤ − t k z k − c . HE QUASI-PROJECTIVITY OF THE MODULI SPACE OF HIGGS BUNDLES 21 If z =
0, then t ≫ q ( t ) <
0. Hence, q ( t ) = t > z = [ A , Φ ] S / ∈ h − ( ) . We claim that the function t f ([ A , t Φ ] S ) is unbounded below as t → ∞ . If this claim is true, then q ( t ) < t ≫
0, and hence q ( t ) = t >
0. Now, we prove the claim. Assuming thecontrary, we may choose a sequence of { t n } ⊂ C ∗ such that t n → ∞ and f ([ A , t n Φ ]) isbounded. By the properness of f , by passing to a subsequence, we may assume that [ A , t n Φ ] converges to some [ B , Ψ ] S . Hence, h ([ A , t n Φ ]) also converges as t n → ∞ . Thisimplies that h ([ A , Φ ] S ) =
0, which is a contradiction.Finally, note that the proof has already shown that ( M × C ) ss = C ∗ · e f − ( ) . (cid:3) Corollary 6.7.
The inclusion e f − ( c ) ֒ → ( M × C ) ss induces a homeomorphism (6.16) e f − ( c ) / U ( ) ∼ −→ ( M × C ) ss / C ∗ = W / C ∗ . Moreover, W / C ∗ is compact.Proof. Since f and the norm k · k on C are proper, f − ( c ) is compact. Therefore, e f − ( c ) / U ( ) is also compact. Moreover, since ( M × C ) ss / C ∗ is Hausdorff, to showthat the map is a homeomorphism, it suffices to show that it is a continuous bijection.The continuity is obvious. By Lemma 6.6, the surjectivity is clear.To show the injectivity, suppose that ([ A , Φ ] , z ) and ([ A , Φ ] , z ) lie in e f − ( c ) and the same C ∗ -orbit. Since each orbit type stratum in M is C ∗ -invariant, they liein Q × C for some stratum Q in M . By Proposition 6.2, e f | Q × C is a moment map forthe U ( ) -action on Q × C with respect to the product Kähler form on Q × C . Hence, ([ A , Φ ] , z ) and ([ A , Φ ] , z ) must lie in the same U ( ) -orbit by general propertiesof moment maps (see [18, Lemma 7.2]). (cid:3) Proof of Theorem A.
Write W = ( M \ h − ( ) × { } ) ∪ ( M × C ∗ ) . Note that it is a disjointunion. Let W ∗ = M × C ∗ and consider the map(6.17) W ∗ → M , ([ A , Φ ] S , z ) z − [ A , Φ ] S .Since it is C ∗ -invariant, it induces a well-defined map ( M × C ∗ ) / C ∗ → M . The injec-tivity is clear. Its inverse is given by [ A , Φ ] S ([ A , Φ ] S , 1 ) .Then, we show that it is a biholomorphism. Since M is normal, M × C is alsonormal. Therefore, both W and W ∗ are normal. As categorical quotients of normalspaces, W ∗ / C ∗ and W / C ∗ are also normal. Moreover, fibers of W ∗ → W ∗ / C ∗ havepure dimension 1. Since M s is pure dimensional, Proposition 2.8 implies that M and hence W ∗ are pure dimensional. Therefore, by Remmert’s rank theorem (see [1,Proposition 1.21]), we conclude that W ∗ / C ∗ is pure dimensional, and(6.18) dim W ∗ / C ∗ = dim W ∗ − dim C ∗ = dim M .Then, by [11, p.166, Theorem], the map W ∗ / C ∗ → M is a biholomorphism.Since W / C ∗ is compact, we have shown that M admits a compactification(6.19) W / C ∗ = M ∪ Z ,where Z = ( M \ h − ( ) × { } ) / C ∗ is of pure codimension 1. (cid:3) Finally, we prove the following result that will be used later. Note that W inherits astratification from M × C . More precisely, W is a disjoint union of Q W = W ∩ ( Q × C ) where Q ranges in the stratification of M . Moreover, if necessary, we may also refinethis stratification into connected components. The following shows that how Q W / C ∗ fits together in M . Proposition 6.8.
Let π : W → M be the quotient map. (1) Each π ( Q W ) is a closed complex subspace of M , where the closure is taken in W. (2) Each π ( Q W ) is a locally closed complex subspace of M , and its closure is precisely π ( Q W ) . (3) If π ( Q W ) ∩ π ( S W ) = ∅ , then π ( Q W ) ⊂ π ( S W ) . (4) M is a disjoint union of π ( Q W ) . (5) The restriction π : Q W → π ( Q W ) is the analytic Hilbert quotient of Q W by C ∗ .Moreover, the inclusion ( e f | Q W ) − ( c ) ֒ → Q W induces a homeomorphism (6.20) ( e f | Q W ) − ( c ) / U ( ) ∼ −→ Q W / C ∗ . Proof.
Fix a stratum Q in M . By Proposition 2.8, Q W is a closed complex subspaceof W that is also C ∗ -invariant. Therefore, [16, §1(ii)] implies that π ( Q W ) is a closedcomplex subspace of M . Moreover, the restriction π : Q W → π ( Q W ) is also an analyticHilbert quotient. This proves ( ) .Since Q W is open in Q W , and π : Q W → π ( Q W ) is an open map, π ( Q W ) is open in π ( Q W ) . Moreover, the continuity of π shows that π ( Q W ) ⊂ π ( Q W ) . Since π ( Q W ) isclosed in M , we have π ( Q W ) = π ( Q W ) . This proves ( ) .If π ( Q W ) ∩ π ( S W ) = ∅ for some stratum S in M , then π ( Q W ) ∩ π ( S W ) = ∅ .Since π : W → M is a geometric quotient, and both Q W and S W are C ∗ -invariant, weconclude that Q W ∩ S W = ∅ . Therefore, Q W ⊂ S W , and hence π ( Q W ) ⊂ π ( S W ) . Thisshows ( ) .Obviously, M is a union of π ( Q W ) as Q ranges in the stratification of M . Since each Q W is C ∗ -invariant, and π : W → M is a geometric quotient, it is a disjoint union. Thisproves ( ) .Finally, ( ) immediately follows from Corollary 6.7 and the fact that Q W is C ∗ -invariant. (cid:3) Projectivity.
In this section, we will prove ( ) and ( ) in Theorem D. Let us startwith the construction of a line bundle on M = W / C ∗ . Note that the C ∗ -action on M lifts to the line bundle L → M . This can be seen as follows. The C ∗ -action on Ω ( g C E ) lifts to the trivial line bundle by letting C ∗ act on the fiber trivially. By constructionof the line bundle L → C , the C ∗ -action on C lifts to L . Since the G C -action and the C ∗ -action commutes, we see that the C ∗ -action on L descends to L which covers the C ∗ -action on M . Then, consider the trivial line bundle over C . The C ∗ -action on C liftsto the trivial line bundle by letting C ∗ act on the fiber trivially. Moreover, we equipthe trivial line bundle with a Hermitian metric determined by(6.21) | s | = exp ( − πχ ) ,where s ( z ) = ( z , 1 ) is a section of the trivial line bundle, and χ ( z ) = k z k is a Kählerpotential for the standard Kähler form on C .Now, we pullback the trivial line bundle on C and the line bundle L → M to M × C , and denote the resulting line bundle by L C . Moreover, we equip the linebundle L C → M × C with the product of the pullback Hermitian metrics, and the C ∗ -action on M × C lifts to L C . We will still use the letter h to denote the resultingHermitian metric on L C . Proposition 6.9.
The Hermitian metric h on L C is smooth along Q × C for every stratum Qin M . Moreover, its curvature on Q × C is precisely − π √− ω Q × C .Proof. By Proposition 5.2, the curvature of the Hermitian metric on L along Q is − π √− ω Q . By the construction of h on L C , we see that the curvature of h along Q × C must be − π √− ω Q × C . (cid:3) HE QUASI-PROJECTIVITY OF THE MODULI SPACE OF HIGGS BUNDLES 23
By construction, if p = ([ A , Φ ] S , z ) ∈ M × C , then ( C ∗ ) p acts trivially on ( L C ) p . Asa consequence, we obtain the following. Proposition 6.10.
The canonical line bundle L C → M × C descends to M . In other words,there is a line bundle L → M such that π ∗ L = L C | W , where π : W → W / C ∗ = M is thequotient map.Proof. This follows from Proposition 4.1. (cid:3)
Since the Hermitian metric h on L C is preserved by the U ( ) -action, Corollary 6.7implies that h induces a continuous Hermitian metric h on L → M . Although h issmooth along Q W for each stratum Q in M , h may not be smooth along π ( Q W ) . Thatsaid, we note that Q W is a Kähler manifold with a C ∗ -action such that the induced U ( ) -action is Hamiltonian with respect to the Kähler form ω Q W on Q W . Hence, by[27, Theorem 2.10] and Proposition 6.8, we may further stratify π ( Q W ) by C ∗ -orbittypes. Since the curvature of ( L C , h ) on Q W is − π √− ω Q W , by [27, Lemma 2.16], weconclude the following. Proposition 6.11.
For each stratum Q in M , π ( Q W ) admits a C ∗ -orbit type stratificationsuch that each stratum S is a locally closed Kähler submanifold of π ( Q W ) with Kähler form ω S . Moreover, the Hermitian metric h on L is smooth along S, and the its curvature on S isprecisely − π √− ω S . Now we are ready to prove that the line bundle L → M is ample. The first step isthe following. Lemma 6.12.
The Chern current c ( L , h ) of ( L , h ) is positive, where (6.22) c ( L , h ) = √− π ∂∂ log | s | h , and s is any local holomorphic section of L that is nowhere vanishing.Proof. By Proposition 6.10, for every open subset U of M , we may choose a C ∗ -equivariant holomorphic section s of L C over π − ( U ) that is nowhere vanishing,where π : W → M is the quotient map. Then, we define(6.23) v = − π log | s | h .Since v is U ( ) -invariant, by Corollary 6.7, the restriction of v to π − ( U ) ∩ e f − ( c ) in-duces a well-defined continuous function v : U → R . If Q = M s , then Proposition 6.8and 6.11 imply that π ( Q W ) is open in M and that π ( Q W ) admits a C ∗ -orbit type strat-ification. Moreover, if S is the top-dimensional stratum, then S is open and dense in π ( Q W ) . By Proposition 6.11 again, the restriction of v to S is a Kähler potential forthe Kähler form on S so that v | S is strictly plurisubharmonic. Since v is alreadycontinuous, and M is normal, the extension theorem of plurisubharmonic functions(see [10]) implies that v : U → R is plurisubharmonic. Since c ( L , h ) = √− ∂∂ v , wesee that c ( L , h ) is positive. (cid:3) Then, the key result to show that L is ample is the following. Lemma 6.13.
For every closed irreducible complex subspace Y of M with dim Y > , therestriction of the line bundle L → M to Y is big.Proof. By ( ) in Proposition 6.8, there is a natural partial order among π ( Q W ) , where Q ranges in the stratification of M . We define π ( Q W ) ≤ π ( S W ) if π ( Q W ) ⊂ π ( S W ) . If Y is a closed irreducible complex subspace, Y must intersect some π ( Q W ) . Wechoose π ( Q W ) to be the largest one with respect to the partial order ≤ just men-tioned. By ( ) in Proposition 6.8 again, π ( Q W ) is open in M \ ∪ π ( S W ) > π ( Q W ) π ( S W ) .Therefore, π ( Q W ) ∩ Y is open in Y . By Proposition 6.11, π ( Q W ) admits a C ∗ -orbittype stratification. Similarly, we may further choose a stratum S in π ( Q W ) such that Y ∩ π ( Q W ) ∩ S = Y ∩ S is open in Y ∩ π ( Q W ) . Therefore, Y reg ∩ S is also open in Y ,where Y reg is the smooth locus of Y .Now we consider the restriction of L to Y . We will use [21, Theorem 1.3] to showthat L | Y is big. By taking a desingularization of Y , we may assume that Y is a compactcomplex manifold. Clearly, the Chern current c ( L , h ) is still positive. Therefore,by Lebesgue’s decomposition theorem, the absolutely continuous part c ( L , h ) ac of c ( L , h ) is also positive. Hence,(6.24) Z Y c ( L , h ) dim Yac ≥ Z Y reg ∩ S c ( L , h ) dim Yac > Y reg ∩ S is a complex submanifold of S andhence Kähler. By Proposition 6.11, c ( L , h ) is the Kähler form on S . Therefore, therestriction of c ( L , h ) dim Yac to Y reg ∩ S is precisely the volume form on Y reg ∩ S . Hence,the last inequality in Equation (6.24) holds. (cid:3) Proof of Theorem D.
Note that ( ) in Theorem D is already proved in Proposition 5.1.To prove ( ) , we use Grauert’s criterion of ampleness for a line bundle over a compactcomplex space (see [9]). Therefore, we need to show that the restriction of L to any ir-reducible closed complex subspace Y with dim Y > Y . Let Y be an irreducible closed complex sub-space of M with dim Y >
0. By Lemma 6.13, L | Y is big. Hence, it admits a nontrivialholomorphic section. Such a section must vanish somewhere on Y . Otherwise, L | Y isholomorphically trivial and cannot be big. Therefore, L is ample, and M is projective.To see that M is quasi-projective, let us recall that M = M ∪ Z , where Z is a closedcomplex subspace of M . Moreover, let i : M → P N be a projective embedding. ByRemmert’s proper mapping theorem, i ( Z ) is a closed complex subspace of P N . ByChow’s theorem, both i ( M ) and i ( Z ) are Zariski closed in P N so that i ( M ) is Zariskiopen in i ( M ) . By definition, M is quasi-projective.Finally, we show ( ) . It suffices to show that the line bundle L → W ∗ / C ∗ is iso-morphic to L → M via the biholomorphism W ∗ / C ∗ → M described in the proof ofTheorem A. By definition, the total space of the line bundle L C → M × C is L × C . Ifwe restrict L C to W ∗ = M × C ∗ , we obtain the following commutative diagram(6.25) L × C ∗ / / (cid:15) (cid:15) L (cid:15) (cid:15) M × C ∗ / / M ,where the top horizontal map is given by ( v , z ) z − · v . Therefore, the diagram (6.25)defines a map ( L C | W ∗ ) / C ∗ → L covering the biholomorphism W ∗ / C ∗ → M . Finally,by the proof of Proposition 4.1, it is easy to verify that the total space L of the linebundle L → W ∗ / C ∗ is precisely ( L C | W ∗ ) / C ∗ . Therefore, we have obtained a bundlemap L → L that is an isomorphism on each fiber. (cid:3) HE QUASI-PROJECTIVITY OF THE MODULI SPACE OF HIGGS BUNDLES 25 R eferences [1] A. Andreotti and W. Stoll. Analytic and algebraic dependence of meromorphic functions . Lecture Notes inMathematics, Vol. 234. Springer-Verlag, Berlin-New York, 1971.[2] M. F. Atiyah and R. Bott. The Yang-Mills equations over Riemann surfaces.
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Comm. Anal. Geom. , 16(2):283–332, 2008. Y ue F an , D epartment of M athematics , U niversity of M aryland , C ollege P ark , MD, 20742, USA E-mail address ::