An Analytic Application of Geometric Invariant Theory
aa r X i v : . [ m a t h . C V ] O c t AN ANALYTIC APPLICATION OF GEOMETRICINVARIANT THEORY
NICHOLAS BUCHDAHL AND GEORG SCHUMACHER
Abstract.
Given a compact K¨ahler manifold, Geometric Invariant The-ory is applied to construct analytic GIT-quotients that are local modelsfor a classifying space of (poly)stable holomorphic vector bundles con-taining the coarse moduli space of stable bundles as an open subspace.For local models invariant generalized Weil-Petersson forms exist on theparameter spaces, which are restrictions of symplectic forms on smoothambient spaces. If the underlying K¨ahler manifold is of Hodge type,then the Weil-Petersson form on the moduli space of stable vector bun-dles is known to be the Chern form of a certain determinant line bundleequipped with a Quillen metric. It gives rise to a holomorphic line bun-dle on the classifying GIT space together with a continuous hermitianmetric. Introduction
The aim of this article is to apply methods of Geometric Invariant The-ory to Deformation Theory and construct a classifying space M GIT for(poly)stable vector bundles on compact K¨ahler manifolds (Theorem 8), con-cluding previous work from [Bu-Sch]. The space M GIT carries a naturalcomplex analytic structure. The underlying topological space is, by themethods of Atiyah, Hitchin, and Singer [AHS, §
6] a Hausdorff space. Thecoarse moduli space of stable holomorphic vector bundles is contained in theclassifying space as the complement of an analytic subspace.Given a semi-universal but not universal deformation of a holomorphicvector bundle say, one cannot expect that the action of the automorphismgroup G of the given object on the linear space V of infinitesimal defor-mations provides an action on the parameter space that is compatible withthe respective family. However, this property holds in the case of polystablevector bundles (Theorem 5), which was shown in [Bu-Sch]. It is emphasizedthat due to Theorem 7, (also from [Bu-Sch]) it is possible to speak about“(poly)stable points” referring to (poly)stable fibers or (poly)stable pointswith respect to group actions so that methods of deformation theory matchthe GIT approach.The analogous problem for moduli of cscK manifolds was solved by Dervan-Naumann [D-N]. The role of our Theorem7 was played by the result ofSz´ekelyhidi [Sz].The action of the reductive group G on the vector space V yields a GIT-quotient V //G . On the other hand the parameter space S has been realizedas a closed analytic subspace of a neighborhood U ⊂ V of the origin together with a restricted action of G . There exists a G -saturated open subset e U ⊃ U and a closed, G -invariant, analytic subspace e S ⊂ e U (cf. Theorem 1).In fact the algebraic approach to construct a GIT-quotient of e S cannotbe applied. However, the analyticity of the image of e S in V //G followsfrom Kuhlmann’s generalization of the Remmert proper mapping theorem(Theorem 6). The image e S// an G is called analytic GIT-quotient. It is aclosed analytic subset of the open set e U // an G ⊂ V //G .In case the analytic structure on S is weakly normal, it follows immedi-ately that the structure sheaf on the quotient consists of invariant holomor-phic functions, or more generally the weak normalization of e S// an G carriesthe G -invariant holomorphic functions on the weak normalization of e S sothat a globalization is constructed in the weakly normal category.For the general case (cf. the approach by Dervan-Naumann [D-N]) oneobserves that Heinzner’s complexification theorem implies that e S is Stein,and Roberts’ theorem on G -sheaves [Ro] yields that holomorphic functionson the analytic GIT-quotient are exactly the G -invariant functions on thepull-back to e U . In order to glue local analytic GIT-quotients together (The-orem 8) there exists Snow’s theorem [Sn], who also proves the existence of acategorical quotient of a Stein space by a reductive group. In our situationa simple direct application of Luna’s slice theorem for the action on V ispossible (Theorem 4) by restricting an algebraic slice to the given analyticsubspace. The result is the classifying space M GIT of isomorphism classesof polystable holomorphic vector bundles.Fiber integral formulas were used in [F-S] to define a generalized Weil-Pe-tersson form on the moduli space of compact K¨ahler manifolds with constantscalar curvature, and to construct a determinant line bundle together witha Quillen metric, whose curvature is, up to a numerical constant, equalto the Weil-Petersson form. For moduli of stable vector bundles such aconstruction ultimately goes back to Donaldson [Do]. For details see e.g.[Bi-Sch]. Using secondary Bott-Chern classes and the Donaldson invariantone can find K¨ahler potentials in the case of holomorphic vector bundles,whereas in the case of cscK manifolds the Mabuchi functional plays a similarrole [D-N].There are various ways to extend determinant line bundles. The generalmethod is to extend the Weil-Petersson form as a positive current from thecomplement of an analytic set into its closure, provided the given familiesextend as singular families of compact manifolds or as coherent sheavesresp. This was done in [Sch] by means of C -estimates for Monge-Amp`ereequations and in [Bi-Sch] using Donaldson invariants respectively. Oncesuch extensions exist, the determinant line bundles together with the Quillenmetrics can be extended as singular, hermitian (holomorphic) line bundlesafter blowing up the boundary if necessary [Sch]. Such results apply tocertain compactifications of the moduli spaces.Less qualitative and more specific solutions are desirable. Dervan andNaumann succeeded in extending the determinant line bundle for analyticGIT spaces of cscK-manifolds in [D-N] arriving at line bundles on theseclassifying spaces. N ANALYTIC APPLICATION OF GEOMETRIC INVARIANT THEORY 3
For a semi-universal deformation of a polystable vector bundle we use thefiber integral for the Weil-Petersson form ω W P (cf.[Bi-Sch]) which togetherwith Theorem 5 and 7 implies that is the restriction of a symplectic form ona smooth ambient space. If X is a Hodge manifold, more can be said. Tech-niques involving the Riemann-Roch-Hirzebruch formula by Bismut, Gillet,and Soul´e provide a determinant line bundle together with a Quillen met-ric on a local parameter space such that the Chern form is equal to theWeil-Petersson form on the stable locus. Also at polystable points the re-sult yields the L -inner product of harmonic Kodaira-Spencer tensors. Theconstruction is compatible with descending to local analytic GIT-quotientsand the gluing of local models, and it provides the classifying space witha Weil-Petersson form that possesses a continuous ∂∂ -potential. Then thegeneral methods of Dervan and Naumann [D-N] are applicable.2. Analytic GIT-quotient
For the convenience of the reader we recall notions and facts from geo-metric invariant theory as far as these will be used here. The primary refer-ences include the work of Mumford-Fogarty-Kirwan, Ness, Hoskins, Kirwan[MFK, Nes, Ho, Kir] and Thomas [Th] – for projective and affine GIT-quotients and moduli of vector bundles over Riemann surfaces see New-stead’s book and article [New, New1].Let the reductive Lie group G act linearly on a finite dimensional complexvector space V . A point v ∈ V \{ } is called unstable for the group actionif 0 ∈ V is contained in the closure of the orbit of v , otherwise it is called semistable . A semistable point is called polystable if its G -orbit is closed,and stable if it is polystable with a finite isotropy group. (The point 0 ∈ V is not part of this classification – in the present situation it is meaningful todefine it as polystable).The existing GIT-quotient V //G of the affine space V is an affine spaceequipped with the algebra C [ V ] G of G -invariant regular functions. Further-more there exists a morphism p : V → V //G .2.1.
Restricted actions of reductive groups on complex spaces.
Ac-tions of reductive groups on analytic space germs have been investigated (cf.G. M¨uller’s article [M¨u]), however, a global approach is necessary. Applica-tions of the Kempf-Ness Theorem [KN] will be needed first:Assume that the group G is the complexification of a maximal com-pact subgroup K , and let ρ : G → GL ( V ) be a representation. By as-sumption the vector space V carries a hermitian inner product with norm m , and corresponding moment map µ for the action of G on V . In thiscontext the point 0 ∈ V is considered to be in V ps (with µ (0) = 0). Theorem (Kempf-Ness [KN]) . (i) The polystable locus V ps is equal to G · µ − (0) . (ii) Every G -orbit in V ps contains only one K -orbit of µ − (0) . (iii) In particular the GIT-quotient
V //G can be identified with the settheoretic quotient V ps /G , NICHOLAS BUCHDAHL AND GEORG SCHUMACHER (iv) and the space µ − (0) /K equipped with the quotient topology is home-omorphic to the GIT-quotient V //G . We note the following consequence:
Remark 1.
Let U ⊂ V be an open ball around ∈ V with respect tothe norm m . Assume that K = SU( V ) ∩ G . Then the image of U under p : V → V //G is an open neighborhood of p (0) ∈ V //G .Proof.
The set U is K -invariant and µ − (0) ∩ U is open in µ − (0). (cid:3) For moduli theoretic applications the following situation is of interest.Let a reductive group G act linearly on a complex vector space V withpolystable locus V ps = { } . Let 0 ∈ U ⊂ V be an open subset, and0 ∈ S ⊂ U be a closed analytic subset such that for g ∈ G , s ∈ S with g · s ∈ U the point g · s is contained in S . An equivalence relation on S isdefined by s ′ ∼ s ′′ if and only if there exists g ∈ G such that s ′′ = g · s ′ . Definition 1 (Restricted group action) . Let S be a closed analytic subsetof an open subset U ⊂ C N with ∈ U , where dim T S = N . Let G be areductive group with a linear action on the vector space V := T S .Given the action of G on V a restricted action of G on the pair ( S, U ) isdefined as an equivalence relation of the above kind on S . A restricted group action on a pair (
S, U ) can be globalized.
Theorem 1.
There exists a closed G -invariant analytic subset e S := S g ∈ G g · S of the G -invariant open set e U := S g ∈ G g · U ⊂ V such that e S ∩ U = S . In general e U ( V , and e S need not be affine. In fact, by Remark 1 the set W = p ( U ) is an open neighborhood of p (0). Notation.
Given a restricted group action on a pair ( S, U ) the tilde notationsuch as e S ⊂ e U will be consistently used for the globalized spaces.Proof of Theorem 1. Let g ∈ G be given so that U ∩ g · U = ∅ . The set S ∪ g · S is locally analytic in U ∪ g · U . It is an analytic subset of U ∪ g · U if it is closed. Let S be the closure in U ∪ g · U . Take x ∈ S \ S . Then x ∈ g · S . Near this point, on a certain neighborhood U ( x ) of x the set g · S is given by certain holomorphic equations f = . . . = f m = 0. Consider( g − · U ( x )) ∩ U and the pulled back equations. Since S is invariant in therestricted sense, these equations must vanish at x so that x ∈ g · S . Theargument is symmetric in U and g · U , and the same argument holds for sets g · U and g · U .Finally, the analyticity of ∪ g g · S ⊂ e U follows, if this set is closed in e U . Let s ∈ ∪ g g · S ⊂ e U , say s = lim g j · s j ∈ g j · U . Then g − j · s = lim j g − j g j · s j ∈ U ∩ S so that s ∈ g j S ⊂ e S . (cid:3) In the algebraic setting an immediate argument shows that the ideal sheafof functions that vanish on the image p ( e S ) in the open subset W = p ( e U ) ⊂ V //G is coherent as a certain ideal of G -invariant functions on preimagesof open subsets. In the analytic setting a different approach is needed, N ANALYTIC APPLICATION OF GEOMETRIC INVARIANT THEORY 5 namely first prove the analyticity of p ( e S ) by Kuhlmann’s generalization ofRemmert’s proper mapping theorem (cf. also the book by Whitney) [Ku,W, R]. Definition 2.
Let f : X → Y be a continuous mapping of locally compact,Hausdorff topological spaces with countable topology. The map f is called semi-proper , if for any point of Y there exists an open neighborhood W anda compact set L ⊂ X such that f ( L ) ∩ W = f ( X ) ∩ W. Obviously restrictions of semi-proper maps to closed subsets need not besemi-proper in general.
Theorem (Semi-proper mapping theorem, [Ku, W, R]) . Let f : X → Y bea semi-proper holomorphic map of reduced complex spaces. Then f ( X ) ⊂ Y is an analytic subset. The following situation is given, e S (cid:31) (cid:31) ❃❃❃❃❃❃❃❃ (cid:31) (cid:127) / / e U p | e U (cid:15) (cid:15) (cid:31) (cid:127) / / V p (cid:15) (cid:15) W (cid:31) (cid:127) / / V //G where p is rational (with restriction to a classically open subset e U ). As inTheorem 1 the set U is a convex neighborhood of 0 of the form U = U c = { x ; m ( x ) < c } for some c >
0, and e U , W = p ( U ) = p ( e U ) in the above sense. Theorem 2.
The canonical holomorphic maps p | e U : e U → W and the re-striction p | e S : e S → W are semi-proper. In particular p ( e S ) ⊂ W ⊂ V //G isa closed analytic subset of W .Proof. The Kempf-Ness Theorem will be applied. Let w ∈ W . There exists u ∈ V ss with minimal norm within its orbit such that p ( u ) = w . It followsfrom the construction of e U and the convexity of U that u ∈ U . Choose0 < a < b < c such that u ∈ U a , and accordingly w is contained in theopen neighborhood W a = p ( U a ). Pick the compact set L = U b . Then p ( L ) ∩ W a = p ( e U ) ∩ W a .Now the semi-properness of p | e S : e S → W is shown. Let w ∈ W . Let0 < a < b < c such that w ∈ W a = p ( U a ) ⊂ W = p ( U ). We claim that p ( e S ) ∩ W a = p ( U b ∩ e S ) ∩ W a . Namely, let p ( e s ) ∈ W a for some e s ∈ e S . Then there exists some b s ∈ V ss with G · e s = G · b s and b s of minimal norm within its G -orbit. Hence b s ∈ U a ⊂ U b .Furthermore b s ∈ G · e s ⊂ e S = e S by Theorem 1, Altogether p ( e s ) ∈ p ( U b ∩ e S ),where U b ∩ e S = U b ∩ S is compact. (cid:3) Let p : V → V //G be the quotient of a smooth (or maximal) affine space V by a reductive group, W ⊂ V //G be an open Stein subset, and e U = p − ( W ). NICHOLAS BUCHDAHL AND GEORG SCHUMACHER
Let e S ⊂ e U be a G -invariant closed subset whose image p ( e S ) is a closedanalytic subset of W . Then e S ֒ → e U ֒ → V is G -equivariant. (This is the motivation for the notation ( e S ֒ → e U ֒ → V ) asopposed to ( e S ⊂ e U ⊂ V ).) Definition 3.
The triple ( p ( e S ) ֒ → W ֒ → V //G ) is called analytic GIT-quotient of e S by G , and denoted by ( e S// an G ֒ → e U // an G ֒ → V //G ) . Note that p | e S : e S → e S// an G identifies G -orbits (and their closures). Remark 2.
By Theorem 1 and Theorem 2 any restricted group action inthe sense of Definition 1 gives rise to an analytic GIT-quotient.
Complex structure for analytic GIT-quotients.
An analytic GIT-quotient e S// an G according to Theorem 2 carries the natural complex struc-ture of an analytic subset of the open subset e U // an G ⊂ V //G .Let V be a smooth affine space together with the action of a reductivegroup G such that all points of V are semistable. Denote by p : V → V //G the GIT-quotient. It is known (cf. articles by Neeman [Nee, Nee1], Roberts[Ro], Snow [Sn], and Thomas [Th]) that due to the normality of
V //G the(analytic) structure sheaf satisfies(2.1) O V//G ( W ′ ) = O GV ( p − ( W ′ ))for any (Stein) open subset W ′ ⊂ V //G , where a holomorphic function ϕ on a subset W ′ of the GIT-quotient corresponds to the G -invariant function ϕ ◦ p on p − ( W ′ ) ⊂ e S .The aim is to show that holomorphic functions on open subsets of thequotient e S// an G are exactly the G -invariant holomorphic functions on thepreimages under p .Let W ⊂ V //G be a Stein open subset, and e U = p − ( W ) ⊂ V . Let e S ⊂ e U be a closed, G -invariant, analytic subset. The coherence of the ideal sheafof G -invariant functions vanishing on e S can be shown directly. Remark 3.
Let p ( e S ) = e S// an G ⊂ W = e U // an G ⊂ V //G be an analyticGIT-quotient. Then for the (coherent) vanishing ideals I p ( e S ) ⊂ O V//G | W and I e S ⊂ O V | e U the following holds: I p ( e S ) ( W ′ ) = I e S ( p − ( W ′ )) G , where W ′ ⊂ W is an open Stein set, and where I e S ( p − ( W ′ )) G denotes G -invariant holomorphic functions that vanish on e S .Proof. Let ϕ ∈ I p ( e S ) ( W ′ ). By (2.1) the function ϕ ◦ p is G -invariant andvanishes on p − ( W ′ ∩ p ( e S )).Conversely, let a G -invariant holomorphic function on p − ( W ′ ) be giventhat vanishes on p − ( p ( e S ) ∩ W ′ ). By 2.1 it is of the form ϕ ◦ p , hence itdescends to ϕ with ϕ | W ′ ∩ p ( e S ) = 0. (cid:3) N ANALYTIC APPLICATION OF GEOMETRIC INVARIANT THEORY 7
In this sense by Proposition 3, given a G -invariant analytic set e S ⊂ e U with analytic image p ( e S ) ⊂ W the short exact sequence0 → I p ( e S ) → O V//G | W → O e S// an G → → I e S ( p − ( W ′ )) G → O V ( p − ( W ′ )) G → O e S// an G ( W ′ ) → . Since W ′ ⊂ V // an G is an open Stein subset, (2.2) implies the existence of anatural injective map(2.3) ι : O e S// an G ( W ′ ) ֒ → O e S ( p − ( W ′ )) G . Theorem 3.
The embedding (2.3) is an isomorphism.
If we restrict the claim to the “normal category” i.e. to weakly normalspaces (or to the weak normalization of a space e S ), then the claim of the the-orem follows in an elementary way: A reduced complex space is called weaklynormal if continuous holomorphic functions on the regular locus of an opensubset extend holomorphically to the given open subset. The condition isequivalent to saying that any continuous, weakly holomorphic function (i.e.holomorphic outside a nowhere dense analytic subset) is holomorphic. Weuse the notation b S → S for the weak normalization (cf. the book by L. Kaupand B. Kaup [K-K, § S is weakly normal, then so is e S . The analyticGIT-quotient e S// an G exists and by [K-K, § G -invariant holomorphic functions on e S descend to continuous func-tions that are holomorphic on the locus, where p is of maximal (differentialrank) the structure sheaf of the weak normalization of e S// an G satisfies O \ e S// an G ( W ′ ) = O be S ( p − ( W ′ )) G for Stein open subspaces W ′ ⊂ W .Now the general case of Theorem 3 will be proven. Proposition 1.
Let W ′ ⊂ V //G be a K -invariant Stein open set, e.g. let W ′ = p ( U ) , where U ⊂ V is an open ball around ∈ V . Then p − ( W ′ ) ⊂ V is a Stein open subset. The proof is an immediate consequence of Heinzner’s complexification the-orem – it follows that the set p − ( W ′ ) is equal to the complexification W ′ C in the sense of [He]. (cid:3) In [Ro] Roberts studied coherent sheaves F on a space V with G -actions.These give rise to sheaves p G ∗ F on V //G defined by( p G ∗ F )( W ′ ) := F ( p − ( W ′ ) G for Stein open subsets W ′ (cf. also the article by Heinzner and Loose [H-L]).Roberts constructs an averaging operator for coherent O e U -modules F . L : F e U ( p − ( W ′ )) −→ F e S// an G ( W ′ ) G by means of integration over the compact group K that is a projection onto G -invariant sections. By [Ro, Theorem 3.1] the coherence of p ∗ O G e S follows,and (2.1),(2.2) imply the proof of Theorem 3. (cid:3) NICHOLAS BUCHDAHL AND GEORG SCHUMACHER
Recent references for Higgs bundles on Riemann surfaces include articlesby Fan [F1, F2].3.
Slice theorem for analytic GIT-quotients
Snow showed an analytic slice theorem for the action of a reductive groupon a Stein space together with the existence of a categorical quotient in [Sn].Here the situation is simpler than in the general case.The action of a reductive group G on an affine space V is already given,and restricted to a closed subset of an open (Stein) subspace. We need thedirect application to spaces of the form e S ⊂ e U ⊂ V .A slice for the action on an affine space is given by Luna’s theorem [Lu](see also the exposition by Dr´ezet [D]).3.1. Algebraic case.
Let the reductive group G act on an affine space(or vector space) V . If the G -orbit of a point s ∈ V is closed, then byMatsushima’s theorem the stabilizer subgroup G s is reductive again.Let Y be a G s -invariant affine subspace containing s , and G × Y → V be the canonical G -equivariant map, where the left action of G on itselfdefines an action on G × Y . The left action of G s on G together with theaction of G s on Y is given by γ · ( g, w ) = ( gγ − , γw ) for γ ∈ G s , g ∈ G ,and w ∈ Y . It defines a GIT-quotient ( G × Y ) //G s for which the notation G × G s Y is common usage. Now G × G s Y //G ≃ Y //G s . Altogether there isa commutative diagram G × Y //G s = G × G s Y (cid:15) (cid:15) ψ / / V ⊂ V (cid:15) (cid:15) Y //G s ≃ ( G × G s Y ) //G ψ / / V //G ⊂ V //G where ψ and ψ are induced by the action of G . The spaces V and V //G are the images of ψ and ψ resp. in the sense below. Theorem (Luna, [Lu][D, Thm. 5.3, 5.4]) . Let G be a reductive group actingon an affine variety V with closed orbit G · s . Then there exists a slice Y through s , i.e. a G s -invariant affine subvariety such that the map ψ is ´etaleonto an open, G -invariant subvariety V ⊂ V , and such that ψ is ´etale onto V //G . If V is smooth at s so is Y .Let V ′ ⊂ G × G s Y be a closed G -invariant subvariety. Then there existsa closed G s -invariant subvariety Y ′ of Y such that V ′ = G × G s Y ′ . Note that the dimension of the isotropy subgroups of G is not assumedto be constant.3.2. Application to analytic GIT-quotients.
For any analytic applica-tion the property ´etale will be replaced by locally biholomorphic , and the (an-alytic) structure sheaves of analytic GIT-quotients such as p : e S → e S// an G of reduced complex spaces consist of holomorphic G -invariant functions onpull-backs under the projection p .Let e S ֒ → e U ֒ → V be given as in Section 2.2 where V is a smooth affinespace. In particular, referring to Definition 3, let s ∈ e S be a point with N ANALYTIC APPLICATION OF GEOMETRIC INVARIANT THEORY 9 a closed G -orbit, and let Y be a slice through s for the action of G on V according to the slice theorem. Without loss of generality let e S ⊂ V .In the process the space e U is replaced by a smaller saturated open sub-space that contains s and that is of the form p − ( W ). Accordingly thealgebraic slice Y will be replaced by e Y = e U ∩ Y . The analytic quotients( G × e Y )) // an G s , e Y // an G s , and e Y // an G exist, and the structure sheaves con-sist of G s - and G -invariant functions according to Section 2.2. Altogetherthere exist analytic quotients and locally biholomorphic, surjective maps ψ and ψ (3.1) ( G × Y ) //G s ψ −→ V ∪ ∪ ( G × e Y )) // an G s −→ e U ↓ p ↓ p e Y // an G s → e U // an G ∩ ∩ Y //G s ψ −→ V //G. With the above notation the following theorem holds.
Theorem 4 (Application to analytic GIT-quotients) . Let the reductivegroup G act on a smooth (or weakly normal) affine space V , and as abovelet S ⊂ U ⊂ V be a G -invariant analytic subset with ∈ S . Then for points = s ∈ e S the following holds: Let the orbit G · s ⊂ V be closed. Then (i) There exists the slice e S ∩ e Y for the action of G s on e S , namely alocally biholomorphic map from ( G × ( e S ∩ e Y )) // an G s to e S . (ii) The action of G defines an analytic GIT-quotient ( G × ( e S ∩ e Y ) // an G s ) // an G ≃ ( e S ∩ e Y ) // an G s , and a locally biholomorphic map to e S// an GG × ( e S ∩ e Y ) // an G s ψ −→ e S ↓ p ↓ p ( e S ∩ e Y ) // an G s ψ −→ e S// an G. Proof.
Consider the multiplication map G × e Y → e Y . If g · y ∈ e S , thenobviously y ∈ e S so that y ∈ e S ∩ e Y , hence ψ − ( e S ) = G × ( e Y ∩ e S ) // an G s .This proves (i). The same argument and (3.1) yield (ii). Observe that theanalytic GIT-quotient are images under a projection map equipped with thesheaves of G s -invariant functions. (cid:3) Deformations of holomorphic vector bundles
We will fix some notation now. Let X be a compact complex manifold,which will be equipped later with a K¨ahler form ω X . A holomorphic family( E s ) s ∈ S of holomorphic vector bundles on X parameterized by a (reduced)complex space S is given by a holomorphic vector bundle E over X × S suchthat E s = E| X × { s } . Deformations ξ of a fixed holomorphic vector bundle E are defined over complex spaces ( S, s ) with a distinguished point s ∈ S (or a “pointed” complex space). Such an object consists of a holomorphic family E over S together with an isomorphism χ : E ∼ −→ E s . Accordinglyan isomorphism of two deformations of a bundle E over the same space isan isomorphism of holomorphic families that is compatible with the givenidentifications of central fibers with E . Consequently, a change of χ usuallygenerates a non-isomorphic deformation (unless χ can be extended to thewhole family).During most arguments that involve deformations a parameter space hasto be replaced by a neighborhood of the distinguished point. In this sensespace germs are being used as parameter spaces for deformations.Let β : ( R, r ) → ( S, s ) be a holomorphic map of spaces with distin-guished points, and ξ a deformation of E over ( S, s ). The base change mapassigns to ξ a deformation β ∗ ξ over ( R, r ), which is given by the pull-back β ∗ E of the corresponding family together with the induced isomorphism of E and the fiber β ∗ E| X × { r } = E s . Infinitesimal deformations of a bundle E are by definition deformationsover the “double point” D = (0 , C [ t ] / ( t )) ⊂ ( C , v ∈ T s S correspond exactly to holomorphic maps v : D → S sending theunderlying point to s ∈ S . The space of isomorphism classes of infini-tesimal deformations of E can be identified with the complex vector space H ( X, End ( E )). In this set-up, given a deformation ξ , the Kodaira-Spencermap ρ : T s S → H ( X, End ( E ))sends a tangent vector v to the pull-back v ∗ ξ , which is the “restriction ofthe given deformation to the direction given by the tangent vector v ”.A deformation ξ is called complete , if any other deformation is of the form β ∗ ξ , where β is a base change map – it is called semi-universal , if in additionthe derivative T r β at the distinguished point is uniquely determined.More precisely, let E be a polystable holomorphic vector bundle on( X, ω X ), and ξ a semi-universal deformation of E over a complex analyticspace ( S, s ). In order to have a halfway simple notation deformations arerestricted to the category of reduced spaces first, and ( S, s ) will alwaysdenote a reduced complex space. Concerning the Kodaira-Spencer map therestriction to reduced complex analytic spaces yields values in a linear sub-space of V = H ( X, End ( E )).5. Application to deformations of polystable vector bundles
Let again (
X, ω X ) be a compact K¨ahler manifold. For a polystable vectorbundle the base of a semi-universal deformation ξ in general contains positivedimensional local analytic subsets, namely parts of G -orbits of semistable(but not polystable points). Also, in general, stable points give rise to localanalytic sets, where the fibers are isomorphic. This fact can be illustratedas follows. If the tangent cohomology H q ( X, End ( E )) vanishes for q ≥ E = E polystable and E s stable for some s near s we have 0 Theorem 5 ([Bu-Sch][Theorem 1]) . For any polystable vector bundle ona compact K¨ahler manifold there exists a semi-universal deformation witha parameter space ( S, s ) such that the action of the group of holomorphicautomorphisms Aut ( E ) on the tangent space of S at s extends to an actionon the space germ of S at s that is compatible with the holomorphic familyof vector bundles. The space S has been realized as an analytic subspace of an open neighbor-hood U of s = 0 in the space of infinitesimal deformations H ( X, End ( E ))of the form S = Ψ − (0), where Ψ : U → H ( X, End ( E )) has been con-structed as an Aut ( E )-equivariant map in the restricted sense of Definition 1.For a suitable chosen open set U all fibers of points from S correspond tosemistable vector bundles. In other words, the action of G = Aut ( E ) on V gives rise to a restrictedaction on S ⊂ U ⊂ V in the sense of Section 2. Theorem 6. Let a polystable vector bundle E on a compact K¨ahler manifold ( X, ω X ) be given. Then a semi-universal deformation of E over a space ∈ S ⊂ U ⊂ V together with the reduced action of G = Aut ( E ) on thespace V of infinitesimal deformations gives rise to an analytic GIT-quotient e S// an G which is a closed reduced subspace of an open subset e U // an G of theGIT-quotient V //G . The following correspondence will also be crucial: Theorem 7 ([Bu-Sch, Thm. 3]) . A class s ∈ S = Ψ − (0) ⊂ H ( X, End ( E )) is (poly)stable with respect to the action of Aut ( E ) if and only if the corre-sponding bundle E s is (poly)stable with respect to the given K¨ahler metric. For the property of semi-stability such an analogy does not exist: In gen-eral there may be points s ∈ S \ V ss that correspond to semi-stable bundles(cf. [Bu-Sch, § Change of affine space. In principle the special rˆole of the origin in V with respect to the group action is not intended. Points that are unstablehave the same status as points that are semistable but not polystable. Onecan use the following auxiliary affine space.Set V = C × V . Lemma 1. A given action of G on V is extended to V by the identity on C , i.e. g · ( ζ, v ) := ( ζ, g · v ) . Then the following holds (again after dividingby the ineffectivity kernel). Let ( ζ, v ) ∈ V . (i) G ( ζ,v ) = G v (ii) G · ( ζ, v ) = { ζ } × G · v (iii) Let ζ = 0 , v = 0 , then ( ζ, v ) is G -(poly)stable, if and only v has thisproperty. Let ζ = 0 , then ( ζ, is G -polystable, and all ( ζ, v ) are G -semistable points. Observe that C [ V ] = C [ ζ ] ⊗ C [ V ], and accordingly C [ V ] G = C [ ζ ] ⊗ C [ V ] G so that V //G = C × ( V //G ).Nevertheless in the sequel the original spaces will be used. Analytic GIT-quotient for parameter spaces of holomorphicvector bundles. Given a semi-universal deformation of a polystable vectorbundle, the parameter space S can be chosen so that the triple( S ֒ → U ֒ → V )satisfies the assumptions of Theorem 6.Eventually all points of e S// an G correspond to holomorphic vector bun-dles or equivalence classes of holomorphic vector bundles as in the case ofalgebraic GIT-spaces. Proposition 2. Let E → X × S be a holomorphic family of vector bundlesover a compact K¨ahler manifold ( X, ω X ) parameterized by an irreducible,reduced space Q . Then there exists an analytic subset A ⊂ Q such that allbundles E s for s ∈ Q \ A possess a universal deformation.Proof. Let F be the sheaf of trace-free holomorphic endomorphisms of E .Then the above locus is contained in the support of pr ∗ F , where pr : X × S → S denotes the canonical projection. (cid:3) Given a deformation of a polystable bundle over a space ( S, 0) the set A ⊂ S from Proposition 2 is G -invariant in the restricted sense and givesrise to a GIT-subspace e A// an G ⊂ e S// an G , whose points represent polystable,non-stable vector bundles. Remark 4. The set of polystable points that are not stable in the analyticGIT-quotient e S// an G is a closed analytic subspace. The following fact will be needed. Proposition 3 ([Bu-Sch][Corollary 4.6]) . Let p > im ( X ) . Then the(pointwise) quotient Ψ − (0) /G can be identified with F / G , where F denotesthe space of L p integrable connections near the given Hermite-Einstein con-nection d on E , and G denotes the corresponding complexified gauge group:parameters with isomorphic fibers are actually contained in the same G -orbit. Let S = Ψ − (0) ⊂ U be the base of a semi-universal deformation of apolystable bundle. Two points s ′ , s ′′ of S will be identified, s ′ ∼ s ′′ , if thefibers E s ′ and E s ′′ are isomorphic. By Proposition 3 this is exactly the case,if s ′ = g · s ′′ for some g ∈ G . Conversely, if s ′ , s ′′ ∈ e S then there exist g ′ , g ′′ ∈ G such that g ′ · s ′ , g ′′ · s ′′ ∈ S . If these points of S are equivalent s ′ = g · s ′′ holds for some g ∈ G so that the topological spaces e S/G and S/ ∼ can be identified.After applying the methods from Section 2 to the reduced action of G tothis standard situation the following holds. Lemma 2. Let ξ be a semi-universal deformation of a polystable bundleover ( S, , and η : ( S, → ( S, a holomorphic map (of germs) suchthat η ∗ ξ ≃ ξ . Then η is an isomorphism, whose derivative at the originis the identity. The map η defines a map S → S , where S , S ⊂ S areneighborhoods of the origin that extends to a map e S → e S and descends toan isomorphism of analytic GIT-quotients e S // an G ∼ −→ e S // an G. N ANALYTIC APPLICATION OF GEOMETRIC INVARIANT THEORY 13 Proof. The property of ξ being a semi-universal deformation implies that η is an isomorphism with T η = id T S . Represent η by an isomorphismdenoted by the same letter η : S → S of suitable neighborhoods of theorigin. Note that the construction of the analytic GIT-quotient containsthe fact that the maps e S j → e S j // an G etc. are still surjective when restrictedto S j ⊂ e S j . Due to the construction of η and Proposition 3 the map η isconstant on closures of G -orbits, in particular it is G -invariant. The claimnow follows by Theorem 3. (cid:3) Let again a semi-universal deformation ξ of a polystable bundle over ( S, G on ( S ֒ → U ֒ → V ). Let s ∈ S correspond to a polystable bundle E s .Let R ⊂ U R ⊂ V R denote the base of a semi-universal deformation ζ of E s with restricted group action of H on ( R ֒ → U R ֒ → V R ) with V R = H ( X, End ( E s )). Now there exist base change maps of space germs α :( S, s ) → ( R, 0) and β : ( R, → ( S, s ) such that α ∗ ζ = ξ s and β ∗ ξ s = ζ ,because of completeness and semi-universality of ξ s and ζ resp. By Lemma 2the map α ◦ β induces an isomorphism of neighborhoods of the distinguishedpoint which descends to an isomorphism (near the distinguished point) of e R// an H . Lemma 3. The map β descends to a holomorphic map β : e R// an H → e S// an G. Proof. It is already known that β gives rise to a holomorphic map e β : R → e S// an G . Points with isomorphic fibers are mapped to points with isomorphicfibers so that the topological space R/H is mapped homeomorphically ontoan open subset of the topological space S/G , and points corresponding topolystable bundles i.e. polystable points with respect to the group action of H go to polystable points with respect to G . In particular the holomorphicmap e β is constant on G -orbits. Any holomorphic function on e S// an G ispulled back under β to an H -invariant function on e R . The claim followsagain from Theorem 3. (cid:3) In order to descend the map α : ( S, s ) → ( R, 0) to GIT-quotients, first aslice Q ⊂ S through s for the action of G on S is chosen (and α is restrictedto Q ) so that G × Q// an G s ⊂ e S// an G is an open neighborhood of the imageof s . Lemma 4. The map α : ( S, s ) → ( R, descends to a holomorphic map e S// an G ⊃ e Q// an G s α −→ e R// an H where e Q// an G s ⊂ e S// an G is an open neighborhood of the image of s .Proof. Once the analytic slice is chosen, the proof is the same as the proofof Lemma 3. (cid:3) Note that the analytic slices are not shown to be parameter spaces ofsemi-universal deformations – identification takes place only after passingto the analytic GIT-quotient. Remark 5. By [Bu-Sch, Theorem 4.7] it follows that H = G s . Proposition 4. Let ξ be a semi-universal deformation with parameter space ( S, , let s ∈ S be a polystable point, denote by ξ s the induced deformationof the fiber at s , and let ζ be a semi-universal deformation of ξ s over a basespace ( R, . Let α : ( S, s ) → ( R, and β : ( R, → ( S, s ) be holomorphicmaps of induced space germs such that α ∗ ζ = ξ s and β ∗ ξ s = ζ . Let Q be aslice for the action of G on S through s so that Q// an G s can be identifiedwith a neighborhood of the image of s in S// an G .Let G and H be the automorphism groups of the central fiber and of thefiber at s resp. Then the base change morphisms descend to isomorphisms ofanalytic GIT-quotients near the points and s after replacing the respectivespaces by neighborhoods of the distinguished points: R// an H β ∼ / / Q// an G s α ∼ / / R// an H Proof. The above holomorphic maps were constructed in Lemma 3 andLemma 4. The underlying topological spaces consist of polystable points sothat α and β are homeomorphisms, and on the complements of thin analyticsets these maps are biholomorphic. Moreover, by Lemma 2 the map α ◦ β is an isomorphism on the whole space. This is only possible, if both mapsare isomorphisms (cf.[G-R, Chap. 8 Sect. 2] on one-sheeted coverings). (cid:3) Remark 6. If E s is stable, then H is trivial. Application to moduli of holomorphic vector bundles The aim is to construct a complex space whose local models are analyticGIT-quotients such that the complement of a closed analytic set is the coarsemoduli space of stable holomorphic vector bundles.The local model was constructed in Theorem 6. By general theory, in par-ticular the Kempf-Ness Theorem, and Proposition 2, and by Proposition 3the points of the space correspond to isomorphism classes of polystable vec-tor bundles. An analytic version of Luna’s slice theorem was shown andProposition 4 provides the gluing of local analytic GIT models. Namely,when two such models contain a point with (polystable) isomorphic fibers E say, each of the models is locally isomorphic to the GIT-quotient con-structed from a semi-universal deformation of E .Finally the Hausdorff property of the resulting space follows from argu-ments of Atiyah, Hitchin, and Singer [AHS, § M GIT contains the coarse moduli space of stable holomorphicvector bundles by Remark 6. Theorem 8. Given a compact K¨ahler manifold there exists a complex space M GIT , whose local models are analytic GIT-quotients of parameter spacesof holomorphic vector bundles with a restricted group action.The space M GIT contains the coarse moduli space M of stable holomor-phic vector bundles as an open subspace. The complement M GIT \M is aclosed analytic subspace, whose points correspond to isomorphism classes ofpolystable, non-stable vector bundles. Definition 4. Let K be the class of polystable (including stable) vector bun-dles on the given compact K¨ahler manifold. A reduced complex space N iscalled a classifying space for K , if the following conditions hold. N ANALYTIC APPLICATION OF GEOMETRIC INVARIANT THEORY 15 (i) The points of N correspond to isomorphism classes of polystable,holomorphic vector bundles on the K¨ahler manifold X , (ii) Let E be a holomorphic family of holomorphic vector bundles on X × Z , where Z is a reduced complex space, such that the fiber E z = E| X ×{ z } , of a point z ∈ Z is polystable. Then, after replacing Z bya neighborhood of z if necessary, there exists a unique holomorphicmap ϕ : Z → N such that ϕ ( z ) ∈ N corresponds to the isomorphismclass of E z provided E z is polystable. (iii) If e N is a further reduced complex space satisfying (i) and (ii) withunique holomorphic maps e ϕ : Z → e N for a families over spaces Z in the sense of (ii) , then there exists a unique holomorphic map χ : N → e N such that χ ◦ ϕ = e ϕ . Proposition 5. The space M GIT is a classifying space for the class ofpolystable holomorphic vector bundles on a compact K¨ahler manifold.Proof. The first condition follows from the construction, which identifieslocal models, if the polystable fibers are isomorphic.Let a holomorphic family ζ in the sense of (ii) be given. After replacing Z by a neighborhood of z the given family is isomorphic to the pull-back ψ ∗ ξ of a semi-universal family ξ over a space ( S, s ) where ψ : ( Z, z ) → ( S, s )is the base change map. Let G be the automorphism group of the centralfiber. The analytic quotient p : S → S// an G ⊂ M GIT is a local model, andwe set ϕ = p ◦ ψ . So far ψ is uniquely determined by the deformation ζ onlyon the set of polystable points. If z ∈ Z is any point, and s = ψ ( z ), thenthe image in S// an G corresponds to a polystable point in the closure G · s of the orbit of s which is unique up to the action of K . Let ψ and e ψ betwo choices for the same deformation ζ . Then for any z ∈ Z , and s = ψ ( z ). e s = e ψ ( z ) the bundles E s and E e s are isomorphic. By Proposition 3 the points s and e s are in the same G -orbit, hence the images in S// an G are the same.This shows existence and uniqueness of ϕ .In order to prove (iii) a map χ : M GIT → e N has to be constructed. Thepointwise definition and uniqueness of any such map χ follows because ofcondition (i) holding for both M GIT and e N . The holomorphicity of χ isshown as follows: Let ( S, p −→ S// an G ⊂ M GIT be a local model. Then by(ii) for e N there exists a holomorphic function e ϕ : S → e N so that χ ◦ p = e ϕ pointwise. The map can be extended to the respective G -invariant space e S . e S p / / e ϕ ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ e S// an G ⊂ M GITχ (cid:15) (cid:15) e N Let Q be an open neighborhood of e ϕ (0). Then holomorphic functions on Q ,pulled back to e S are G -invariant, hence holomorphic on χ − ( Q ) ⊂ M GIT byTheorem 3. This shows that χ is holomorphic. For families over arbitraryspaces Z in the sense of condition (ii) the construction is compatible witha base change map ψ : Z → S , where S is the base of a semi-universal deformation. This implies the claim together with the uniqueness of mapsfrom parameter spaces to M GIT and e N resp. (cid:3) Generalized Weil-Petersson metric General theory. The coarse moduli space of stable holomorphic vec-tor bundles on a compact K¨ahler manifold is known to carry a K¨ahler form ω W P (for notation and properties cf. [Bi-Sch]). As a Hermitian metric itis defined as the L -inner product of Kodaira-Spencer tensors that are har-monic with respect to Hermite-Einstein metrics on the given bundle and theK¨ahler form on the fixed manifold.Let {E s } s ∈ S be a holomorphic family of stable vector bundles parameter-ized by a reduced complex space S given by a holomorphic vector bundle E on X × S , such that E s = E| X × { s } . Let h be a Hermitian metric such that h |E s is Hermite-Einstein, and denote by F the curvature form of h , whereΛ( √− F |E s ) = λ · id E s λ = 2 π ( c ( E s ) ∧ ω n − X )[ X ] rank ( E s ) · vol ( X ) . Set E = E s for s ∈ S . Let again ρ : T s S → H ( X, End ( E ))be the Kodaira-Spencer map and v ∈ T s S a tangent vector. Then theHermite-Einstein condition implies immediately that the harmonic repre-sentative A v of ρ ( v ) equals the contraction of the tangent vector with thecurvature form of E resulting in a ∂ -closed (0 , A v = v y √− F | E. This equation also holds for polystable structures, so is meaningful to extendthe Weil-Petersson inner product to polystable points.The above fact yields that the Weil-Petersson form satisfies a general fiberintegral formula, which immediately implies that the result is a positivedefinite, d -closed, real (1 , ω W PS = 12 Z X × S/S tr( F ∧ F ) ∧ ω n − + λ Z X × S/S tr ( √− F ) ∧ ω n . The second term in (7.2) and an unwanted contribution in the first termcancel out. By rescaling the hermitian metric on E both of these can assumedto be equal to zero.7.2. Application to the classifying space M GIT . Let E be a polystablevector bundle with a semi-universal deformation parameterized by a reducedcomplex space S ⊂ U ⊂ V such that E = E s in the situation of Sections 5and 6. The local model for the classifying space is equal to S// an G , and itsunderlying topological space is equal to the quotient ( µ − (0) ∩ S ) /K .By Theorem 5 and Theorem 7 the existence of the semi-universal familyamounts to the construction of a semi-connection on E × U over X × U that is integrable over X × S . Denote by F the curvature form. Thenthe righthand side of the fiber integral (7.2) taken over X × U/U defines asymplectic form ω U of class C ∞ . N ANALYTIC APPLICATION OF GEOMETRIC INVARIANT THEORY 17 Proposition 6. The restriction of the form ω U to S is of type (1 , . Theform ω U | ( µ − (0) ∩ U ) is K -invariant. Its restriction to the stable locus S ′ ⊂ S ⊂ U is the pull-back of the Weil-Petersson form. At polystablepoints the form ω W PS defines the inner product of harmonic Kodaira-Spencertensors. At ∈ S it is positive definite.Proof. At integrable points the curvature F is of type (1 , 1) so that ω U is oftype (1 , 1) on S . At points of µ − (0) the relevant connection is K -invariant,so is F , which shows that the fiber integral has this property. At points of S ∩ µ − (0) equation (7.1) is also valid so that (7.2) defines the Weil-Peterssoninner product. (cid:3) For the restriction of the moduli space M GIT to a (smooth) connectedcomponent M consisting of objects whose semi-universal deformations areunobstructed more can be said. Singular symplectic quotients and locallystratified spaces were studied by Sjamaar and Lerman [S-L]. Here the com-plex structure of the classifying space is of interest. We assume that on M the stable points form an everywhere dense subset. Theorem 9. The Weil-Petersson form on the stable locus of M extendsas a positive (1 , -current ω W P to M that possesses locally a continuous ∂∂ -potential χ .Proof. The classifying space is locally of the form e U // an G with S = U in theabove notation. The space U can be chosen in a way that ω U possesses a ∂∂ -potential χ of class C ∞ . A C ∞ K -invariant function on µ − (0) is givenby integration over the group K which descends to a continuous function χ on the quotient µ − (0) /K . At the open locus of stable points, locallythe action of K is of maximal differentiable rank so that χ is of class C ∞ on the set of stable points and is a potential for the Weil-Petersson formthere. Because of the continuity of local potentials these define a globalextension as a positive current (with positivity being primarily defined onthe normalization). (cid:3) The Weil-Petersson form for projective manifolds. The aim is todefine an extension of the Weil-Petersson form also if the parameter spacesfor polystable bundles are singular. This is possible for projective manifolds X such that ω = c ( L , h L ) for a positive holomorphic line bundle L . Thefollowing general fact about determinant line bundles and Quillen metricsis applicable (cf. eg. [Bi-Sch], in particular (4 . 11) for the definition of thedeterminant line bundle λ equipped with the Quillen metric h Q on the base S for a family of hermitian holomorphic vector bundles on X × S , and (4 . λ, h Q )). The formulas hold for a holomorphic family( E , h ) of hermitian vector bundles of rank r over X × S . The determinantline bundle λ on S is defined in terms of a certain virtual holomorphic vectorbundle:(7.3) λ = det Rp ∗ (cid:16) ( End ( E ) − O r X × S ) ⊗ ( p ∗ L − p ∗ L − ) ⊗ ( N − (cid:17) The Quillen metric h Q depends upon the fiberwise L norms and the zetafunction with respect to the fiberwise Dirac operators/Laplacians of theholomorphic bundles that are involved. In particular the Quillen metric is of class C ∞ . The formula of Bismut, Gillet, and Soul´e (cf. [Bi-Sch] for a listof references) implies [Bi-Sch, (4.13)] so that(7.4) c ( λ, h Q ) = − Z X × S/S td( X × S/S, ω ) · ch( F , h ) ! (1 , , where td( X × S/S ) denotes the relative Todd character form, and where F denotes the virtual bundle on the righthand side of (7.3). For singularspaces S see [F-S]. The fiber integral formula (7.2) for the Weil-Peterssonform together with this formula imply the following fact. Fact. Let a semi-universal deformation of a polystable holomorphic vectorbundle be given with parameter space S . Then on the stable locus S ′ up to afixed positive numerical constant the curvature form of the determinant linebundle ( λ, h Q ) is equal to the pull-back of the Weil-Petersson form ω W PS ′ : c ( λ, h Q ) | S ′ ≃ ω W PS ′ . In general the fiber integral (7.2) only implies that ω W PS is the restrictionof a symplectic form on the smooth ambient space U to the reduced analyticsubspace S . In the case of a Hodge manifold X it possesses a ∂∂ -potential ofclass C ∞ . Now the arguments of Section 7.1 are applicable to the situationwhere semi-universal deformations are not necessarily unobstructed. Theorem 10. Let ( X, ω ) be a Hodge manifold with ω = c ( L , h L ) , anddenote by M GIT the moduli space of polystable vector bundles that canbe deformed locally into stable vector bundles. Then the Weil-Peterssonform on the stable locus extends to M GIT as a positive current with localcontinuous ∂∂ -potentials. The abstract methods of Dervan and Naumann [D-N] are applicable now.We indicate their argument very briefly. The coherent sheaf on the GIT-quotient for a local model defined by G -invariant sections of the respectivedeterminant line bundle becomes invertible, when the determinant line bun-dle is replaced by a finite power given by the number of connected compo-nents of G (cf. [Sj]). Using Kirwan’s stratification of GIT quotients [Kir] theyshow that the continuous potentials for the Weil-Petersson metric give riseto continuous (singular) hermitian metrics on these line bundles, which is ofclass C ∞ on the stable locus using the stratification of the GIT-quotients.Altogether the results of Dervan and Naumann yield the following. Derived from determinant line bundles equipped with Quillen metricsthere exists a holomorphic line bundle λ M on M , equipped with a contin-uous (singular) hermitian metric, which is of class C ∞ on the stable locus.Its curvature current is positive, and equal to the Weil-Petersson form onthe stable locus. References [AHS] Atiyah, Michael; Hitchin, Nigel J.; Singer, Isadore M.: Self-duality in four-dimensional Riemannian geometry. Proc. R. Soc. London. A , 425–461(1978). N ANALYTIC APPLICATION OF GEOMETRIC INVARIANT THEORY 19 [Bi-Sch] Biswas, Indranil; Schumacher, Georg: The Weil-Petersson current for moduliof vector bundles and applications to orbifolds. Ann. Fac. Sci. Toulouse XXV , 231–247 (1987).[D] Dr´ezet, Jean-Marc: Luna’s slice theorem applications. Wi´sniewski, Jaros law A.(ed.), Algebraic group actions quotients. Notes of the 23rd autumn school inalgebraic geometry, Wykno, Pol , September 3—10, 2000. Cairo: Hindawi Pub-lishing Corporation. 39–89 (2004).[F1] Fan, Yue: Construction of he Moduli Space f Higgs Bundles Using AnalyticMethods. arXiv:2004.07182.[F2] Fan, Yue: The Orbit Type Stratification of the Moduli Space of Higgs Bundles.arXiv:2005.13649.[F-S] Fujiki, Akira; Schumacher, Georg: The moduli space of extremal compact K¨ah-lermanifolds and generalized Weil-Petersson metrics. Publ. Res. Inst. Math. Sci. , 101–183 (1990).[G-R] Grauert, Hans; Remmert, Reinhold: Coherent analytic sheaves. Grundlehrender Mathematischen Wissenschaften, . Berlin etc.: Springer-Verlag (1984).[He] Heinzner, Peter: Geometric invariant theory on Stein spaces. Math. Ann. ,631-662 (1991).[H-L] Heinzner, Peter; Loose, Frank: Reduction of Complex Hamiltonian G -Spaces.Geom. Funct. Anal. , 288–297 (1994).[Ho] Hoskins, Victoria: Stratifications associated to reductive group actions on affinespaces. Quart. J. Math. , 1011–1047 (2014).[Kir] Kirwan, Frances C.: Cohomology of quotients in symplectic algebraic geometry.Princeton University Press: Princeton, NJ, 1984.[K-K] Kaup, Ludger; Kaup, Burchard: Holomorphic Functions of Several Variables.Walter de Gruyter, Berlin, New York, 1983.[KN] Kempf, George; Ness; Linda: The length of vectors in representation spaces.Algebraic geometry, Proc. Summer Meet., Copenh. 1978, Lect. Notes Math. , 233–243 (1979).[Ku] Kuhlmann, Norbert: ¨Uber holomorphe Abbildungen komplexer R¨aume. Arch.Math. , 81–90 (1964).[Lu] Luna, Domingo: Slices ´etales. Bull. Soc. Math. France, M´emoire (1973),81–105.[M¨u] M¨uller, Gerd: Reduktive Automorphismengruppen analytischer C -Algebren. J.Reine Angew. Math. , 26–34 (1986).[MFK] Mumford, David; Fogarty, John; Kirwan, Frnces C.: Geometric invariant theory(3rd ed.) Springer, Berlin, 1994.[Nee] Neeman, Amnon: Analytic questions in geometric invariant theory. Contemp.Math. , 11–23 (1989).[Nee1] Neeman, Amnon: The Topology of Quotient Varieties Ann. Math. , 419–459(1985).[Nes] Ness, Linda: A stratification of the null cone via the moment map. With anappendix by David Mumford. Amer. J. Math. , 1281–1329 (1984).[New] Newstead, Peter E.: Introduction to moduli problems and orbit spaces. Lectureson Mathematics and Physics. Mathematics. Tata Institute of Fundamental Re-search 51.[New1] Newstead, Peter E.: Geometric invariant theory. Brambila-Paz, Leticia (ed.) etal., Moduli spaces and vector bundles. London Mathematical Society LectureNote Series , 99–127 (2009).[R] Remmert, Reinhold: Holomorphe und meromorphe Abbildungen komplexerR¨aume. Math. Ann. , 328–370 (1957). [Ro] Roberts, Mark: A Note on Coherent G-Sheaves. Math. Ann. , 573–582(1986).[Sch] Schumacher, Georg: An extension theorem for Hermitian line bundles. Analyticand Algebraic Geometry. 225–237, Hindustan Book Agency, New Delhi, 2017,arXiv:1507.06195.[S-L] Sjamaar, Reyer; Lerman, Eugene: Stratified symplectic spaces and reduction.Ann. Math. , 375–422 (1991).[Sj] Sjamaar, Reyer: Holomorphic slices, symplectic reduction and multiplicities ofrepresentations. Ann. Math. , 87–129 (1995).[Sn] Snow, Dennis: Reductive group actions on Stein spaces. Math. Ann. , 79–97(1982).[Sz] Sz´ekelyhidi, G´abor: The K¨ahler-Ricci flow and K-polystability. Am. J. Math. , 1077–1090 (2010).[Th] Thomas, Richard P.: Notes on GIT symplectic reduction for bundles and va-rieties. In: Surveys in differential geometry Vol. , Int. Press Somerville, MA2006, 221–273.[W] Whitney, Hassler: Complex Analytic Varieties. Addison-Wesley, Reading, Mass.1972.[Wo] Wolpert, Scott.: On Obtaining a Positive Line Bundle From the Weil-PeterssonClass. Amer. J. Math. , 1485–1507 (1985). School of Mathematical Sciences, University of Adelaide, Adelaide, Aus-tralia 5005 Email address : [email protected] Fachbereich Mathematik und Informatik, Philipps-Universit¨at Marburg, Lahn-berge, Hans-Meerwein-Straße, D-35032 Marburg, Germany Email address ::