Polystable bundles and representations of their automorphisms

: Using a quasi-linear version of Hodge theory, holomorphic vector bundles in a neighbourhood of a given polystable bundle on a compact Kähler manifold are shown to be (poly)stable if and only if their corresponding classes are (poly)stable in the sense of geometric invariant theory with respect to the linear action of the automorphism group of the bundle on its space of infinitesimal deformations.


Introduction
The aim of this paper is to shed light on the relationship between the notion of polystability for holomorphic vector bundles on a compact Kähler manifold and the classical GIT notion of polystability for points in a representation space of a reductive Lie group. More speci cally, we study the action of the automorphism group of a polystable bundle on its space of in nitesimal deformations and relate this to an action on the parameter space of a semi-universal deformation. In so doing, it is shown that the relationship between the in nite-dimensional bundle-theoretic notion and the nite-dimensional representation-theoretic notion is more than just analogy.
In the sense of Kuranishi's approach to deformation theory, Miyajima [28] constructed a semi-universal deformation of a holomorphic vector bundle E on a compact (Kähler) manifold from in nitesimal deformations, i.e., from elements of H (X, End E ). A parameter space S is given as the zero-set of a holomorphic map Ψ : N → H (X, End E ), where N ⊆ H (X, End E ) is a certain neighbourhood of the origin.
Deformations of non-stable holomorphic vector bundles can be realised as pull-backs of semi-universal deformations by means of base change maps, but these need not be not uniquely determined and therefore an action of the automorphism group on a semi-universal deformation cannot be expected in general. Nevertheless, a holomorphic map Ψ with the following property exists:

Theorem 1. Let E be a polystable holomorphic vector bundle on a compact Kähler manifold (X, ω). Then there is a holomorphic map Ψ de ned on an open neighbourhood N of zero in H (X, End E ) with values in H (X, End E ) that is equivariant with respect to the action of Aut E . The space S = Ψ − ( ) is the parameter space of a semi-universal deformation of E , and the action of Aut(E ) on H (X, End E ) induces a holomorphic action on the germ of (S, ). Points of S correspond to isomorphic bundles if and only if they lie in the same orbit of Aut E acting on the ambient space H (X, End E ).
Strictly speaking, the (reductive) group with respect to which stability and its concomitant notions are applied (to elements of H (X, End E )) is the quotient of Aut E by the subgroup C * · of non-zero scalar multiples of the identity since that subgroup acts trivially. In general, the ine ectivity kernel of the representation of Aut E may be larger, although in this case no small deformation of E is stable; see the rst of the Remarks at the end of §3.
For bundles near E that are semistable but not polystable, the following analogue of the standard GIT result holds: Theorem 4. Ifᾱ is a polystable point in the closure of the orbit of α ∈ Ψ − ( ) under the action of Aut E , the polystable bundle Eᾱ is isomorphic to the graded object Gr(Eα) associated to a Seshadri ltration of Eα.
The manuscript is organised as follows. In the next section notation is xed, based largely on [23]. The second section contains a construction for a slice of the complexi ed gauge group G acting on the space of hermitian connections on a given bundle, leading to the construction of the function Ψ of Theorem 1.
In the third and fourth sections semistability and polystability are taken into consideration respectively, including the estimates required for the subsequent solving of the Hermite-Einstein equations. The fth section includes proofs of Theorem 2 and of the "if" part of Theorem 3, as well as proving the equivariance of Ψ and the last statement of Theorem 1. In the sixth section, using ellipticity of the equations and by restricting to slices, the action of G on the space of integrable semi-connections is reduced to an action of Aut E on H (X, End E ), this making critical use of the existence of a Hermite-Einstein connection on E . The process of comparing the two notions of polystability is commenced in the seventh section, and this is continued over the course of the eighth section, concluding with the completion of the proof of Theorem 3 in §8.
In §9, consideration is given to those bundles that are semistable but not polystable, particularly those for which the closure of their orbit under G is the polystable bundle E , with the conclusions summarised by Theorem 4. The eleventh and nal section presents several general remarks and observations concerning the methods and results in the paper.

Preliminaries
Following Kobayashi [23], a semi-connection on a complex vector bundle E is a C-linear map ∂ on di erentiable local sections of E taking values in Λ , ⊗ E and satisfying the ∂-Leibniz rule; here Λ p,q is the space of (p, q)-forms on X. A semi-connection has natural prolongations Λ ,q ⊗ E → Λ ,q+ ⊗ E, and it is by de nition integrable if ∂ • ∂ = . A semi-connection on a bundle induces semi-connections on associated bundles in the usual ways, these being integrable if the original semi-connection is integrable. The set of semi-connections on E is an a ne space.
Let ∂ be an integrable semi-connection on E. Then every semi-connection ∂ on the bundle can be written ∂ = ∂ + a ′′ for some unique ( , )-form a ′′ with coe cients in End E, and the integrability condition for ∂ is ∂ a ′′ + a ′′ ∧ a ′′ = . The notation, which will be used throughout, is derived from the often-used convention to denote by a ′ and a ′′ respectively the ( , )-and ( , )-components of a -form a, where that -form may take values in some vector bundle.
The group G of (di erentiable) complex automorphisms of E acts on the a ne space of semi-connections as a "complex gauge group". This action, which preserves the integrability condition, is denoted by g · ∂ := g • ∂ • g − . A holomorphic structure is de ned by an integrable semi-connection, and two such structures are isomorphic if and only if they lie in the same orbit of G. By virtue of the Newlander-Nirenberg theorem, this view of holomorphic structures is equivalent to the more usual one of holomorphic vector bundles being described by systems of holomorphic transition functions.
Denote by A p,q (E) the global smooth (p, q)-forms with coe cients in E. For an integrable semi-connection ∂ de ning a holomorphic structure E , the Dolbeault cohomology groups H q ∂ (X, E) = ker ∂ : A ,q (E) → A ,q+ (E) / im ∂ : A ,q− (E) → A ,q (E) are denoted by H q (X, E ), these being nite-dimensional spaces with H (X, End E ) being by de nition the space of in nitesimal deformations of E .
Analysis of the small deformations of E is achieved with the introduction of metrics on both the bundle E and the manifold X. A hermitian metric h is xed once and for all on the bundle E, which is henceforth denoted E h . The group G is the complexi cation of the group U of unitary gauge transformations. The hermitian structure on E h gives a one-to-one correspondence between semi-connections ∂ and hermitian connections d = ∂ + ∂ on E h , and if d = d + a for some skew-adjoint a ∈ A (End E h ), then ∂ = ∂ + a ′ and ∂ = ∂ + a ′′ where a = a ′ + a ′′ and a ′ = −(a ′′ ) * . Henceforth, all connections are taken to be hermitian.
So τ is ∂-harmonic if and only if it lies in the kernel of the ∂-Laplacian ′′ = ∂ ∂ * + ∂ * ∂. In general, there is an L -orthogonal decomposition where H ,q = H ,q ( ∂) is the space of ∂-harmonic ( , q)-forms. Here, notation has been abused in that A ,q (E h ) is no longer denoting the space of smooth sections of E h , but rather the space of global sections of Λ ,q ⊗ E h that are square integrable, and the closures on the right are the closures in L of the images under ∂ and ∂ * of the spaces of smooth global sections. Standard elliptic regularity implies that the ∂-harmonic sections are smooth, at least if the connections are. This abuse of notation will be employed throughout, so that A ,q (E h ) will always denote a space of global sections of Λ ,q ⊗E h but with the degree of di erentiability and/or integrability to be speci ed in the respective context. Sobolev spaces of functions are denoted by L p k , meaning all weak derivatives up to and including those of order k lie in L p . Having xed a base connection on E h once and for all, the spaces of L p k elements of A ,q (E h ) acquire norms that make them Banach spaces.
Henceforth, a number p > n (for n = dim X) will be xed, so by the Sobolev embedding theorem there are compact embeddings L p ⊂ C and L p ⊂ C . By standard elliptic theory on compact manifolds, for an integrable connection d = ∂ + ∂ there is a constant C > (depending upon d) such that where Π ,q τ is the L -orthogonal projection of τ ∈ A ,q (E h ) in H ,q ( ∂). Connections on E h will be permitted to have coe cients in L p , and the fact that an integrable L p connection de nes a holomorphic structure in the usual way follows from Lemma 8 of [5]. When not indicated by a subscript on the norm symbol, τ will always mean τ L . If d is an integrable semi-connection and g ∈ G, the Dolbeault cohomology groups de ned by ∂ and by g · ∂ are isomorphic, the isomorphism induced by mapping a ∂-closed ( , q)-form τ ∈ A ,q (E h ) to the (g · ∂)closed ( , q)-form g τ. This isomorphism does not preserve harmonic representatives in general, unless g ∈ U in which case it also preserves L norms.
Subsequently in this paper it will be useful to consider connections that are not integrable, in which case the Dolbeault cohomology groups are not de ned. However, one can still de ne the spaces H ,q ( ∂) of ∂-harmonic ( , q)-forms as null spaces of the appropriate Laplacians, these still being nite-dimensional spaces consisting only of smooth forms (if ∂ is itself smooth).
If dω = , the formal adjoints ∂ * and ∂ * have alternative expressions in terms of the Kähler identities:

A neighbourhood of a holomorphic bundle
As in the previous section, let X be a compact complex manifold and let E be a holomorphic vector bundle on X. A construction of a semi-universal deformation has been given by Forster and Knorr [14] using power series methods. More in the spirit of Kuranishi's construction [24] is Miyajima's construction [28] (cf. also [15]). Either way, there is a holomorphic function Ψ de ned in a neighbourhood of ∈ H (X, End E ) such that Ψ − ( ) is a complete family of small deformations of E . In this section, the construction of a particular such function Ψ will be presented in a manner to suit the purposes of the remainder of the paper. The entire discussion is essentially an n-dimensional version of the -dimensional case presented in §6.4 of [12].
Fix a positive ( , )-form ω on X, which at this stage is not assumed to be d-closed. Let E h be the complex bundle underlying E equipped with a xed hermitian structure, and let d be a connection on E h inducing the holomorphic structure E .
There is a number ϵ > depending on d with the property that for any integrable hermitian with µ and ν respectively orthogonal to the kernels of ∂ and ∂ * . Then so on taking inner products with ν and µ respectively it follows that where · is the L norm. Since p > n, the Sobolev embedding theorem gives sup |a| ≤ C a L p for some constant C independent of a and d , so after adding the last two inequalities it follows that Since µ and ν are orthogonal to ker ∂ and ∂ * respectively, ellipticity of ′′ implies that µ ≤ C ∂ µ and ν ≤ C ∂ * ν for some constants C , C , so τ ≤ C a L p τ for some new constant C = C (d ), giving the stated result.
This proposition clearly implies the well-known and standard semi-continuity of cohomology. It also has the following useful consequence, resulting from equality of dimensions of cohomology groups: If a ′′ ∈ A , (End E h ) and g ∈ G, then g · ( ∂ + a ′′ ) = ∂ − ∂ g g − + ga ′′ g − . The map maps into the subspace of A , (End E h ) orthogonal to the kernel of ∂ , and its linearisation at ( , ) in the Gdirection is A , (End E h ) γ → − ′′ γ. This is an isomorphism from the space of L p sections in A , (End E h ) orthogonal to ker ∂ = H , to the space of L p sections in A , (End E h ) orthogonal to ker ∂ . The implicit function theorem for Banach spaces now implies: This is a complex analogue of xing a unitary gauge for hermitian connections near a given such connection, corresponding to the linear operation of projecting a ( , )-form orthogonal to the range of ∂ . If ∂ + a ′′ is an integrable semi-connection with a ′′ L p su ciently small as dictated by this lemma, after applying an appropriate complex gauge transformation so that the new semi-connection ∂ +ã ′′ satis es ∂ * ã ′′ = , Corollary 2.4 gives an estimate of the form ã ′′ L p ≤ C Π , ã ′′ L . Consequently, the well-known result that if H (X, End E ) = then every small deformation of E is isomorphic to E follows immediately. A simple but pertinent example is given by that of a trivial bundle on P .
It follows from Corollary 2.4 that for integrable semi-connections in the "good" complex gauge lying in a su ciently small neighbourhood of ∂ in L p , the projection onto the ∂ -harmonic component is a homeomorphism onto a closed subset of an open neighbourhood of in H , , where H ,q denotes H ,q ( ∂ , End E h ) in this section. This closed subset is the zero-set of the holomorphic function Ψ mentioned in the introduction, as will now be discussed.
If a ′′ ∈ A , (End E h ) is ∂ -closed, the semi-connection ∂ + a ′′ is not integrable in general, since a ′′ ∧ a ′′ need not be zero. But one might attempt to perturb a ′′ in such a way that the corresponding perturbed semiconnection is integrable.
The derivative of the map is an L p section in A , (End E h ). Since and ∂ * (a ′′ + ∂ * β) ∧ (a ′′ + ∂ * β) lies in L p (again using p > n and the Sobolev embedding theorem), it follows that Π ⊥ τ lies in L p , and that the composition (a ′′ , β) → Π ⊥ τ is continuous with respect to the L p × L p topology on the domain and the L p topology on the codomain. By ellipticity of ′′ on A , (End E h ), there is a constant K such that sup X |e| ≤ K for any e ∈ H , with e L = . If ∂ a ′′ = , then ∂ (a ′′ ∧ a ′′ ) = , which implies that and hence that the L p norm of Π ⊥ ∂ a ′′ + a ′′ ∧ a ′′ ) is uniformly bounded by a constant multiple of the L p norm of a ′′ ∧ a ′′ in this case. Another application of the implicit function theorem now yields: Proposition 2.6. There exist ϵ > and C > with the properties that for any ∂ -closed a ′′ ∈ A , (End E h ) satisfying a ′′ Remarks.

1.
If β is as in this lemma, then a ′′ + ∂ * β L p is uniformly bounded by a constant multiple of a ′′ L p . In fact, ifã ′′ := a ′′ + ∂ * β, then τ := ∂ ã ′′ +ã ′′ ∧ã ′′ satis es weakly, so by elliptic regularity it follows that τ in fact lies in L p . Hence by Lemma 2.3, if a ′′ L p (and hence ã ′′ L p ) is su ciently small, there is a constant C = C(d ) such that τ L p ≤ C Π , τ L . 2. Proposition 2.6 remains valid even if d is not integrable-all that is required is a uniform C bound on F , (d ). The proof as given only needs minor modi cation by noting that ∂ τ involves an extra term
From (1.2) there is a constant c > depending only on d such that any α ∈ A , with ∂ α = = ∂ * α satis es α L p ≤ c α . Thus there is a number δ > depending only on d such that α L p < ϵ if α < δ, and for such α there is a unique β ∈ A , (End E h ) orthogonal to ker ∂ * with β L p ≤ C ′ α for which the De ne the function Ψ on the set of ∂ -harmonic forms α ∈ A , (End E h ) with α < δ that takes values in H , ( ∂ , End E h ) by Then the zero set of Ψ parameterises precisely the integrable connections in the good complex gauge in this L p neighbourhood of d , in the sense of de ning a semi-universal deformation. By xing orthonormal bases for each of H , and H , and working in these bases, it is immediate that Ψ is holomorphic. As will be shown later (Corollary 4.3), in the special case that d is Hermite-Einstein, Ψ is equivariant with respect to the action of Aut(E ) on these spaces, a consequence of the third remark above.

A neighbourhood of a semistable bundle
With the same notation and conventions as in the previous section, we assume from now on that (X, ω) is Kähler. The results in this section on bounds of slopes of subsheaves, (which can be seen as analogues of the standard fact that Hermite-Einstein implies semistable), may be of independent interest.
where d A and d B are the induced hermitian connections on A and B, and β ∈ A , (Hom Since β is a ( , )-form, i tr β ∧ β * is a non-positive ( , )-form and therefore i tr F A ≤ tr (Π A iF E Π A ). Applying ω n− ∧ and integrating over X, it follows that c (A)· [ω n− ] is bounded above by a xed multiple of F E L . Now if A is only a subsheaf of E of rank a > with torsion-free quotient B, replace E with Λ a E, A by the maximal normal extension of Λ a A in Λ a E (which is a line bundle) and then after blowing up the zero set of the induced section of Hom(det A, Λ a E) and resolving singularities, there is again an upper bound on the degree of the desingularised subsheaf in terms of the L norm of F E on the blowup. This upper bound depends on the metric used on the blowup, but as in § §2, 3 of [5], there is a family ωϵ of such metrics converging to the pullback of ω, and the resulting limit then gives the same bound: deg(A) is bounded above by a xed multiple of F(d) L for any subsheaf A ⊂ E with torsion-free quotient.
To obtain uniform bounds on c (A) , suppose again initially that A is a subbundle of E. For notational simplicity, let f := i tr F A and g := i tr (Π A F E Π A ), so from the preceding arguments, g−f ≥ as hermitian forms on TX. The space H , R (X) of real harmonic ( , )-forms is nite dimensional, so by picking an orthonormal basis, it is apparent that there is a constant C > depending on ω such that −C ω ≤ φ ≤ C ω for any φ ∈ H , as real ( , )-forms pointwise on X. Applying ω n− ∧ and integrating over X, it follows that and by allowing φ to vary over the harmonic ( , )-forms of norm , it follows that for some new constant C ′ independent of d E . This implies the second statement of the lemma when A is a sub-bundle.
In the case that A is only a subsheaf rather than a subbundle, the same method as earlier can be used to reduce to the case of a line subbundle on a blowup of X. It need only be checked that for metrics of the kind ωϵ mentioned earlier, lim ϵ→ c (L) L (ωϵ) = for any line bundle L on the blowup that is trivial o the exceptional divisor, which is straight-forward to verify. Proof. If not, there is a sequence (a j ) ∈ A (End E h ) with a j L + a j L → and d j := d + a j integrable such that the holomorphic bundle E j de ned by d j is not semistable, so there is a subsheaf A j ⊂ E j with torsion-free quotient that strictly destabilises E j . Passing to a subsequence, it can be assumed that the ranks of the sheaves A j are constant, a say. The hypotheses on a j imply that F(d j ) L is uniformly bounded and therefore so too is F(d j ) L , and consequently Lemma 3.1 yields a uniform upper bound on deg(A j ). Since deg(A j ) is also uniformly bounded below, these bounds together with the bounds on F(d j ) L then give uniform bounds on the L norms of the harmonic representatives of the forms representing c (A j ) in H dR (X), and hence there is a convergent subsequence. Since the image of H (X, Z) in H dR (X) is discrete, this convergent subsequence must be eventually constant, so after passing to another subsequence, it can be assumed that c (A j ) is constant, c say. Since X is Kähler, Pic c (X) is a compact torus, so after passing to another subsequence, it can be supposed that det A j converges to a holomorphic line bundle L on X. Since (det A j ) * ⊗ Λ a E j has a non-zero holomorphic section for each j, so too does L * ⊗ Λ a E , and therefore Λ a E is strictly destabilised by L. But by Theorem 2 of [17], semistability of E implies that of Λ a E , giving the desired contradiction.

Remark.
It is worth noting that this result is rather delicate in that the Kähler class must be xed. For example, every non-split extension of the form → O(− , ) → E → O( , ) → on P × P is strictly stable with respect to ω t := π * ω + t π * ω for t > , is semistable but not polystable for t = , and is strictly unstable if < t < . The result also fails in the non-Kählerian case, at least when the degree fails to be topological. For example, if L is a non-trivial holomorphic line bundle on an Inoue surface with L ⊗ L trivial, the direct sum of L with the trivial line bundle is polystable with respect to every Gauduchon metric, every small deformation is again a direct sum, and of these, the generic one is strictly unstable with respect to every Gauduchon metric. In particular, semistability is not a Zariski-open condition in this setting. In this case, the automorphism group of E = L ⊕ acts trivially on H (X, End E ), so each of its orbits is closed.

A neighbourhood of a polystable bundle
Families of holomorphic bundles near a polystable but not stable bre di er considerably from families near a stable bre. Fibres near to a stable bre are always stable, but the analogous fact does not hold (in general) in the polystable case: the implicit function theorem fails.
A holomorphic bundle E that is a non-split extension → A → E → B → by semistable bundles A, B of the same slope is semistable but not polystable, and cannot be separated from the direct sum A ⊕ B in the quotient topology on the space of integrable semi-connections modulo the complex gauge group. The corresponding holomorphic one-parameter family of extensions yields a family of holomorphic vector bundles being mutually isomorphic except for the direct sum. Consequently a coarse moduli space cannot exist. Given this, it makes sense to focus on polystable structures in a neighbourhood of a given polystable bundle, in which case a great deal more can be said than in the previous section. The same objects and de nitions as in the last section are used here, but now E is assumed to be a polystable holomorphic bundle.
The following is a minor generalisation of a well-known result essentially due to Kobayashi [21]. Although its proof is elementary, it is presented here for the reason that in some respects, it is the pivotal result used in the paper. Proof. The equation ∂s = implies ∂ s, s = s, ∂s , using the convention that ·, · is conjugate-linear in the rst variable. Therefore ∂∂ s, s = ∂s ∧ , ∂s + s, F , (d)s . Applying iΛ, it follows ′′ |s| + |∂s| = , so integration over X gives ∂s = .
Note that it is not assumed that d should be integrable.
For a connection d on E h whose central curvature F(d ) is a constant multiple of the identity, the central curvature of the connection induced by d on End E h is identically zero, so Lemma 4.1 implies: This corollary yields the equivariance property of the function Ψ asserted at the end of §2:
The proof of Proposition 3.2 combined with Lemma 4.1 have the following useful consequence, which simplies a number of subsequent arguments:

de nes a semistable holomorphic bundle E with the property that any subsheaf A ⊂ E with µ(A) = µ(E) and with E/A torsion-free is a sub-bundle of E.
Proof. If not, then from the proof of Proposition 3.2, there is a sequence of integrable semi-connections ∂ j = ∂ + a ′′ j with a ′′ j L p → such that the corresponding holomorphic bundle E j has a subsheaf A j with rk A j = a independent of j, µ(A j ) = µ(E j ) = µ(E ) for all j, and with E j /A j torsion-free but not locally free. Hence Λ a A j is a rank subsheaf of Λ a E j , and the bundle (det A j ) * ⊗ Λ a E j has a holomorphic section that has a zero. After passing to a subsequence, the Hermite-Einstein connections on the line bundles (det A j ) * can be assumed to converge to de ne a holomorphic line bundle L on X, and compactness of the embedding of L p in C implies that a subsequence of the integrable connections de ning (det A j ) * ⊗ Λ a E j converges uniformly in C to an Hermite-Einstein connection on L ⊗ E . After scaling the sections of (det A j ) * ⊗ Λ a E j so as to have L norm , the convergence of the connections implies that a subsequence of the rescaled sections converges weakly in L (say) and strongly in L to a holomorphic section of L ⊗ Λ a E of norm . From the Cauchy integral formula integrated over a poly-annulus P, a sequence of holomorphic functions converging in L on P is converging uniformly on compact subsets inside P, so if each term in the sequence has a zero there, then so too does the limit. But the limiting section is a holomorphic section of a bundle of degree zero that admits a connection with F = , so it is either identically zero or nowhere zero.
The conclusion of Lemma 4.1 can be strengthened to give a perturbed version; integrability of d is again not required.

Proposition 4.5. Suppose d is a connection on
There is a constant ϵ = ϵ(d ) > such that any connection da = d + a with a = a ′ + a ′′ , ∂ * a ′′ = and a L p < ϵ has the property that any section Proof. Suppose ∂ s + a ′′ s = for some section s ∈ A , (E h ), and write s = s + s where ∂ s = and s is orthogonal in L to the kernel of ∂ . By Lemma 4.1, d s = . Then Using the L inner product, this implies that For notational convenience, the group Aut E consisting of the ∂ -closed elements of G will be denoted by Γ henceforth. An equivalence relation ∼ is de ned on S = Ψ − ( ) by s ′ ∼ s ′′ if s ′′ = γ · s ′ for some γ ∈ Γ. In this sense there exists a (pointwise) quotient Ψ − ( )/Γ without assuming a genuine group action Γ × S → S, which does not exist in general (because of points being pushed outside the domain of Ψ by some elements of Γ). Proof. The continuity of the map is clear.
Since p > n, the Sobolev embedding theorem gives uniform C estimates on g and g − , So if a is su ciently close to a in L p , Proposition 2.6 now gives a uniquely determined α ∈ H , and β ∈ (ker ∂ * ) ⊥ ⊆ A , (End E h ) such that a ′′ = α + ∂ * β , with ∂ * ∂ a ′′ + a ′′ ∧ a ′′ ) = and with β L p ≤ C α . Thus for some new constant C = C(d ), In summary, given α ∈ H , su ciently close to , for each a ∈ A , (End E h ) su ciently close to a in L p there exists g ∈ G and α ∈ H , such that d + a is of the form d + a = g − · (d + a ) for a ′′ . If a L p + a L p < ϵ with ϵ as in Proposition 4.5, apply that proposition to the connection on End E h = Hom(E h , E h ) de ned by d + a on one side and d + a on the other, with s being the section g ∈ Hom(E h , E h ). It follows from that result that d g = (so g ∈ Γ) and that a g = ga . Writing a ′′ j = α j + ∂ * β j with β j determined by Proposition 2.6, orthogonality of the decomposition implies that α g = gα ; that is, α , α ∈ H , represent the same point in the quotient H , /Γ.
Note that again, the integrability of the connections d + a j is not critical; the essential ingredient is the "quasi-integrability" condition implicit in the statement of Proposition 2.6.
The main application of Proposition 4.5 will be to the connections on End E h induced by connections d on E h with F(d ) a scalar multiple of the identity, as in following theorem. Here [x, y] = xy − yx is the usual Lie bracket on endomorphisms, and the reader is alerted to the fact that the hypotheses on σ di er slightly in 1. and 2., although the conclusions are essentially the same.

Theorem 4.7. Let d be a connection on E h with i F(d ) = λ , and let d + a be a connection with a
There is a constant ϵ > depending only on d with the property that if a L p < ϵ then the following hold for any endomorphism σ ∈ A , (End E h ): Setting β = in 1. it follows from that case that d σ = = [α, σ]. Since σ commutes with α, it also commutes with α ∧ α, implying that As in the proof of 1., because β is orthogonal to ker ∂ * , so too is [ After applying ω n− ∧ to both sides and integrating over X, the fact that (σ * σ − σσ * ) lies in ker ∂ together with the fact that Λ(α * ∧ α + α ∧ α * ) is orthogonal to ker ∂ imply that [σ * , α] L = . Remark.
Given a ′′ ∈ A , (End E h ) with a ′′ L p small, Lemma 2.5 yields a uniquely determined g ∈ G near such that ∂ * (g · a ′′ ) = , where g · a ′′ := ga ′′ g − − ∂ gg − . Also, Proposition 2.6 yields a uniquely determined and g in G with both β and g − su ciently close to zero in L p , the element a ′′ of A , (End E h ) obtained by applying the operation of Lemma 2.5 to g · a ′′ + ∂ * β followed by the operation of Proposition 2.6 to the result of that has the form a ′′ = γa ′′ γ − for some automorphism γ ∈ Γ near , where Γ here and subsequently denotes the ∂ -closed elements of G. All of this follows using the analysis of Theorem 4.7.
The content of Theorem 4.7 implies and is implied by a corresponding result for connections on bundles of degree zero. For the purposes of transparency of the proof, the theorem was presented in terms of endomorphisms, but the alternative result is the following: There is a constant ϵ > depending only on d with the property that if a L p < ϵ then the following hold for any section s in A , (E h ): 2. If ∂ s + αs = and β ∈ (ker ∂ * ) ⊥ and ∂ * ( ∂ a ′′ + a ′′ ∧ a ′′ ) = then d s = and α s = = β s; 3. If ∂ s + a ′′ s = and iΛ(α ∧ α * + α * ∧ α) ∈ (ker ∂ ) ⊥ then d s = and αs, ( ∂ * β)s and α * s are all zero.  Hom(A , B ) Hom(B , A )).)

2.
As will be discussed in the next section, the condition that iΛ(α * ∧ α + α ∧ α * ) should be orthogonal to ker ∂ implies that the form α is an element of H , that is polystable with respect to the action of Γ in the sense of geometric invariant theory, and in that context, the last statement of the theorem can also be interpreted as a holomorphic condition on E determined by an algebraic condition on α. This is a manifestation of one half of the "local" Hitchin-Kobayashi correspondence that is at the heart of this paper, made precise in Theorem 5.4 below. The other half, or "converse" of this is the focus of attention in §6, §7 and §8.
The proof of the last statement of Theorem 1 can now be completed: Proof. The "only if" part of the statement has been proved in the last paragraph of the proof of Corollary 4.6. For the converse, given γ ∈ Γ such that α ′ = γαγ − , take the connection on Hom(E h , E h ) induced by d + α − α * and d + α ′ − α ′* and apply 2. of Corollary 4.8, substituting γ for s (and Hom(E h , E h ) for E h ).

Γ-stability and ω-stability.
For the convenience of the reader this section commences with a summary of notions and facts from geometric invariant theory as far as these will be used here. The primary references include [29], [31], [20] and [34].
Recall that when a reductive Lie group G acts linearly on a nite-dimensional complex vector space V, a point v ∈ V\{ } is unstable for the action if ∈ G · v, is semistable if ∉ G · v, is polystable if v is semistable and G · v is closed, and is stable if v is polystable and the isotropy subgroup of v is nite.
Fixing a positive hermitian form on V, in the closure of each orbit there is a point of smallest norm, unique up to the action of the compact subgroup of G preserving the hermitian form. For each v ∈ V, the derivative at ∈ G of the function G g → g · v gives a function µ : V → g * , the moment map, and its zeros on the set of semistable points are precisely the points of smallest norm in the closed orbits.
The Hilbert-Mumford criterion states that a point v ∈ V is stable if and only if it is stable for everyparameter subgroup in G. A -parameter subgroup is given by a homomorphism χ : C * → G, giving a representation of C * on V. The irreducible representations of C * are all -dimensional, so V splits as a direct sum of -dimensional subspaces V j on each of which C * acts with a given weight w j ∈ Z: If v has -dimensional orbit under χ, it is clear that this orbit is closed if and only if C * t → χ(t) · v ∈ V is proper, which in turn is equivalent to the condition that the maximum weight w and minimum weight w of χ on the non-zero components of v in its decomposition into irreducibles should di er in sign. Since t → χ(t − ) is another -parameter subgroup of G for which the maximum and minimum such weights are respectively −w and −w , the criterion reduces to the condition that v is a stable point for the action of G if and only if w is negative for every -parameter subgroup of G. In practice, this is the condition that lim t→ log χ(t) · v / log |t| < for every -parameter subgroup χ of G, which is an analogue of the numerical condition in the de nition of stability for a holomorphic vector bundle on a compact Kähler manifold.
Consider now the situation discussed in the previous section: (X, ω) is a compact Kähler n-manifold and E h is a complex vector bundle over X equipped with a xed hermitian structure. The group G of complex automorphisms of E h acts on the a ne space A of hermitian connections on E h , preserving the subspace of integrable such connections. The group G is the complexi cation of the group U of unitary gauge transformations.
A connection d ∈ A that is integrable and has curvature F(d ) satisfying i F(d ) = λ is a minimum of the Yang-Mills functional. By Lemma 4.1, the group Γ ⊂ G of complex gauge transformations xing ∂ is the same group as the group that xes d ; these are the holomorphic automorphisms of the holomorphic structure E de ned by d . The group Γ = Aut(E ) acts on the space H (X, End E ) of in nitesimal deformations of E by conjugation, and since each element of Γ is covariantly constant with respect to d , this action preserves the harmonic subspaces H ,q = H ,q ( ∂ , End E h ). From Corollary 4.3 the function Ψ of (2.4) is equivariant with respect to the action of Γ on H , (at least, near ), from which it follows that Ψ − ( ) ⊆ H , is invariant under Γ. The linearisation of the action of Γ on H ,q at ∈ Γ is given by the Lie bracket, The group Γ is the complexi cation of the subgroup U(Γ) of d -closed unitary automorphisms of E h , so is a reductive Lie group.
Proof. Given self-adjoint δ ∈ H , , let γ t := e tδ for t ∈ R. Then with α t := γ t αγ − t , di erentiation with respect to t givesα t = [δ, α t ] = e tδ [δ, α]e −tδ , using the fact that δ commutes with e tδ . Consequently Thus any critical point of t → α t is a minimum, and since Re α t , [δ, (with the complex structure J(α) = i α). Using the de nition in [31], to prove that this is the case, it must be shown that m is equivariant with respect to the action of SU(Γ) on H , and that dm δ (α) = i ω(X δ ,α) for each skew-adjoint δ ∈ H , and eachα ∈ H , , where X δ is the vector eld on H , determined by δ, this having the value [δ, α] at α ∈ H , . (The factor of i in front of ω comes from (5.1), included to make m(α) self-adjoint.)

Lemma 5.2.
Suppose that X ω n = . Then for any u ∈ H , and v ∈ A , (End E h ) in L , tr (u * Π , v) = X tr (u * v) dV.
Proof. If φ, ψ ∈ H , are arbitrary, they are both covariantly constant and therefore so too is φ * ψ. Hence the trace of this endomorphism is constant, which implies that tr (φ * ψ) = X tr (φ * ψ) dV = φ, ψ . Consequently, if e , . . . , em is an L -orthonormal basis for H , , the endomorphisms e , . . . , em are pointwise orthonormal. Writing u = j e j , u e j and Π , v = k e k , v e k , it follows that It follows from this lemma that if v ∈ A , (End E h ) is in L and u ∈ SU(Γ), then Π , (uvu − ) = u(Π , v)u − , which implies that the map m of (5.1) is indeed equivariant with respect to the action of SU(Γ). Furthermore, thinking of α as depending di erentiably on a parameter t ∈ R and di erentiating at t = , forα : Hence for δ ∈ H , , using Lemma 5.2 it follows that Thus: Proof. The proof is by induction on the rank r of E h , the case r = being elementary.
Suppose rst that Eα is stable but α is not Γ-polystable. Then the orbit of α under Γ is not closed, and indeed, the in mum of γ · α over γ ∈ Γ is not attained in that orbit. Let α ∈ H , be a point of smallest norm in the closure of the orbit of α under Γ, unique up to conjugation by unitary elements of Γ. So there is a sequence γ j ∈ Γ with det γ j = for all j such that γ j · α is decreasing to α but γ j is not bounded. Let β ∈ A , (End E h ) be the form orthogonal to ker ∂ * determined by Proposition 2.6 such that ∂ + α + ∂ * β is an integrable semi-connection de ning the holomorphic structure Eα that is L p -near to E , and let a ′′ = α + ∂ * β.
If α j := γ j · α then α j is decreasing. From Proposition 2.6 it follows that β j L p is uniformly bounded, where If a ′′ j := α j + ∂ * β j = γ j · a ′′ , it follows that a ′′ j L p is uniformly bounded, so after passing to a subsequence if necessary, the forms a ′′ j converge weakly in L p and strongly in C to a limit a ′′ . The limiting connection d +a is integrable, de ning a holomorphic structure Eα on E h . After rescaling γ j toγ j with γ j = , a subsequence of the automorphismsγ j converges to a non-zero limitγ with γ = and detγ = , this de ning a holomorphic map from the holomorphic structure Eα de ned by d +a to the holomorphic structure de ned by d + a . Since the latter can be assumed to be semistable (by Proposition 3.2), there is a non-zero holomorphic map from Eα to a semistable bundle of the same degree and rank that is not an isomorphism, and this contradicts the assumption that Eα is stable. Therefore α is Γ-polystable. If α is polystable but not stable with respect to the action of Γ, the isotropy subgroup Γα ⊂ Γ of α has dimension greater than one. Then it follows from 2. of Theorem 4.7 that Eα is polystable but not stable, a contradiction.
Suppose now that Eα is polystable but not stable. Then Eα splits into a direct sum of stable subbundles, each of the same slope. In terms of connections, the bundle-with-connection (E h , d +a) has a unitary splitting into a direct sum of irreducible unitary bundles-with-connection. Orthogonal projection onto any of these subbundles (followed by inclusion) is a holomorphic endomorphism of Eα, and by Theorem 4.7, such an endomorphism is in fact covariantly constant with respect to d and also commutes with α and β. Thus the bundle-with-connection (E h , d ) has a unitary splitting into a direct sum of subbundles-with-connection, each of which de nes a polystable subbundle of E of the same slope. If (E h , d ) = j (B j , d ,j ) is this last splitting, then i F(d ,j ) = λ and for some skew-adjoint a j ∈ A (End B j ), (E h , d + a) = j (B j , d ,j + a j ) corresponds to the splitting of Eα into stable components. The compatibility of the splittings implies that each a j is of the form a j = a ′ j + a ′′ j with a ′′ j = α j + ∂ * ,j β j where α j is a ∂ ,j -harmonic ( , )-form with coe cients in End B j . By the inductive hypothesis, each α j is stable with respect to the action of Γ j , the group of ∂ ,j -closed automorphisms of B j . Hence there exists such an automorphism ρ j such that ρ j ·α j =:α j is a zero of the moment map, which means that Π , j iΛ(α j ∧α * j +α * j ∧α j ) = , where Π , j is the orthogonal projection onto ker ∂ ,j . But this implies that there is an automorphism σ ∈ Γ such thatα := σ · α satis es Π , iΛ(α ∧α * +α * ∧α) = . Therefore α is polystable.
The last statement of the proposition follows immediately from Theorem 4.7.
The group Γα acts brewise on E h , splitting each bre into a direct sum of irreducible Γα-invariant subspaces. These subspaces together form subbundles, namely the intersections of the eigen-bundles associated with the elements of Γα. The fact that Γα is closed under adjoints implies that the splitting of E h into Γα-irreducible subbundles is a unitary splitting. For |t| su ciently small and |s| ≤ , the connections d t,s = d + a t,s given by a ′′ t,s = tα + s ∂ * (tα) preserve these splittings, and restrict to irreducible unitary connections on each of the Γα-irreducible subbundles.
As mentioned in the second of the Remarks towards the end of §4, Theorem 5.4 is one half of a local version of the Hitchin-Kobayashi correspondence: ω-(poly)stability of Eα implies Γ-(poly)stability of α, where the latter term means polystable with respect to the action of Γ in the sense of geometric invariant theory. The more di cult task is to establish the other half; that is, the converse, and this e ectively involves solving di erential equations. This will be the subject of the next three sections.

Connections with constant central curvature
As in previous sections, let (X, ω) be a compact Kähler n-manifold and let E h be a xed complex r-bundle equipped with a xed hermitian metric; all conventions and notations from previous sections are also retained. In this section, the study of §2 into an L p neighbourhood of a given (hermitian) connection will be continued but the focus is now is on the central component F = ΛF of the curvature F rather than the ( , )component. As previously, G is the group of complex automorphisms of E h , with U ⊂ G the subgroup preserving the given hermitian metric. Let A denote the space of hermitian connections d = ∂ + ∂ on E h , so the action of G on A is given by Unless otherwise stated, elements of G are assumed to lie in L p and elements of A to lie in L p . Projection to the central component of the curvature de nes a function Φ on G × A with values in the space of self-adjoint endomorphisms of E h lying in L p given by and it is the properties of this function and its derivatives with respect to each of its arguments on which the analysis concentrates in this section.
Let d = ∂ + ∂ be a connection on E h , and let a = a ′ + a ′′ be an element of A (End E h ) with a ′ = −(a ′′ ) * . If da = d + a, the curvature of this connection is 3) It follows that if a = a(t) depends di erentiably on the real parameter t, then where · denotes di erentiation with respect to t. The action of G on A has the explicit form (6.5) If g = g(t) also depends di erentiably on t, then by direct calculation from (6.5), it follows that Hence, from (6.4) and (6.6), it follows that Applying iΛ to both sides and recalling that ∂ * = iΛ ∂ and ∂ * = −iΛ∂ on -forms, Since Λ( ∂∂ + ∂ ∂) = F, writing σ :=ġg − and decomposing into self-adjoint and skew-adjoint components gives the following conclusion, which will be used frequently in this section:

Lemma 6.1. Let a = a(t) ∈ A (End E h ) be a di erentiable -real parameter family of skew-adjoint forms, and let g = g(t) be a di erentiable -real parameter family of complex automorphisms of E h . If d = ∂ + ∂ is a connection on E h and da = d + a, then
where σ :=ġg − and σ± := (σ ± σ * ).
Taking a ∈ A (End E h ) to be independent of t, it follows from (6.9) that the linearisation of the map G g → i F(g · da) = Φ(g, da) at a connection g · d is In particular, if g = and a = , the linearisation at ( , d ) ∈ G × A of the action of G on the space of L p (hermitian) connections is an isomorphism from the space of L p self-adjoint sections of End E h that are orthogonal to the d -closed sections to the space of self-adjoint L p sections of End E h that are again orthogonal to the kernel of d .
From now on, let d be a connection with i F(d ) = λ , so by Corollary 4.2, ker d = ker ∂ . In general, the map G× A (End E h ) (g, a) → Φ(g, da) = i F(g · da) takes values in the self-adjoint endomorphisms of E h , but it does not map into the space orthogonal to ker ∂ . However, if Π , is the L projection of A , (End E h ) onto ker ∂ and Π , ⊥ = − Π , is the projection onto the orthogonal complement, then for any skew-adjoint L p section a ∈ A (End E h ), the composition maps the Banach space of self-adjoint L p sections in A , (End E h ) orthogonal to ker ∂ into the Banach space of such sections lying in L p , and when a = , this map has the same linearisation at as the earlier map. The implicit function theorem for Banach spaces then yields: Proposition 6.2. There exists ϵ > depending only on d with the property that for each skew-adjoint section Furthermore, there is a constant C depending only on d such that φ satis es The power series expansion in φ at φ = of i F(exp(φ) · da) is, to rst order, where the term R (φ) involves products of φ and its rst and second derivatives with respect to da with at least two such factors, but where the second-order derivatives appear linearly and the rst-order derivatives appear at most quadratically. Consequently, if a satis es the hypotheses of Proposition 6.2 and if φ ∈ A , (End E h ) satis es the inequality in the statement of that proposition, then If φ is self-adjoint and if ∂ * a ′′ = , the formula (6.10) simpli es somewhat. In this case, it is useful to project both sides into ker ∂ and (ker ∂ ) ⊥ , respectively giving If φ = φ(a) is the endomorphism of Proposition 6.2, the left-hand side of (6.11)(b) vanishes, giving the equation that e ectively determines φ(a). Since Λd a = now, the uniform estimate on φ provided by that proposition then implies φ(a) L p ≤ C a L p , so the remainder term R (φ) appearing in (6.11) is uniformly bounded by a constant multiple of a L p , something that is also true of the other terms on the second line of (6.11)(a).
If σ ∈ A , (End E h ) is ∂ -closed, then ∂ σ = by Corollary 4.2, and where R(a) L p ≤ C a L p for some constant C = C(d ).
The term Π , Λ(α * ∧ α + α ∧ α * ) is O( α ) in general, whereas if β is as in Proposition 2.6, all the other terms on the right of (6.14) are O( α ). But if Π , Λ(α * ∧ α + α ∧ α * ) vanishes (as considered in §5) then the whole of the right-hand side of (6.14) is O( α ) and the connection exp(φ(a)) · (d + a) is very close to having central curvature equal to −i λ . Given that φ(a) is orthogonal to ker ∂ , it can be hoped that a small perturbation by an element of ker ∂ will yield a connection with i F ≡ λ . For γ ∈ Γ, so since exp(φ + δ) is close to exp(φ) exp(δ) for small φ ∈ (ker ∂ ) ⊥ and small δ ∈ ker ∂ , an alternative is to perturb α by conjugation with an element of Γ close to . Such an argument will involve an application of the inverse function theorem in nite dimensions, for which purpose the variation in F as α is varied in this way must be determined.
With 'sk' denoting skew-adjoint and 'sa' denoting self-adjoint, consider rst the function G de ned on for Φ as in (6.2)).
If G := Π , G and G := Π , ⊥ G, the conclusion of Proposition 6.2 is that for a ∈ A sk (End E h ) with a L p < ϵ, G (φ(a), a) ≡ , where a → φ(a) is the function speci ed in that proposition, the existence of which is guaranteed by the implicit function theorem.
If a moves in a di erentiable -parameter family a(t), then it follows that where D G , D G are respectively the partial derivatives of G with respect to its rst and second arguments andȧ denotes the derivative with respect to t as before. The implicit function theorem implies that if (ψ, a) is su ciently close to ( , ), then (D G ) (ψ,a) is an isomorphism from the space of self-adjoint elements of A , (End E h ) orthogonal to ker ∂ lying in L p to the space of self-adjoint elements of A , (End E h ) orthogonal to ker ∂ lying in L p , so The variation in G(φ(a), a) at a is therefore given by whereȧ =ȧ ′ +ȧ ′′ , with the derivatives D G and D G obtained by projecting into ker ∂ and its orthogonal complement respectively. Similarly, the partial derivative of G with respect to its rst variable ψ at (φ(a), a) is obtained from (6.9) by substitutingȧ = and σ = (de ψ /dt) e −ψ ψ=φ(a) for a -parameter family ψ(t) into that formula, so (6.17) again with the derivatives of G and G obtained by taking L projection into ker ∂ and its orthogonal complement. Note that σ is not self-adjoint in general, but satis es σ * = e −φ(a) σe φ(a) . Thus the second term on the right of (6.17) is not zero in general, unless i F(d b ) = λ .

Lemma 6.4.
Under the hypotheses of Proposition 6.2, suppose in addition that ∂ * a ′′ = . Then there is a Proof.
The assumption that ∂ * a ′′ = implies that Λd a = , so the bound of Proposition 6.2 implies that φ L p ≤ C a L p for some uniform constant C. By the Sobolev embedding theorem, there is a similar such bound on the C norm of φ, so (given that a L p is su ciently small), an arbitrary endomorphism ψ ∈ A , (End E h ) will satisfy a pointwise bound of the form ψ − e φ ψe −φ ≤ C a L p |ψ|, which can be seen by orthogonally diagonalising φ at the point in question. Since ( ∂ e φ )e −φ is uniformly bounded in C by a multiple of a L p , it follows that b ′′ = e φ a ′′ e −φ − ( ∂ e φ )e −φ also satis es a pointwise bound of the form |b ′′ − a ′′ | ≤ C a L p , and therefore ( ∂a e φ )e −φ = a ′′ − b ′′ satis es this bound; similarly, e −φ ∂a e φ also satis es such a bound. Consequently, for any χ ∈ A , (End E h ), there is a uniform pointwise bound of the form for some new uniform constant C. Taking χ =ȧ in (6.16), it follows that (D G) (φ,a) (ȧ) ≤ C a L p | ∂ * ȧ| + |ȧ| pointwise. (6.20) It follows from this that the L norm of (D G) (φ,a) (ȧ) is uniformly bounded above by a constant multiple of a L p ∂ * ȧ L + ȧ L , which implies the same such bound for its orthogonal projection onto ker ∂ . Since ker ∂ is nite dimensional, the L norm on this space is equivalent to any other, so it follows from (6.20) that there is uniform bound of the form From the proof of Proposition 6.2 using the implicit function theorem, the operator (D G ) (φ,a) is an isomorphism from the space of self-adjoint L p sections of A , (End E h ) orthogonal to ker ∂ to the same such space of L p sections, so For a section σ ∈ A , (End E h ), (6.17) gives for which the rst term is annihilated by Π , , estimation of the second term gives The a priori L p bound on a and the estimate on φ(a) from Proposition 6.2 implies that b L p is uniformly bounded above by a multiple of a L p , so Π , ∂ * b ∂ b σ L ≤ C a L p σ L . Taking adjoints, this same estimate applies to Π , ∂ * b ∂ b σ * to give Combining this with the estimates of (6.22) and (6.20) gives (6.18).
In accordance with the strategy outlined following the statement of Lemma 6.3, Lemma 6.4 provides su cient information to analyse the variation of i F(e φ(a) ·da) as a ′′ = α+ ∂ * β(α) is varied according to H , α → γαγ − for γ ∈ Γ, at least when the connections da are su ciently near to d .

Γ-polystable locally implies ω-polystable
Retaining all of the notation and de nitions of the previous section, suppose now that γ t ∈ Γ is a family depending di erentiably on the real variable t, with γ = andγ t | t= = δ ∈ H , . Suppose that a = a ′ + a ′′ satis es the hypotheses of Proposition 6.2 as well as ∂ * a ′′ = , and let a t : δ], so ∂ * ȧ ′′ = and Lemma 6.4 gives an estimate of the contribution of the rst term on the right of (6.15) to the variation in i F in terms of [a ′′ , δ] L p . But since δ is d -closed, its C norm is bounded by a uniform multiple of its L norm so [a ′′ , δ] L p ≤ C a L p δ for C = C(d ), and therefore that contribution is uniformly bounded above by a constant multiple of a L p δ , (the fourth power coming from (6.18)). It follows that if a L p is su ciently small, the dominant term in the variation (6.15) of i F is that coming from (D G ) (φ ,a) [a ′′ , δ]+ , given by the projection of (6.16) onto ker ∂ , provided that this is appropriately non-degenerate as a function of δ.
As in the proof of Lemma 6.4, (6.19) gives a bound Noting thatȧ ′′ = −[a ′′ , δ] and ∂ a δ = [a ′′ , δ], it follows that By taking adjoints and usingȧ ′ = +[a ′ , δ * ], the same estimates apply to the other term in (6.16) with δ replaced by δ * . Then if δ is taken to be self-adjoint, by combining these estimates with those of Lemma 6.4 the following conclusion is reached: With a ′′ and δ as in this proposition, write a ′′ = α + ∂ * β where α ∈ H , and β ∈ A , (End E h ). Given that Assuming that β satis es the uniform bound of Proposition 2.6, from Proposition 6.2 it follows that there is a bound of the form a L p ≤ C α for some constant C = C(d ). Hence for some new constant C = C(d ), the bound of Lemma 6.4 implies a bound of the form This estimate has been derived under the assumption that β satis es the uniform bound given in Proposition 2.6. In particular, it applies if β = β (α) where β (α) is the unique element of A , (End E h ) speci ed in that result, but more generally, it also applies if β = s β (α) where s ∈ [ , ]. This observation facilitates some homotopy arguments to follow.
The leading term on the right of (6.14) is the negative of m(α) = Π , iΛ(α∧α * +α * ∧α). Fixing α temporarily, this gives a map Γ → H , give by Γ γ → m(γαγ − ), and the derivative of this map at γ = is given by If mα : H , → H , is the R-linear function on the right of (7.2) and Lα : H , → H , is the C-linear function Lα(δ) := [α, δ], it is clear that the kernel of Lα is contained in that of mα. In fact, by direct calculation, for δ ∈ H , with δ = δ+ + δ− and (δ±) * = ±δ±, Thus when acting on the self-adjoint elements of H , , mα and Lα have the same kernel. Since mα is itself self-adjoint as an R-linear map H , → H , , it follows that mα maps the space of self-adjoint elements of H , that are orthogonal to ker Lα isomorphically into the same space. At this point, arguments are simpli ed if it is assumed that α is stable with respect to the action of Γ, not just polystable. Given that α is Γ-polystable, the assumption of stability is equivalent to the condition that ker Lα = span { }. For g ∈ G, (1.1) implies tr i F(g · (d + a)) = tr i F(d + a) + i ∂∂ log det(g * g) = rλ + iΛd tr a + i Λ ∂∂ log det(g * g) , so if ∂ * a ′′ = then tr i F(g · (d + a)) = rλ + ′ log det(g * g) .
If V is a hermitian vector space, the real vector space End sa (V) of self-adjoint endomorphisms has a canonically induced orientation. This can be seen by induction on dim V, given that the space of self-adjoint endomorphisms of V ⊕ C is canonically isomorphic to End sa (V) ⊕ V ⊕ R. So the space of self-adjoint automorphisms of V, which is a real closed submanifold of Aut(V), is orientable. Similarly, the space of trace-free self-adjoint endomorphisms of V ⊕ C is canonically isomorphic to End sa (V) ⊕ V, so the space of self-adjoint automorphisms of determinant , which is a real closed submanifold of Aut(V), is orientable. The function Γ γ → m(γαγ − ) restricted to the space Γ sa of self-adjoint elements of Γ of determinant maps Γ sa into the space (H , ) sa of trace-free self-adjoint elements of H , ; that is, into its tangent space at ∈ Γ sa , and (given that α is Γ-stable) its derivative at is an isomorphism between T Γ sa and (H , ) sa .
Replacing α in (7.1) with tα for t > su ciently small (depending on α), it follows from that estimate that once t is su ciently small, the kernel of the R-linear function (D Φ) acting on the trace-free self-adjoint elements of H , is zero, and this holds for any such t and any sβ(tα) with |s| ≤ where β(α) ∈ A , (End E h ) is the section speci ed by Proposition 2.6. Now view Φ in Proposition 7.1 as a function of γ ∈ Γ and ∂ * -closed a ′′ ∈ A , (End E h ), taking values in the trace-free self-adjoint elements of H , . When restricted to those a ′′ ∈ A , (End E h ) of the form a ′′ = tα + s ∂ * β (tα) for < t ≤ and ≤ s ≤ and self-adjoint γ near of determinant , there is an induced map Γ sa → (H , ) sa , for which, from (7.1), the derivative with respect to its rst variable γ at is injective once t is su ciently small, where "su ciently" depends upon α; more speci cally, on the rst non-zero eigenvalue of L * α L α .
The assumption on α implies that it is polystable with respect to the action of Γ. Assume initially that α is stable with respect to this action. If t > is su ciently small and s ∈ [ , ], (D Φ) ( ,a ′′ s,t ) gives an isomorphism between the tangent space to Γ sa at and (H , ) sa . The manifold Γ sa is orientable, and the degree of Φ(−, a ′′ s,t ) at λ is independent of s and su ciently small t, and is therefore equal to the degree at λ for such t and s = . If φ t := φ(a ,t ), then by (6.14), Lemma 6.3 and the remarks following that lemma, for δ ∈ H , near , there is a function R = R (t, δ) with R (t, δ) L p ≤ Ct such that, for γ δ := exp(δ), Since both Φ − λ and mα take their values in the space of trace-free self-adjoint elements of H , , so too does R . Since mα is an isomorphism on this space and R (t, δ)/t is uniformly bounded as t → , it follows that λ is in the range of Φ(−, a ′′ ,t ) for t su ciently small, and indeed that the degree of Φ(−, a ′′ ,t ) at λ is precisely . Therefore, for s ∈ [ , ] and t > su ciently small (depending on α), there exists γ ∈ Γ such that i F exp(φ(γ · a s,t )) · γ · (d + a s,t ) = λ .
The invertibility of mα on the space (H , ) sa of trace-free self-adjoint elements of H , implies that there is a solution γ = exp(δ(t)) to i F exp(γ · a s,t )) · γ · (d + a s,t ) = λ depending continuously on t, and from (7.4), the section δ(t) satis es a bound of the form δ(t) ≤ Ct /c α where c α is the lowest eigenvalue of L * α L α acting on (H , ) sa and C = C(d , α). Then uniform bounds on exp(δ(t)) give estimates on exp(δ(t)) · α, and together with the estimates of Proposition 6.2, a uniform L p bound on φ of the form Ct /cα follows for some new constant C depending on d and α.
If now α is assumed to be polystable but not stable, then the isotropy subgroup Γα has dimension greater than one. Hence there are non-zero trace-free elements of H , commuting with α, and by 2. of Theorem 4.7, so too do their adjoints, as they all do with β(tα) for any (small) t > . Hence there are non-zero trace-free self-adjoint elements of H , commuting with α and β(tα) for any such t, and therefore there are non-zero trace-free self-adjoint endomorphisms of E h that are covariantly constant with respect to d + a s,t for all s, t. Any such endomorphism gives a unitary splitting of the bundle and connections, and these splittings are all compatible with one another, including the splitting of (E h , d ). With respect to such a splitting, the form α splits into a collection of endomorphism-valued ( , )-forms on X, each of which is ∂-harmonic with respect to the induced connection. The "o -diagonal" components of α with respect to such a splitting are zero, since α commutes with the trace-free endomorphisms determining the splitting. Since α is of minimal norm in its orbit under γ, each of the "diagonal" components of α must be of minimal norm under the action of the subgroup of Γ that is the automorphism group of the corresponding component of (E h , d ). Hence each of these components de nes an element of the corresponding space that is polystable with respect to the action of the corresponding automorphism group, so it follows by induction on the rank r that for t su ciently small, for each of these new bundles with connection, once t > is su ciently small there is a complex gauge transformation that gives a new connection with i F a scalar multiple of , (the case r = being elementary). Since the splitting of (E h , d ) is unitary and i F(d ) = λ , the relevant scalar in all cases is λ. For each summand, the estimates on the corresponding endomorphisms δ and φ imply an estimate of the required form on the direct sum connection, verifying the last statement of the theorem.
A direct proof of Theorem 7.2 that does not use induction on rank appears to be possible, but raises a number of interesting representation-theoretic questions. Proof. By Theorem 7.2 and the results of Kobayashi and Lübke, the holomorphic bundle de ned by d + a s,t is polystable. If it is not stable, then there exists a non-zero trace-free holomorphic endomorphism of this bundle. By Theorem 4.7, this endomorphism is covariantly constant with respect to d and commutes with α, these facts contradicting the assumption that α is stable with respect to the action of Γ.

Remarks.
1. From the viewpoint of deformation theory, an unobstructed in nitesimal deformation is (poly)stable with respect to the action of Γ if and only if there is a -complex parameter family of (poly)stable holomorphic structures whose tangent at E is the given in nitesimal deformation. Of course, in the case that the latter is polystable but not stable, there may be families of bundles that are semistable but not polystable with that tangent vector, which will often be the case if H (X, End E ) vanishes. (Direct sums of non-isomorphic line bundles of degree zero on a torus provide an example when this is not the case.)

2.
A relatively straightforward application of the continuity method applied to the assignment t → g t ∈ G solving i F(g t · (d + a s,t ) = λ gives a more quantitative version of Theorem 7.2, namely that the dependence of t on α stated in the theorem can be made explicit if the constant C there is replaced by by C α /cα where C is a constant depending only on d and where cα is the rst non-zero eigenvalue of L * α L α : H , → H , . In the interests of brevity, an explicit proof will not be given.
The following proposition is the companion uniqueness result to Theorem 7.2 (existence). The proof of the rst statement is based on the proof of Corollary 9 in [11]: Proposition 7.4. Suppose α ∈ H , , a ′′ = α + ∂ * β, a = −(a ′′ ) * + a ′′ , and a L p < ϵ where ϵ > is as in Theorem 4.7. If i F(g ·(d +a)) = λ = i F(g ·(d +a)), then g = u g γ for some u ∈ U and γ ∈ Γα. Furthermore, there exists g ∈ G with i F(g · (d + a)) = λ satisfying the conditions that g = g * is positive, det g ≡ , and Π , (g * g ) ∈ H , is orthogonal to the space of trace-free elements of ker Lα, with these conditions determining g uniquely up to conjugation by unitary elements of Γα.
Proof. Let da := d + a and set d b := g · da, so for g := g g − it follows that . After a unitary change of gauge applied to g · da, it can be supposed that g is positive selfadjoint, with g = exp(v) for some self-adjoint v. If y ∈ R and with dy := exp(y v) · d b , by (6.9) the function dy v ≥ and is therefore a non-decreasing function on R. Since it attains the value at both y = and y = , it must be constant on [ , ] with derivative identically . Hence d b v = , which implies that ∂a(g − g ) = . By 1. of Theorem 4.7, γ := g − g is dcovariantly constant and commutes with α. Thus ug = g γ for some u ∈ U and γ ∈ Γα.
To prove the second statement, note that Γ acts freely on G by right multiplication, as does the closed subgroup Σα ⊂ Γ of elements in Γα of unit determinant. Given a xed g ∈ G there is a constant c = c(g) such that c γ ≤ gγ ≤ c − γ , so there exists γ ∈ Σα minimising gγ over all γ ∈ Σα. The Euler-Lagrange equation for this functional on Σα is Π(g * g ) = , where Π is L -orthogonal projection onto the space of trace-free elements in ker Lα, the Lie algebra of Σα.
Suppose now that g , g ∈ G are as in the statement of the proposition, with g = ug γ for some u ∈ U and some γ ∈ Γα. Suppose in addition that both g and g have unit determinant, are both positive and self-adjoint, and that Π , (g * j g j ) is orthogonal to the trace-free elements of ker Lα for j = , . Then for any trace-free ϕ ∈ ker Lα, and using the fact that the trace of a covariantly constant endomorphism is constant, Therefore γ * γ is a multiple of , and this multiple must be since = det u det γ. Thus γ ∈ U(Γα), the group of unitary elements in Γ commuting with α. Then since g and g are both self-adjoint, g = g * g = γ * g * g γ = (γ − g γ) , and positivity implies g = γ − g γ. From g = ug γ, it then follows that u = γ − . Theorem 7.2 gives a condition under which a connection near d has a connection with central component of the curvature equal to a scalar multiple of the identity in its orbit under G, but the de ciency of the result is that how near to d the connection must be depends on the connection itself, dictated by the relative sizes of the eigenvalues of L * α L α . This issue is addressed in the next section.

The local Hitchin-Kobayashi correspondence
As stated at the end of the previous section, the objective of this section is to remove the dependency of Theorem 7.2 on α other than through α . That is, retaining all of the notion of that section, the objective is to prove the following result: is as in Proposition 2.6, then there exists g ∈ G with i F(g · (d + a)) = λ , where a = a ′ + a ′′ for a ′′ = α + ∂ * β.
The approach to proving this result is to ensure that the analysis is performed in a su ciently small neighbourhood of d where the connections are well-approximated by their linearisations, which has the e ect of reducing the problem to a nite-dimensional question that is naturally attacked using the methods of classical geometric invariant theory. Before commencing the proof of the theorem, there are several remarks and observations that simplify matters considerably.
First, consider the case in which the rank r of the bundle E h is . Then α is a harmonic ( , )-form on X, β must be zero since α ∧ α = , and the connection da = d + (α − α * ) has curvature F(d ), which already satis es the condition i F = λ. Thus g ≡ solves the equation. In the general case, if i F(g · (d + a)) = λ , then on taking the trace of both sides it follows that iΛ tr (F(d ) + d a + a ∧ a) + ∂ ∂ log det(g * g)) = r λ, which implies that iΛ ∂ ∂ log det(g * g) ≡ and hence that det(g * g) is constant. After rescaling g by a constant, it can therefore be assumed that | det g| ≡ .
Second, given that α is polystable with respect to the action of Γ, it may be assumed without loss of generality that α is of minimal norm in its orbit under Γ, and therefore iΛ(α ∧ α * + α * ∧ α) is orthogonal to ker ∂ , by Lemma 5.1.
Third, as it was for the proof of Theorem 7.2, if α is polystable but not stable, precisely the same argument using induction on r that was employed at the end of the proof of Theorem 7.2 reduces the problem to the case when α is stable with respect to the action of Γ. Given this, the uniqueness result Proposition 7.4 implies that the only freedom in choice of g is that of replacing g by ug for u ∈ U.
Hitherto, little use has been made of unitary gauge freedom U u → u · d for a connection d, as this is subsumed into the complex gauge freedom G g → g · d. But since the equation i F(d) = λ is invariant Suppose now that α ∈ H , satis es m(α) = and γ ∈ Γ. Using a Cartan decomposition of Γ into Γ = U T U where T is a maximal complex torus and U = U(Γ), it follows from the left and right unitary invariance of the norm and the unitary equivariance of m that γ may be assumed to lie in T; that is, γ = diag(t , . . . , tm) for some t j ∈ C * .
Instead of working directly with α, it is more convenient to work with τ := αγ − , so m(τγ) = and it must be shown that γτ − τγ ≤ C m(γτ) . This will follow if it can be shown that for some constant C, using here the fact that the and norms on H , are equivalent in this representation.
Observe that and by (8.5), for each xed i the second term on the right sums to zero on application of j . Similarly, and again (8.5) implies that the second term on the right sums to zero on application of j . So (8.6) is equivalent to After renumbering, it can be assumed that That this is true is a consequence of the following:

Remark.
There is an alternative approach to Lemma 8.3 in terms of general theory. Namely, for α ∈ H , one can study the downwards gradient ow for the function Γ γ → γα γ − , for which the relevant ODE iṡ It is easily checked that the ow t → α t is the same as the downwards gradient ow for H , α → m(α) . Modulo reparameterisation, the latter covers the downwards gradient ow for P(H , ) [α] → m(α) α , for which P(H , ) [α] → m(α)/ α is the moment map for the action of Γ on P(H , ) ( [31]). An unpublished theorem of Duistermaat using the Łojasiewicz inequality presented in [25] shows that this ow de nes a (strong) deformation retract of the set of Γ-polystable points onto the zero set of the moment map (analogous to the result of Neeman [30] in the algebraic setting), and Lemma 8.3 follows quite easily from this; see also § §3,4 of [9].

Non-stability
Theorem 8.1 is a version of the Hitchin-Kobayashi correspondence for bundles in an L p neighbourhood of a polystable bundle, but it does not provide much detail in the case of connections and/or classes that are not polystable. Whereas non-zero elements of H , may be unstable with respect to the action of Γ-that is, zero is in the closures of their orbits, Proposition 3.2 states that there are no strictly unstable bundles near E , so the correspondence between the two di erent notions of stability is not perfect, (although this is more a distinction between (semi)stability in the a ne versus projective settings). However, it is nevertheless true that the interrelation between the two notions goes further than just that described by Theorem 8.1, as will be seen in this section. All notation from earlier sections continues to be retained.
In general, if E is an arbitrary torsion-free semistable sheaf that is not stable, there is a non-zero subsheaf S ⊂ E with µ(S) = µ(E) and with torsion-free quotient Q = E/S for which µ(Q) = µ(E). Both S and Q are necessarily semistable, and if S is of maximal rank, then Q is stable. Iterating this process yields a ltration of E, = S ⊂ S ⊂ S ⊂ · · · ⊂ S k = E such that the successive quotients are all torsion-free and stable. Any such ltration is known as a Seshadri ltration or sometimes a Jordan-Hölder ltration, and although it is not unique, the graded object Gr(E) = k j= (S j /S j− ) is unique after passing to the double-dual. In the current setting of holomorphic structures E near to E , Proposition 3.2 states that E is semistable whilst Proposition 4.4 states that any destabilising subsheaf of E is a subbundle. In this case therefore, there is a Seshadri ltration of E de ned by subbundles, so the graded object Gr(E) associated to E is a polystable holomorphic structure on E h .
Recall from the proof of Lemma 3.1 that if A is a holomorphic subbundle of E with quotient B, then in a unitary frame for A and B, a hermitian connection d E on E and its curvature F E have the form where now β ∈ A , (Hom(B, A) By Proposition 4.5, every holomorphic endomorphism of Gr(E) is also ∂ -closed, and is therefore covariantly constant with respect to d , by Corollary 4.2. Thus the automorphisms g t ∈ G can even be taken to lie in Γ. The following result gives something of a converse to this observation, albeit in a rather special case. In its hypotheses, how small is "su ciently small" is determined by the connection d , so that Corollary 2.4 is applicable.
Lemma 9.1. Let d + a be an integrable connection on E h with a L p su ciently small, and suppose that a ′′ = α + ∂ * β for some α ∈ H , and β ∈ A , (End E h ). Then the following are equivalent: 1. For any ϵ > there exists γ ∈ Γ such that γαγ − < ϵ; 2. For any ϵ > there exists g ∈ G such that g · (d + a) − d L p < ϵ.
Proof. The implication 1. ⇒ 2. follows immediately from Corollary 2.4. For the converse, assume ϵ > is smaller than the number speci ed in Lemma 2.5 and let g ∈ G be an automorphism such that g · (d + a) − d L p < ϵ. Applying Lemma 2.5 to the semi-connection g · ( ∂ + a ′′ ) yields a unique φ ∈ A , (End E h ) orthogonal to ker ∂ such that d +ã := exp(φ) · g · (d + a) satis es ∂ * ã ′′ = , with ã ′′ bounded by a xed multiple of ϵ. Applying Proposition 4.5 to the connection on Hom(E h , E h ) induced by d +ã and d + a and the section exp(φ)g of this bundle, it follows that if ϵ is su ciently small then exp(φ)g =: γ must be ∂ -closed with a ′′ γ = γã ′′ . Then ifã ′′ =α + ∂ * β , orthogonality of the decompositions gives γ − αγ =α, and α is bounded by a xed multiple of ϵ since ã ′′ L p is.
Consider now an integrable connection d + a, with a L p assumed to be appropriately small and with a ′′ = α + ∂ * β for some α ∈ H , and some β ∈ A , (End E h ) orthogonal to the kernel of ∂ * . Under the action of Γ on H , , there is a pointᾱ ∈ H , of smallest norm in the closure of the orbit of α unique up to conjugation by unitary elements in Γ, and this is a Γ-polystable point (if not zero). Sinceᾱ is in the closure of the orbit of α and each element near in this orbit lies in the analytic set Ψ − ( ), there is a unique sectionβ ∈ A , (E h ) such that ∂ +ā ′′ := ∂ +ᾱ + ∂ * β is integrable, so by Theorem 8.1 the corresponding holomorphic bundle E is polystable. The following is the main result of this section, this being Theorem 4 of the introduction: The proof, which is principally by induction on the rank r of E h (with the initial case r = being self-evident) proceeds in several stages, corresponding to three cases: 1. That α = ; 2. That α is not zero and is not Γsemistable; and 3. That α is Γ-semistable. The rst is the totally degenerate case for which α = : Proof. If Π , a ′′ = , then it follows from Corollary 2.4 that a ′′ = , and therefore a = . Conversely, if E E , then there exists g ∈ G such that g · d = d + a, or equivalently, ∂ g + a ′′ g = . Applying Proposition 4.5 to the connection on Hom(E h , E h ) induced by d and d + a, it follows that d g = = a ′′ g, so a ′′ = .
Before moving on to the other two stages of the proof of Theorem 9.2, consider rst some general features applicable in all cases. Let d + a be as above with a ′′ = α + ∂ * β. Choose a sequence (γ j ) in Γ with det γ j = for every j such that γ j αγ − j → inf γ∈Γ γαγ − as j → ∞, so after passing to a subsequence if necessary, it can be assumed that α j := γ j αγ − j converges to someᾱ ∈ H , . If β j := γ j β γ − j and a ′′ j := α j + ∂ * β j , then d + a j = γ j · (d + a) is an integrable connection de ning a holomorphic structure isomorphic to E, with ∂ * a ′′ j = . By Corollary 2.4, a ′′ j L p is uniformly bounded independent of j, so after passing to another subsequence if necessary, the connections d +a j can be assumed to converge weakly in L p and strongly in C (say) to a limiting connection d +ā, with a ∈ L p . Elliptic regularity combined with integrability of the connection together with the equation ∂ * ā ′′ = imply thatā is in fact smooth. Indeed, using the analysis of §1, the forms β j can be assumed to be converging in L p to a limit in A , (End E h ) that is orthogonal to ker ∂ * , and by the uniqueness statement of Proposition 2.6, this limit must be the formβ mentioned earlier, withā ′′ =ᾱ + ∂ * β .
Since det γ j = for every j, it follows that if γ j is uniformly bounded then a subsequence can be found converging to some γ ∈ Γ, and then d +ā = γ · (d + a). This is the case considered in the previous section when α ∈ H , is a Γ-polystable point. So it may be supposed that γ j is not uniformly bounded, and after replacing γ j by γ j / γ j , these may be assumed to converge to some γ ∈ H , with γ = and det γ = . It may also be assumed without loss of generality that γ j is self-adjoint and positive for each j, so γ is also self-adjoint and non-negative.
The equation γ j · (d + a) = d + a j is equivalent to ∂ j γ j = where d j is the connection on Hom(E h , E h ) induced by d + a and d + a j , these connections converging to the connection on this bundle induced by d +a and d +ā. So γ de nes a non-zero holomorphic map from E to E, this map having determinant . Since γ must be of constant rank on X, its kernel K is a holomorphic subbundle of E, necessarily of the same slope as that of E h (because E and E are semistable of the same slope). Thus E may be expressed as an extension by holomorphic semistable bundles → K → E → Q → , where Q := E/K.
Consider now Case 2. of Theorem 9.2, namely when α is non-zero and is not Γ-semistable. By de nition, zero is in the closure of the orbit of α under Γ, soᾱ = and therefore E = E by Proposition 9.3. For notational convenience, set E := K = ker γ and E := Q = E/K. Since γ is self-adjoint, E can be identi ed with E ⊥ ⊂ E h as a unitary bundle. The holomorphic structures on E and E are those induced from E as holomorphic sub-and quotient bundles. But since E = ker γ and γ is a d -closed self-adjoint endomorphism of the holomorphic bundle E , both E and E have hermitian connections induced from d , with respect to which the connections are Hermite-Einstein with the same Einstein constant as E . These connections will be denoted by d , , d , respectively, so d = d , ⊕ d , using self-evident notation.
The limit γ of the (rescaled) automorphisms γ j is d -closed and satis es γ α = = γ β and also γ a ′′ = , so in terms of the splitting E h = E ⊕ E , whereγ =γ * has non-zero determinant. Since γ is d -closed, the connection on E induced by the connection d + a (i.e., as a quotient of E) is the same as the connection on this bundle induced by d (i.e., as a subbundle of E ), so the holomorphic bundle E is isomorphic to a direct sum of stable summands of E . The connection on E induced by d + a is identi ed with d , + a , with a = α + ∂ * , β . By the inductive hypothesis (of Theorem 9.2), under the action of Γ = Aut(E (d , )), there are connections in the of the Yang-Mills equations on a compact Kähler manifold for which ∂F , (d) = = ∂ F(d) (which includes some self -dual solutions on compact surfaces), these bearing some formal similarities to solutions of the Seiberg-Witten equations.

5.
At its heart, the proof of Proposition 3.2 is a manifestation of a very coarse compactness property of stable bundles, a desirable property used to great e ect in gauge theory. Moduli spaces of stable holomorphic bundles on a Kähler surface can fail to be compact in two ways, one re ecting the degeneration from stable to polystable and the other in terms of the concentration of curvature of Hermite-Einstein connections. The former is the subject of this paper, whereas the latter is considered in [6]. Although the failure of moduli spaces of stable bundles on compact Kähler surfaces to be compact can be controlled to some extent as described in that reference, in higher dimensions there is less control on the degeneration and one is forced to consider compacti cations in terms of sheaves ( [2]). The Bogomolov inequality (c − (r − )c / r) · ω n− ≥ for semistable sheaves and bundles does not provide su cient control on subbundles in dimensions greater than .

6.
As alluded to in the introduction, there are profound relationships between the theory of stable holomorphic vector bundles on compact Kähler manifolds and the theory of constant scalar curvature Kähler metrics, these relationships mediated by geometric invariant theory. In that the former theory is a quasilinear analogue of the latter (in the sense of partial di erential equations), it can be hoped that the results here may provide useful directions for the further investigation of moduli of compact complex manifolds and their geometries.

7.
To conclude on an even more speculative note, in view of the critical importance of Yang-Mills theory and of representation theory in contemporary physics, it might also be hoped that our results may provide deeper insight into the nature of elementary particles and their interactions.

Con ict of interest:
The authors state no con ict of interest.