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Moduli spaces of bundles over Riemann surfaces and the Yang-Mills stratification revisited

Refinements of the Yang-Mills stratifications of spaces of connections over a compact Riemann surface are investigated. The motivation for this study was the search for a complete set of relations between the standard generators for the cohomology of the moduli spaces of stable holomorphic bundles of rank n and degree d when n and d are coprime and n>2.

C(n, d), and to relate these stratifications to natural refinements of the notion of the Harder-Narasimhan type of a holomorphic bundle over Σ.
The motivation for this study was the search for a complete set of relations among the standard generators for the cohomology of the moduli spaces M(n, d) when the rank n and degree d are coprime and n > 2 [11]. The cohomology rings of the moduli spaces M(n, d) have been the subject of much interest over many years; see for example [1,3,7,8,14,16,17,20,21,29,31,32,38,40,41] among many other pieces of work. In the case when n = 2 we now have a very thorough understanding of the structure of the cohomology ring [2,21,37,41]. For arbitrary n it is known that the cohomology has no torsion and formulas for computing the Betti numbers have been obtained, as well as a set of generators for the cohomology ring [1,6,7,16,42]. When n = 2 the relations between these generators can be explicitly described and in particular a conjecture of Mumford, that a certain set of relations is a complete set, was proved some years ago [2,21,27,37,41]. However less is known about the relations between the generators when n > 2, and the most obvious generalisation of Mumford's conjecture to the cases when n > 2 is false, although a modified version of the conjecture (using 'dual Mumford relations' together with the original Mumford relations) is true for n = 3 [9]. There is, however, a further generalisation of Mumford's relations, and the attempt by Richard Earl and the author to prove that this set is indeed complete was the original stimulus for studying refinements of the Yang-Mills stratification, although a different application has now appeared [11,19].
The layout of this paper is as follows. §1 recalls background material on moduli spaces of bundles and different versions of Mumford's conjecture, while §2 reviews the Morse stratification of the normsquare of a moment map and some refinements of this stratification. §3 studies the structure of subbundles of semistable bundles over Σ which are direct sums of stable bundles all of the same slope, and this is used in §4 to define two canonical refinements of the Harder-Narasimhan filtration of a holomorphic bundle over Σ, and thus to construct two refinements of the Yang-Mills stratification. In the next two sections the stratification defined in §2 is applied to holomorphic bundles over Σ; its indexing set is studied in §5 and the associated strata are investigated in §6. This stratification corresponds to a third refinement of the Harder-Narasimhan filtration whose subquotients are all direct sums of stable bundles of the same slope. The relationship between these three filtrations is considered in §7, and §8 provides a brief conclusion.

Background material on moduli spaces of bundles
When n and d are coprime, the generators for the rational cohomology 1 of the moduli space M(n, d) given by Atiyah and Bott in [1] are obtained from a (normalised) universal bundle V over M(n, d) × Σ. With respect to the Künneth decomposition of H * (M(n, d) × Σ) the rth Chern class c r (V ) of V can be written as where {1}, {α j : 1 ≤ j ≤ 2g}, and {ω} are standard bases for H 0 (Σ), H 1 (Σ) and H 2 (Σ), and (1.1) a r ∈ H 2r (M(n, d)), b j r ∈ H 2r−1 (M(n, d)), f r ∈ H 2r−2 (M(n, d)), for 1 ≤ r ≤ n and 1 ≤ j ≤ 2g. It was shown by Atiyah and Bott [1,Prop. 2.20 and p.580] that the classes a r and f r (for 2 ≤ r ≤ n) and b r j (for 1 ≤ r ≤ n and 1 ≤ j ≤ 2g) generate the rational cohomology ring of M(n, d).
Since tensoring by a fixed holomorphic line bundle of degree e gives an isomorphism between the moduli spaces M(n, d) and M(n, d + ne), we may assume without loss of generality that (2g − 2)n < d < (2g − 1)n.
This implies that H 1 (Σ, E) = 0 for any stable bundle of rank n and degree d [34,Lemma 5.2], and hence that π ! V is a bundle of rank d − n(g − 1) over M(n, d), where π : M(n, d) × Σ → M(n, d) is the projection onto the first component and π ! is the K-theoretic direct image map. It follows that c r (π ! V ) = 0 for r > d − n(g − 1). Via the Grothendieck-Riemann-Roch theorem we can express the Chern classes of π ! V as polynomials in the generators a r , b j r , f r described above, and hence their vanishing gives us relations between these generators. These are Mumford's relations, and they give us a complete set of relations when n = 2. We can generalise them for n > 2 as follows.
The relations between the generators a r , b j r , f r obtained in this way for 0 <n < n and d n + 1 >d n > d n and nn(g − 1) − dn +dn < r < nn(g + 1) − dn +dn (with a little more care taken whenn andd are not coprime) are the ones we consider. They are essentially Mumford's relations when n = 2. To show that these form a complete set of relations, a natural strategy is to consider the Yang-Mills stratification which was used by Atiyah and Bott to obtain their generators for the cohomology ring.
Recall that a holomorphic vector bundle E over Σ is called semistable (respectively stable) if every holomorphic subbundle D of E satisfies µ(D) ≤ µ(E), (respectively µ(D) < µ(E)), where µ(D) = degree(D)/rank(D) is the slope of D. Bundles which are not semistable are said to be unstable. Note that semistable bundles of coprime rank and degree are stable. Let E be a fixed C ∞ complex vector bundle of rank n and degree d over Σ. Let C be the space of all holomorphic structures on E and let G c denote the group of all C ∞ complex automorphisms of E. Atiyah and Bott [1] identify the moduli space M(n, d) with the quotient C ss /G c where C ss is the open subset of C consisting of all semistable holomorphic structures on E. The group G c is the complexification of the gauge group G which consists of all smooth automorphisms of E which are unitary with respect to a fixed Hermitian structure on E. We shall write G for the quotient of G by its U (1)-centre and G c for the quotient of G c by its C * -centre. There are natural isomorphisms H * (M(n, d)) ∼ = H * (C ss /G c ) = H * (C ss /G c ) ∼ = H * G c (C ss ) ∼ = H * G (C ss ) between the cohomology of the moduli space and the G-equivariant cohomology of C ss , since the C * -centre of G c acts trivially on C ss , while G c acts freely on C ss and G c is the complexification of G.
In order to show that the restriction map H * G (C) → H * G (C ss ) is surjective, Atiyah and Bott consider the Yang-Mills (or Atiyah-Bott-Shatz) stratification of C. This stratification {C µ : µ ∈ M} is the Morse stratification for the Yang-Mills functional on C, but it also has a more explicit description. It is indexed by the partially ordered set M consisting of all the Harder-Narasimhan types of holomorphic bundles of rank n and degree d, defined as follows. Any holomorphic bundle E over M of rank n and degree d has a filtration 0 = E 0 ⊂ E 1 ⊂ · · · ⊂ E P = E of subbundles such that the subquotients Q p = E p /E p−1 are semi-stable for 1 ≤ p ≤ P and satisfy where d p and n p are the degree and rank of Q p and µ(Q p ) is its slope. This filtration is canonically associated to E and is called the Harder-Narasimhan filtration of E.
We define the type of E to be µ = (µ(Q 1 ), ..., µ(Q P )) ∈ Q n where the entry µ(Q p ) is repeated n p times. The semistable bundles have type µ 0 = (d/n, ..., d/n) and form the unique open stratum. The set M of all possible types of holomorphic vector bundles over Σ provides our indexing set, and if µ ∈ M the subset C µ ⊆ C is defined to be the set of all holomorphic vector bundles over Σ of type µ. A partial order on M with the property that the closure of the stratum indexed by µ is contained in the union of all strata indexed by µ ′ ≥ µ is defined as follows. Let σ = (σ 1 , ..., σ n ) and τ = (τ 1 , ..., τ n ) be two types; then The gauge group G acts on C preserving the stratification which is equivariantly perfect with respect to this action, which means that its equivariant Thom-Gysin sequences is the complex codimension of C µ in C and U µ is the open subset of C which is the union of all those strata labelled by µ ′ ≤ µ; we also have Atiyah and Bott show that the stratification is equivariantly perfect by considering the composition of the Thom-Gysin map H with restriction to C µ , which is multiplication by the equivariant Euler class e µ of the normal bundle to C µ in C. They show that e µ is not a zero-divisor in H * G (C µ ) and deduce that the Thom-Gysin maps H j−2dµ G (C µ ) → H j G (U µ ) are all injective. So putting this all together Atiyah and Bott obtain inductive formulas for the G-equivariant Betti numbers of C ss , and they also conclude that there is a natural surjection . Thus generators of the cohomology ring H * (BG) give generators of the cohomology ring M(n, d).
The classifying space BG can be identified with the space Map d (Σ, BU (n)) of all smooth maps f : Σ → BU (n) such that the pullback to Σ of the universal vector bundle over BU (n) has degree d. If we pull back this universal bundle using the evaluation map then we obtain a rank n vector bundle V over BG × Σ. If further we restrict the pullback bundle induced by the maps to C ss ×{e}×Σ for some e ∈ EG then we obtain a G-equivariant holomorphic bundle on C ss × Σ. The C * -centre of G acts as scalar multiplication on the fibres, and the associated projective bundle descends to a holomorphic projective bundle over M(n, d) × Σ. In fact this is the projective bundle of a holomorphic vector bundle V over M(n, d) × Σ which has the universal property that, for any [E] ∈ M(n, d) representing a bundle E over Σ, the restriction of V to By a slight abuse of notation we define elements a r , b j r , f r in H * (BG; Q) by writing where, as before, ω is the standard generator of H 2 (Σ) and α 1 , ..., α 2g form a fixed canonical cohomology basis for H 1 (Σ). In fact the ring H * (BG) is freely generated as a graded super-commutative algebra over Q by the elements , and the proof that the Yang-Mills stratification is equivariantly perfect leads to completeness criteria for a set of relations to be complete. Let R be a subset of the kernel of the restriction map H * G (C) → H * G (C ss ). Suppose that for each unstable type µ = µ 0 there is a subset R µ of the ideal generated by R in H * G (C) such that the image of R µ under the restriction map is zero unless ν ≥ µ and when ν = µ contains the ideal of H * G (C µ ) generated by the equivariant Euler class e µ of the normal bundle to the stratum C µ in C. Then R generates the kernel of the restriction map H * G (C) → H * G (C ss ) as an ideal of H * G (C). In fact Atiyah and Bott could have replaced the Yang-Mills stratification with a coarser stratification of C and obtained equivalent results. For any integers n 1 and d 1 let S n1,d1 be the subset of C consisting of all those holomorphic structures with Harder-Narasimhan filtration 0 = E 0 ⊂ E 1 ⊂ · · · ⊂ E s = E where E 1 has rank n 1 and degree d 1 . We shall say that such a holomorphic structure has coarse type (n 1 , d 1 ). Then S n1,d1 is locally a submanifold of finite codimension is an open subset of C(n − n 1 , d − d 1 ). Moreover the equivariant Euler class e n1,d1 of the normal to S n1,d1 in C is not a zero divisor in H * G (S n1,d1 ), so the stratification of C by coarse type S n1,d1 : 0 < n 1 < n, is equivariantly perfect. This means that we can modify our completeness criteria, so that for each pair of positive integers (n,d) with 0 <n < n andd n > d n we require a set of relations whose restriction in H * G (S n1,d1 ) is zero when d 1 /n 1 <d/n or d 1 /n 1 =d/n and n 1 <n, and when (n 1 , d 1 ) = (n,d) equals the ideal of H * G (Sn ,d ) generated by the equivariant Euler class en ,d of the normal to Sn ,d in C.
It is easy enough to prove that if γ ∈ H G(n,d) * (C(n,d) s ) whered/n > d/n and if r > nn(g − 1) +nd − nd then the image of the slant product c r (−π ! (V * ⊗ V))\γ under the restriction map (C(n,d) s ) whered/n > d/n and if r = nn(g − 1) +nd − nd + 1 + j, then it turns out that the image of the slant product c r (−π ! (V * ⊗ V))\γ under the restriction map is the product (−a 1 ) j LD(γ)en ,d of the equivariant Euler class en ,d of the normal bundle to Sn ,d in C with the image of γ under the Lefschetz duality map LD and j copies of minus the generator a (1)) ⊗ H * G(n,d) (C(n,d) s ) which comes from the copy of the polynomial ring H * (BU (1)). The proof of this is based on Porteous's Formula (as in Beauville's alternative proof [2] of the theorem of Atiyah and Bott that the classes a r , b j r , f r generate H * (M(n, d)); cf. [39] and [12]), which allows us to deduce that the Poincaré dual of ∆ s × U (n,d) in . In other words the restriction of cn2 (g−1)+1 (−π ! (V * ⊗V 1 )) to C(n,d) s × C(n,d) ss × U (n,d) is the image of 1 under the G(n,d) × G(n,d) × G(n − n, d −d)-equivariant Thom-Gysin map associated to the inclusion of ∆ s × U (n,d)) in C(n,d) s × C(n,d) ss × U (n,d)). We can express the higher Chern classes of −π ! (V * ⊗ V 1 ) in a similar way [11] by using Fulton's Excess Porteous Formula [13]. Recall that given η ∈ H * G(n,d) (C(n,d) ss ) ⊗ H * G(n−n,d−d) (C(n −n, d −d)) we wish to find a relation whose restriction to C(n 1 , d 1 ) ss × U (n 1 , d 1 ) is zero when d 1 /n 1 <d/n or d 1 /n 1 =d/n and n 1 < n, and when (n 1 , d 1 ) = (n,d) equals ηen ,d .
We have found such a relation when η lies in the image of the Lefschetz duality map LD which maps H G(n,d) * (C(n,d) s ) to H * G(n,d) (C(n,d) ss ) and thus into and more generally when η has the form η = (−a 1 ) j LD(γ), for some element γ of H G(n,d) * (C(n,d) s ). Whenn andd are coprime this gives us all η ∈ H * G(n,d) (C ss ). Moreover ) is generated as a module over H * G (C) by H * G(n,d) (C(n,d) ss ), so whenn andd are coprime, we have now obtained the relations we need from the slant products and a little more work reduces the range of r andd (see [11] for more details).
This deals with the case whenn andd are coprime, but the completeness criteria have not yet been shown to hold for pairsn andd with common factors. This was the original motivation for considering further modifications to the Yang-Mills stratification. The difficulty with using the Yang-Mills stratification itself, or the stratification of C by coarse type, is that in each case, although n and d are chosen to be coprime so that semistability and stability coincide for n and d, in the construction of the stratification othern andd appear for which semistability and stability do not coincide.

Stratifying a set of semistable points
In this section we shall describe briefly how to stratify the set X ss of semistable points of a complex projective variety X equipped with a linear action of a complex reductive group G, so that the set X s of stable points of X is an open stratum (see [15,22,30,34] for background and [28] for more details).
We assume that X has some stable points but also has semistable points which are not stable. In [23,25] it is described how one can blow X up along a sequence of nonsingular G-invariant subvarieties to obtain a G-invariant morphismX → X whereX is a complex projective variety acted on linearly by G such thatX ss =X s . The setX ss can be obtained from X ss as follows. Let r > 0 be the maximal dimension of a reductive subgroup of G fixing a point of X ss , and let R(r) be a set of representatives of conjugacy classes of all connected reductive subgroups R of dimension r in G such that Z ss R = {x ∈ X ss : R fixes x} is non-empty. Then R∈R(r) GZ ss R is a disjoint union of nonsingular closed subvarieties of X ss . The action of G on X ss lifts to an action on the blow-up of X ss along R∈R(r) GZ ss R which can be linearised so that the complement of the set of semistable points in the blow-up is the proper transform of the subset φ −1 (φ(GZ ss R )) of X ss where φ : X ss → X//G is the quotient map (see [23] 7.17). Moreover no semistable point in the blow-up is fixed by a reductive subgroup of G of dimension at least r, and a semistable point in the blow-up is fixed by a reductive subgroup R of dimension less than r in G if and only if it belongs to the proper transform of the subvariety Z ss R of X ss . If we repeat this process enough times, we obtain π :X ss → X ss such that X ss =X s . Equivalently we can construct a sequence X ss (R0) = X ss , X ss (R1) , . . . , X ss (Rτ ) =X ss where R 1 , . . . , R τ are connected reductive subgroups of G with and if 1 ≤ l ≤ τ then X (R l ) is the blow up of X ss (R l−1 ) along its closed nonsingular subvariety GZ ss Similarly, X//G =X ss /G can be obtained from X//G by blowing up along the proper transforms of the images Z R //N in X//G of the subvarieties GZ ss R of X ss in decreasing order of dim R.
If 1 ≤ l ≤ τ then we have a G-equivariant stratification {S β,l : (β, l) ∈ B l × {l}} of X (R l ) by nonsingular G-invariant locally closed subvarieties such that one of the strata, indexed by (0, l) ∈ B l × {l}, coincides with the open subset X ss (R l ) of X (R l ) . Here B l is a finite subset of a fixed positive Weyl chamber t + in the Lie algebra t of a maximal compact torus T of G. In fact β ∈ t + lies in B l if and only if β is the closest point to 0 in the convex hull in t of some nonempty subset of the set of weights {α 0 , . . . , α n } for the linear action of T on X (R l ) .
There is a partial ordering on B l given by γ > β if ||γ|| > ||β||, with 0 as its minimal element, such that if β ∈ B l then the closure in X l of the stratum S β,l satisfies If β ∈ B l and β = 0 then the stratum S β,l retracts G-equivariantly onto its (tranverse) intersection with the exceptional divisor E l for the blow-up X (R l ) → X ss (R l−1 ) . This exceptional divisor is isomorphic to the projective bundle P(N l ) over GẐ ss R l , whereẐ ss R l is the proper transform of Z ss R l in X ss (R l−1 ) and N l is the normal bundle to GẐ ss R l in X ss R l−1 . The stratification {S β,l : β ∈ B l } is determined by the action of R l on the fibres of N l over Z ss R l (see [23] §7). There is thus a stratification {Σ γ : γ ∈ Γ} of X ss indexed by defined as follows. We take as the highest stratum Σ R1 the nonsingular closed subvariety GZ ss R1 whose complement in X ss can be naturally identified with the complement X (R1) \E 1 of the exceptional divisor E 1 in X (R1) . We have GZ ss where N 1 is the normaliser of R 1 in G, and Z ss R1 is equal to the set of semistable points for the action of N 1 , or equivalently for the induced action of N 1 /R 1 , on Z R1 , which is a union of connected components of the fixed point set of R 1 in X. Moreover since R 1 has maximal dimension among those reductive subgroups of G with fixed points in X ss , we have Z ss R1 = Z s R1 where Z s R l denotes the set of stable points for the action of N l /R l on Z R l for 1 ≤ l ≤ τ .
Next we take as strata the nonsingular locally closed subvarieties , and then we take the intersection of X ss (R1) \E 1 with GZ ss R2 . This intersection is GZ s R2 where Z s R2 is the set of stable points for the action of N 2 /R 2 on Z R2 , and its complement in X ss (R1) \E 1 can be naturally identified with the complement in X (R2) of the union of E 2 and the proper transformÊ 1 of E 1 .
The next strata are the nonsingular locally closed subvarieties , and the stratum after these is GZ s R3 . We repeat this process for 1 ≤ l ≤ τ and take X s as our final stratum indexed by 0.
The given partial orderings on B 1 , . . . , B τ together with the ordering in the expression (2.2) above for Γ induce a partial ordering on Γ, with R 1 as the maximal element and 0 as the minimal element, such that the closure in X ss of the stratum Σ γ indexed by γ ∈ Γ satisfies It is possible to describe the strata Σ γ in more detail. Either Σ γ is GZ s R l , for some l, or else it is S β,l \(E l ∪Ê l−1 ∪ ... ∪Ê 1 ) for some l and β ∈ B l \{0}, in which case by [22] §5 we have where Y ss β,l fibres over Z ss β,l with fibre C m β,l for some m β,l > 0, and P β is a parabolic subgroup of G with the stabiliser Stab(β) of β under the adjoint action of G as its maximal reductive subgroup. Here the fibration p β : Y ss β,l → Z ss β,l sends y to a limit point of its orbit under the complex one-parameter subgroup of R l generated by β. Moreover β,l ∩ E l fibres over Z ss β,l with fibre C m β,l −1 (see [23] Lemmas 7.6 and 7.11). Thus β,l \E l fibres over Z ss β,l with fibre C m β,l −1 ×(C\{0}). Furthermore if π l : E l ∼ = P(N l ) → GẐ ss R l is the projection then Lemma 7.9 of [23] tells us that when x ∈Ẑ ss R l the intersection of S β,l with the fibre π −1 l (x) = P(N l,x ) of π l at x is the union of those strata indexed by points in the adjoint orbit Ad(G)β in the stratification of P(N l,x ) induced by the representation ρ l of R l on the normal N l,x to GẐ ss R l at x. Careful analysis [28] shows that we can, if we wish, replace the indexing set B l \{0}, whose elements correspond to the G-adjoint orbits Ad(G)β of elements of the indexing set for the stratification of P(N l,x ) induced by the representation ρ l , by the set of their N l -adjoint orbits Ad(N l )β. Then we still have (2.4) -(2.6), but now if q β : P β → Stab(β) is the projection we have . This process gives us a stratification {Σ γ : γ ∈ Γ} of X ss such that the stratum indexed by the minimal element 0 of Γ coincides with the open subset X s of X ss . We shall apply this construction to obtain a stratification of C ss , and thus inductively to refine the Yang-Mills stratification {C µ : µ ∈ M} of C by Harder-Narasimhan type.

Direct sums of stable bundles of equal slope
In the good case when n and d are coprime, then C ss = C s and M(n, d) = C ss /G c is a nonsingular projective variety. When n and d are not coprime, we can use the description of M(n, d) as the geometric invariant theoretic quotient C//G c to construct a partial desingularisationM(n, d) of M(n, d). From this construction we can use the methods described in §2 to obtain a stratification of C ss with C s as an open stratum, and thus (by considering the subquotients of the Harder-Narasimhan filtration) obtain a stratification of C refining the stratification {C µ : µ ∈ M}. To understand this refined stratification we need to use the description in [26] of the partial desingularisationM(n, d).
In fact in [26]M(n, d) is not constructed using the representation of M(n, d) as the geometric invariant theoretic quotient of C by G c , although (as is noted at [26, p.246]) this representation of M(n, d) would lead to the same partial desingularisation. Instead in [26] M(n, d) is represented as a geometric invariant theoretic quotient of a finite-dimensional nonsingular quasi-projective variety R(n,d) by a linear action of SL(p; C) where p =d +n(1 − g) withd >> 0. We may assume thatd >> 0, since tensoring by a line bundle of degree l gives an isomorphism of M(n, d) with M(n,d +nl) for any l ∈ Z. By [34,Lemma 5.2] if E is a semistable bundle over Σ of rankn and degreed >n(2g − 1) where g is the genus of Σ, then E is generated by its sections and H 1 (Σ, E) = 0. If p =d +n(1 − g), this implies that dim H 0 (Σ, E) = p and that there is a holomorphic map h from Σ to the Grassmannian G(n, p) ofn-dimensional quotients of C p such that the pullback Let R(n,d) be the set of all holomorphic maps h : Σ → G(n, p) such that E(h) = h * T has degree d and the map on sections C p → H 0 (Σ, E(h)) induced from the quotient bundle map C p × Σ → E(h) is an isomorphism. Ford >> 0 this set R(n,d) has the structure of a nonsingular quasi-projective variety and there is a quotient E of the trivial bundle of rank p over R(n,d) × Σ satisfying the following properties (see [34, §5]).
(i) If h ∈ R(n,d) then the restriction of E to {h} × Σ is the pullback E(h) of the tautological bundle T on G(n, p) along the map h : Σ → G(n, p).
(ii) If h 1 and h 2 lie in R(n,d) then E(h 1 ) and E(h 2 ) are isomorphic as bundles over Σ if and only if h 1 and h 2 lie in the same orbit of the natural action of GL(p; C) on R(n,d).
(iii) If h ∈ R(n,d) then the stabiliser of h in GL(p; C) is isomorphic to the group Aut(E(h)) of complex analytic automorphisms of E(h).
If N >> 0 then R(n,d) can be embedded as a quasi-projective subvariety of the product (G(n, p)) N by a map of the form h → (h(x 1 ), . . . . , h(x N )) where x 1 , . . . , x N are points of Σ. This embedding gives a linearisation of the action of GL(p; C) on R(n,d). If N >> 0 and d >> 0 then we also have the following: (iv) The point h ∈ R(n,d) is semistable in the sense of geometric invariant theory for the linear action of SL(p; C) on R(n,d) if and only if E(h) is a semistable bundle. If h 1 and h 2 lie in R(n,d) ss then they represent the same point of R(n,d)//SL(p; C) if and only if gr(E(h 1 )) ∼ = gr(E(h 2 )), and thus there is a natural identification of M(n, d) with R(n,d)//SL(p; C) (see for example [34, §5]).
It is shown in [24] that the Atiyah-Bott formulas for the equivariant Betti numbers of C ss can be obtained by stratifying R(n, d) instead of C(n, d), and in fact throughout this paper we could work with either R(n, d) or C(n, d). In particular properties (i) to (iv) above imply that the analysis in [26] of the construction of the partial desingularisation ofM(n,d) asR(n,d)//SL(p; C) applies equally well if we work withC//G c .
To describe the construction ofM(n, d), first of all we need to find a set R of representatives of the conjugacy classes of reductive subgroups of SL(p; C) which occur as the connected components of stabilisers of semistable points of R(n, d). In fact it is slightly simpler to describe the corresponding subgroups of G = GL(p; C), and since the central one-parameter subgroup of GL(p; C) consisting of nonzero scalar multiples of the identity acts trivially on R(n, d), finding stabilisers in GL(p; C) is essentially equivalent to finding stabilisers in SL(p; C). By [26, pp. 248-9] such conjugacy classes in GL(p; C) correspond to unordered sequences (m 1 , n 1 ), ..., (m q , n q ) of pairs of positive integers such that m 1 n 1 + ... + m q n q = n and n divides n i d for each i. A representative R of the corresponding conjugacy class is given by where D 1 , . . . , D q are all semistable and D i has rank n i and degree d i , while GZ s R consists of all those holomorphic structures E with where D 1 , . . . , D q are all stable and not isomorphic to one another, and D i has rank n i and rank d i . Moreover the normaliser N of R in GL(p; C) has connected component Sym(#{i : m i = j and n i = k}) where Sym(b) denotes the symmetric group of permutations of a set with b elements. Furthermore in terms of the notation of §2, if R = R l then a holomorphic structure belongs to one of the strata Σ β,l with β ∈ B l \{0} if and only if it has a filtration 0 where D 1 , . . . , D q are all stable and not isomorphic to one another, and D i has rank n i and rank d i [26, p. 248]. Thus to understand the refined Yang-Mills stratification of C, we need to study refinements . . , D q not isomorphic to each other and all stable of the same slope as E, such that any other subbundle of the same form r not isomorphic to each other and all stable of the same slope as E, satisfies r ≤ q and (after permuting the order of D 1 , ..., D q suitably) D j ∼ = D ′ j for all 1 ≤ j ≤ r and the inclusion If we choose a subbundle of E of maximal rank among those of the required form, this follows immediately from → E are injective bundle homomorphisms, then there exist nonnegative integers n 1 , ..., n k and linear injections i j : C mj → C nj and i ′ j : C m ′ j → C nj for 1 ≤ j ≤ k and an injective bundle homomorphism γ from to E such that α and β both factorise through γ via the injections i j and i ′ j for 1 ≤ j ≤ k in the obvious way.
Proof: Consider the kernel of shows that this kernel is a subsheaf of the domain of α ⊕ β which has the same slope as E, D 1 , ..., D q , and D ′ 1 , ..., D ′ r . Induction on Lemma 5.1] and the obvious projection from C m1 onto C m1−1 shows that such a subsheaf is in fact of the form is the canonical subbundle of this form associated to E as in Proposition 3.1, and without loss of generality we assume D j ∼ = D ′ j for 1 ≤ j ≤ k for some 0 ≤ k ≤ min{q, r} and that there are no other isomorphisms between the bundles D 1 , ..., D q and D ′ 1 , ..., D ′ r .
Definition 3.4. With the notation above we set for any semistable bundle E.
Remark 3.5. Of course, by the Jordan-Hölder theorem, given any filtration

Maximal and minimal Jordan-Hölder filtrations
Recall that the Harder-Narasimhan filtration of a holomorphic bundle E over Σ is a canonical filtration In the last section we saw that any semistable bundle E has a canonical maximal subbundle of the form where D 1 , . . . , D q are not isomorphic to each other and are all stable of the same slope as E. This subbundle is nonzero if E = 0, since any nonzero semistable bundle is either stable itself or it has a proper stable subbundle of the same slope.
Therefore any semistable bundle E has a canonical filtration whose subquotients are direct sums of stable bundles, which is defined inductively so that where D 1 , ..., D q are stable nonisomorphic bundles all of the same slope as E with nonnegative integers m ij for 1 ≤ i ≤ q and 1 ≤ j ≤ r, and E j /E j−1 is the maximal subbundle of E/E j−1 of this form. If, moreover, we assume that r j=1 m ij > 0 for all 1 ≤ i ≤ q then the filtration (4.1), the bundles D i and integers m ij (for 1 ≤ i ≤ q and 1 ≤ j ≤ r) and the decompositions (4.2) are canonically associated to E up to isomorphism of the bundles D i , the usual action of GL(m ij ; C) on C mij and the obvious action of the permutation group Sym(q) on this data. We can generalise the definition to the case when E is not necessarily semistable, by applying this construction to the subquotients of the Harder-Narasimhan filtration of E. This gives us a canonical refinement of the Harder-Narasimhan filtration such that each subquotient E j /E j−1 is the maximal subbundle of E/E j−1 which is a direct sum of stable bundles all having maximal slope among the nonzero subbundles of E/E j−1 . We shall call this refinement of the Harder-Narasimhan filtration the maximal Jordan-Hölder filtration of E.
Definition 4.1. Let s, q 1 , ..., q s and r 1 , ..., r s be positive integers and let d k , n ik and m ijk (for 1 ≤ i ≤ q k , 1 ≤ j ≤ r k and 1 ≤ k ≤ s) be integers satisfying under the action of the product of symmetric groups Σ q1 × ... × Σ q k on the set of such sequences, and let I = I(n,d) denote the set of all such orbits, for fixedn and d. Given [d, n, m] ∈ I(n,d) let s([d, n, m] = s and let S maxJH [d,n,m] denote the subset of C consisting of those holomorphic structures on our fixed smooth bundle of rank n and degreed whose maximal Jordan-Hölder filtration where D 1 , ..., D q are semistable bundles all having the same slope as E, but this filtration is not necessarily the maximal Jordan-Hölder filtration of E. If the bundles D 1 , ..., D q are not all stable or if two of them are isomorphic to each other, then If, on the other hand, D 1 , ..., D q are all stable and not isomorphic to each other, then the maximal Jordan-Hölder filtration of E is of the form or n ′ = n and 1 ≤ r ′ ≤ r and Using (1.2), we can then extend this partial order to I so that Proof: (cf. [1, §7]) The rank and degree are the only C ∞ invariants of a vector bundle over Σ. Thus we may choose a C ∞ isomorphism of our fixed C ∞ bundle E over Σ with a bundle of the form Let Y [n,m] be the subset of C ss consisting of all semistable holomorphic structures E on E for which the subbundles are holomorphic for 1 ≤ j ≤ r, and for which there are nonisomorphic stable holomorphic structures D 1 , . . . , D q on the C ∞ bundles D 1 , . . . , D q such that the and finally for each 1 ≤ j ≤ r the quotient E j /E j−1 is the maximal subbundle of E/E j−1 isomorphic to a direct sum of stable bundles of the same slope as E/E j−1 . Let G c [n, m] be the subgroup of the complexified gauge group G c consisting of all C ∞ complex automorphisms of E which preserve the filtration of E by the subbundles for all nonnegative integers k and l 1 , ..., l r . Since the filtration (4.1) and decompositions of E 1 /E 0 ,...,E r /E r−1 are canonically associated to E up to the actions of these general linear and permutation groups, we have As in [1, §7] we have that Y [n,m] is an open subset of an affine subspace of the infinite-dimensional affine space C and the injection m] , let End ′ E be the subbundle of EndE consisting of holomorphic endomorphisms of E preserving the maximal Jordan-Hölder filtration (4.1) and decomposition (4.2) up to isomorphism of the bundles D i and the vector spaces C mij . Let End ′′ E be the quotient of EndE by End ′ E. The normal to the G c -orbit of E in C can be canonically identified with H 1 (Σ, EndE) (see [1, §7]) and the image of T E Y [n,m] in this can be canonically identified with the image of the natural map which fits into the long exact sequence of cohomology induced by the short exact sequence of bundles Thus we get an isomorphism We have short exact sequences so since End ′′ E = EndE/End ′ E we get short exact sequences the short exact sequence (4.5) becomes From the sequences (4.4) and (4.6) it follows that the rank of End ′′ E is Thus by induction on r we have Since D 1 , . . . , D q all have the same sloped/n as E we have deg(End ′′ E) = 0. Therefore by Riemann-Roch Moreover the short exact sequences (4.4) and (4.6) give us long exact sequences of cohomology .., D q are nonisomorphic stable bundles all of the same slope as E 1 , and is the maximal subbundle of E/E 1 which is a direct sum of stable bundles all of the same slope as E/E 1 . Since E 1 and E/E 1 have the same slope, it follows from Corollary 3.3 that By choosing an open cover U of Σ such that the filtration 0 = E 0 ⊂ E 1 ⊂ E 2 ⊂ · · · ⊂ E r = E is trivial over each U ∈ U, and describing E in terms of upper triangular transition functions on and thus by induction on r we have In particular this tells us that the image in C of the derivative of the injection (4.3) has constant codimension, and it follows as in [1, §7] (see also [1, § §14 and 15]) that the subset S [n,m] is locally a complex submanifold of C of finite codimension given by (4.7).
Thus F j /F j−1 is a direct sum of stable bundles all of the same slope, and moreover F j−1 is the minimal subbundle of F j such that F j /F j−1 is a direct sum of stable bundles all of which have minimal slope among quotients of F j . Applying this construction with E replaced by E ′ , we find that every holomorphic bundle E over Σ has a canonical filtration which we will call the minimal Jordan-Hölder filtration of E, with the property that if 1 ≤ j ≤ t then F j−1 is the minimal subbundle of F j such that F j /F j−1 is a direct sum of stable bundles all of which have minimal slope among quotients of F j .
The minimal and maximal Jordan-Hölder filtrations of a bundle do not necessarily coincide. For example, consider the direct sum E ⊕ F of two semistable bundles with maximal Jordan-Hölder filtrations 0 = E 0 ⊂ E 1 ⊂ ... ⊂ E t = E and 0 = F 0 ⊂ F 1 ⊂ ... ⊂ F s = F where without loss of generality we may assume that s ≤ t. If E and F have the same slope, then it is easy to check that the maximal Jordan-Hölder filtration of E ⊕ F is that is, it is the direct sum of the maximal Jordan-Hölder filtrations of E and F with the shorter one extended trivially at the top. Similarly the minimal Jordan-Hölder filtration of E ⊕ F is the direct sum of the minimal Jordan-Hölder filtrations of E and F with the shorter one extended trivially at the bottom. Thus if the minimal and maximal Jordan-Hölder filtrations of E and F coincide (which will be the case if, for example, each of the subquotients E j /E j−1 and F j /F j−1 are stable) but these filtrations are not of the same length, then the minimal Jordan-Hölder filtration  [d,n,m] denote the subset of C consisting of those holomorphic structures on our fixed smooth bundle of rankn and degreed whose minimal Jordan-Hölder filtration is of the form 0 = E 0,1 ⊂ E 1,1 ⊂ · · · ⊂ E r1,1 = E 0,2 ⊂ E 1,2 ⊂ · · · · · · ⊂ E rs−1,s−1 = E rs,0 ⊂ · · · ⊂ E rs,s = E . . , D q k k are nonisomorphic stable bundles with rank(D ik ) = n ik and deg(D ik ) = d ik and E j−1,k is the minimal subbundle of E j,k such that E j,k /E j−1,k is a direct sum of stable bundles of slope d k /n k .

More indexing sets
In this section we will consider the indexing set Γ for the stratification {Σ γ : γ ∈ Γ} of C ss defined as in §2.
If γ ∈ Γ then by (2.2) either γ = 0 or γ = R l or γ ∈ B l \{0} × {l} for some 1 ≤ l ≤ τ . If γ = 0 then Σ γ = C s , while by [26, pp.248-9] ] ∈ I ss with r = 1 and q = q 1 , such that where m i = m i1 , and Σ R l consists of all those holomorphic structures E with where D 1 , . . . , D q are all stable of sloped/n and not isomorphic to one another. In order to describe the strata {Σ β,l : β ∈ B l \{0}} more explicitly, we need to look at the action of R l on the normal N R l to GZ ss R l at a point represented by a holomorphic structure E of the form (5.2), and to understand the stratification on P(N R l ) induced by this action of R l . If we choose a C ∞ isomorphism of our fixed C ∞ bundle E with (C m1 ⊗ D 1 ) ⊕ · · · ⊕ (C mq ⊗ D q ), then we can identify C with the infinite-dimensional vector space Ω 0,1 (End((C m1 ⊗ D 1 ) ⊕ · · · ⊕ (C mq ⊗ D q ))) and the normal to the G c -orbit at E can be identified with H 1 (Σ, EndE), where EndE is the bundle of holomorphic endomorphisms of E [1, §7]. If δ j i denotes the Kronecker delta then the normal to GZ ss R can be identified with where End ′ ⊕ E is the quotient of the bundle EndE of holomorphic endomorphisms of E by the subbundle End ⊕ E consisting of those endomorphisms which preserve the decomposition (5.2). The action of R l = q i=1 GL(m i ; C) on this is given by the natural action on C mi 1 mi 2 −δ i 2 i 1 identified with the set of m i1 × m i2 matrices if i 1 = i 2 and the set of trace-free m i1 × m i1 matrices if i 1 = i 2 ; its weights α are therefore of the form α = ξ − ξ ′ where ξ and ξ ′ are weights of the standard representation of R l on ⊕ q i=1 C mi (see [26, pp.251-2] noting the error immediately before (3.18)).
Any element β of the indexing set B l is represented by the closest point to 0 of the convex hull of some nonempty set of these weights, and two such closest points can be taken to represent the same element of B l if and only if they lie in the same Ad(N l )-orbit, where N l is the normaliser of R l (see [22] or [28]). By (3.3) the orbit of β under the adjoint action of the connected component of N l is just its Ad(R l )-orbit, and so by (3.4) the Ad(N l )-orbit of β is the union where π 0 (N l ) is the product of permutation groups Sym(#{i : m i = j and n i = k}).
We can describe this indexing set B l more explicitly as follows. Let us take our maximal compact torus T l in R l to be the product of the standard maximal tori of the unitary groups U (m 1 ),..., U (m q ) consisting of the diagonal matrices, and let t l be its Lie algebra. Let M = m 1 + ... + m q and let e 1 , ..., e M be the weights of the standard representation of T l on C m1 ⊕ ... ⊕ C mq . We take the usual invariant inner product on the Lie algebra u(p) of U (p) given by A, B = −trAB t and restrict it to T l . Since R l is embedded in GL(p; C) by identifying ⊕ q i=1 (C mi ⊗ C pi ) with C p , it follows that e 1 , ..., e M are mutually orthogonal and ||e  for some λ β ij ∈ R for (i, j) ∈ S such that λ β ij ≥ 0 and (i,j)∈S λ β ij = 1. Replacing S with its subset {(i, j) ∈ S : λ β ij > 0} we may assume that λ β ij > 0 for all (i, j) ∈ S. Moreover clearly if S = {e i − e j } has just one element then β = e i − e j , and we then take J = {(1, 0), (1, 1)} with ∆ (1,0) = {j} and ∆ (1,1) = {i}, so we can assume without loss of generality that λ β ij < 1 for all (i, j) ∈ S. Since β = 0 we can also assume that the convex hull of {e i − e j : (i, j) ∈ S} does not contain 0.
In order to find the closest point to 0 of the convex hull of {e i − e j : (i, j) ∈ S} we minimise || subject to the constraints that λ ij ≥ 0 for all (i, j) ∈ S and (i,j)∈S λ ij = 1. Since the weights e 1 , ..., e M are mutually orthogonal, we have Using the method of Lagrange multipliers, we consider If (i, j) ∈ S then i = j and (j, i) ∈ S since the convex hull of {e i − e j : (i, j) ∈ S} does not contain 0, so Thus β = (i,j)∈S λ β ij (e i − e j ) where for each (i, j) ∈ S we have either λ β ij = 0 or λ β ij = 1 (both of which are ruled out by the assumptions on S) or where λ is independent of (i, j) ∈ S.
From S we can construct a directed graph G(S) with vertices 1, ..., M and directed edges from i to j whenever (i, j) ∈ S. Let ∆ 1 ,..., ∆ N be the connected components of this graph. Then {e i − e j : (i, j) ∈ S} is the disjoint union of its subsets {e i − e j : (i, j) ∈ S, i, j ∈ ∆ h } for 1 ≤ h ≤ N , and {e i − e j : (i, j) ∈ S, i, j ∈ ∆ h } is contained in the vector subspace of t l spanned by the basis vectors {e k : k ∈ ∆ h }. Since these subspaces are mutually orthogonal for 1 ≤ h ≤ N , it follows that is the closest point to 0 of the convex hull of {e i − e j : (i, j) ∈ S, i, j ∈ ∆ h } for 1 ≤ h ≤ N , and without loss of generality we assume that β h is nonzero when 1 ≤ h ≤ L and zero when L < h ≤ N . Note that then where a sum over the empty set is interpreted as 0.
By defining ∆ h,m =∆ h,m−l1(h) for an appropriate integer l 1 (h), we can arrange that the function ǫ defined in the statement of the proposition takes values in the interval [−1/2, 1/2), and then by amalgamating those ∆ h for which ǫ(h) takes the same value and rearranging them so that ǫ is a strictly decreasing function, we can assume that the required conditions on ǫ are satisfied, and we have It suffices to show that β is the closest point to 0 of the convex hull of {e i − e j : (i, j) ∈ S}, where S is the set of ordered pairs (i, j) with i, j ∈ {1, ..., M } such that e i ∈ ∆ h,m and e j ∈ ∆ h,m+1 for some (h, m) ∈ J such that (h, m + 1) ∈ J. For this, it is enough to prove firstly that β lies in the convex hull of {e i − e j : (i, j) ∈ S} and secondly that (e i − e j ).β = ||β|| 2 (or equivalently that (e i − e j ).β = 1) for all (i, j) ∈ S. The latter follows easily from the choice of S: if (i, j) ∈ S then there exists (h, m) ∈ J such that (h, m + 1) ∈ J and e i ∈ ∆ h,m and e j ∈ ∆ h,m+1 , so To show that β lies in the convex hull of {e i − e j : (i, j) ∈ S}, we note that
When it is helpful to make the dependence on β explicit, we shall write δ β : (h, m)).
Note, however, that although given any partition

Balanced δ-filtrations
The last section studied the indexing set Γ for the stratification {Σ γ : γ ∈ Γ} of C ss defined as in §2. In this section we will consider what it means for a semistable holomorphic bundle over the Riemann surface Σ to belong to a stratum Σ γ = Σ β,l , where β is as in Proposition 5.1.
Definition 6.1. We shall say that a semistable bundle E has a δ-filtration with associated function δ : {1, ..., t} → {1, ..., t} if δ is an increasing function such that if 1 ≤ k ≤ t then δ(k) ≥ k and the induced filtration Let G(δ) be the graph with vertices 1, ..., t and edges joining i to j if j − 1 = δ(i) < δ(i + 1). Then the connected components of G(δ) are of the form , then −1/2 ≤ ǫ(h) < 1/2, and the ordering of the components of G(δ) can be chosen so that We shall say that the δ-filtration is balanced if the inequalities in (6.1) are all strict and if if and only if m 1 < m 2 or m 1 = m 2 and h 1 ≤ h 2 ; that is, if the usual ordering on {1, ..., t} is the same as the Hebrew lexigraphic ordering via the pairs (h, m).
Remark 6.2. If β is as in Proposition 5.1 then the proof of that proposition shows that where t and m k i for 1 ≤ i ≤ q and 1 ≤ k ≤ t are as in Definition 5.5 and D 1 , ..., D q are nonisomorphic stable bundles of ranks n 1 , ..., n q and all of the same slope d/n, thenr h,m = rank(E φ(h,m) /E φ(h,m)−1 ). Proposition 6.3. Let β be as in Proposition 5.1, let δ be as in Remark 5.4 and let E be a semistable holomorphic structure on E.
(i) If E represents an element of the stratum Σ β,l then E has a unique balanced and hence where t and m k i for 1 ≤ i ≤ q and 1 ≤ k ≤ t are as in Definition 5.5 and D 1 , ..., D q are nonisomorphic stable bundles of ranks n 1 , ..., n q and all of the same slope d/n.
(ii) Conversely, if E has a balanced δ-filtration 0 = E 0 ⊂ E 1 ⊂ ... ⊂ E t = E as in (i) then E represents an element of the stratum Σ β,l if and only if E has no filtration with the corresponding properties for any β ′ satisfying ||β ′ || > ||β||.
Proof: Recall from (2.7) that and that if E represents an element of Y \E β,l then its orbit under the complex oneparameter subgroup of R l generated by β has a limit point in Z s R l . This limit point is represented by the bundle gr(E) which is of the form where D 1 , ..., D q are nonisomorphic stable bundles of ranks n 1 , ..., n q and all of the same slope d/n. Recall also from [1, § 7] that C is an infinite dimensional affine space, and if we fix a C ∞ identification of the fixed C ∞ hermitian bundle E with q i=1 C mi ⊗ D i then we can identify C with the infinite dimensional vector space in such a way that the zero element of Ω 0,1 (End( q i=1 C mi ⊗D i )) corresponds to the given holomorphic structure on q i=1 C mi ⊗ D i . With respect to this identification, the action of R l = q i=1 GL(m i ; C) on C is the action induced by the obvious action of R l on q i=1 C mi ⊗ D i . The one-parameter subgroup of R l generated by β acts diagonally on C m1 ⊕ ... ⊕ C mq with weights β.e j for j ∈ {1, ..., M } where M = m 1 + ... + m q , and so it acts on .., M } as in Definition 5.5, then any E ∈ Σ β lies in the G c -orbit of an element of This completes the proof of (i), as such an element of Ω 0,1 (End( q i=1 C mi ⊗ D i )) represents a holomorphic structure E on E with a filtration of the required form (uniqueness follows from (6.5) and the fact that Q β,l preserves the filtration), and (ii) is now a consequence of (2.1).
Corollary 6.4. Let β be as in Proposition 5.1, let δ be as in Remark 5.4 and let E be a semistable holomorphic structure on E with a balanced δ-filtration where t and m k i for 1 ≤ i ≤ q and 1 ≤ k ≤ t are as in Definition 5.5 and D 1 , ..., D q are nonisomorphic stable bundles of ranks n 1 , ..., n q and all of the same slope d/n. Then E represents an element of the stratum Σ β,l if and only if, in the notation of Definition 6.1, there is no h ∈ {1, ..., L} and refinement such that the induced filtrations and so E can be given a filtration of the form Proof of Corollary 6.4: This follows from [28] and the proof of Proposition 5.1, which tells us that if β ′ = β is the closest point to 0 of the convex hull of {e i − e j : (i, j) ∈ S ′ } where S ′ is a subset of S, then S ′ can be chosen so that the connected components of the graph G(S ′ ) give us a refinement of the partition {∆ h,m : (h, m) ∈ J} of {1, ..., M } associated to β, which in turn gives us a refinement of the filtration (6.6) with the required properties.
Remark 6.6. Recall from Proposition 6.3 that a semistable bundle E represents an element of the stratum Σ β,l if and only if if has a balanced δ-filtration .., D q are nonisomorphic stable bundles of ranks n 1 , ..., n q and all of the same slope d/n, and moreover E has no balanced δ-filtration with the corresponding properties when β is replaced with β ′ satisfying ||β ′ || > ||β||. From (6.4) we have . This gives us some sort of measure of the triviality of the balanced δ-filtration (6.7); very roughly speaking, the more trivial this filtration, the smaller the size of ||β|| −2 and hence the larger ||β|| becomes.
Let us therefore define the triviality of the balanced δ-filtration (6.7) with associated function δ to be Remarks 5.2 and 5.6 tell us that this is well defined. Thus if E ∈ Σ β,l the balanced δ-filtration (6.7) associated to E by Proposition 10.1 can be thought of as having maximal triviality (according to this measure) among the balanced δ-filtrations of E.
Remark 6.7. Let β be as in Proposition 5.1, let δ be as in Remark 5.4 and let E be a semistable holomorphic structure on E with a δ-filtration .., D q are nonisomorphic stable bundles of ranks n 1 , ..., n q and all of the same slope d/n. Then the proof of Proposition 6.3 and (2.3) shows that E represents an element of Σγ for someγ ∈ Γ satisfyingγ ≥ γ = (β, l) with respect to the partial ordering on Γ defined just before (2.3).  where ≥ denotes in (6.10) the partial order on Γ used in Remark 6.7, whereas in (6.11) it denotes the partial order on I ss described in Remark 4.4.
Remark 6.9. It follows from Proposition 6.3 that if R l is as at (5.1) then a holomorphic structure belongs to . . , D q are all stable of slope d/n and ranks n 1 , . . . , n q and are not isomorphic to one another.

Pivotal filtrations
Let us now consider the relationship between the balanced δ-filtration associated to a semistable bundle E as in Proposition 6.3 and the maximal and minimal Jordan-Hölder filtrations defined in §8.
Indeed, motivated by Proposition 6.3, we can try to carry our analysis of the maximal Jordan-Hölder filtration of a bundle E a bit further. Recall that if 1 ≤ j ≤ t then the subquotient E j /E j−1 is the maximal subbundle of E/E j−1 which is a direct sum of stable bundles all having maximal slope among the nonzero subbundles of E/E j−1 . We can ask whether it is true for every subbundle where D 1 , . . . , D q are all stable of slope d/n and ranks n 1 , . . . , n q and are not isomorphic to one another. So we can then ask whether it is possible to find a canonical refinement of the maximal Jordan-Hölder filtration (7.1) of E with an increasing function δ : {1, ..., u} → {1, ..., u} such that δ(j) ≥ j if 1 ≤ j ≤ u, each subquotient F δ(j) /F j−1 is of the form (7.2) and moreover for every subbundle F of E satisfying F j−1 ⊆ F ⊂ F j and slope(F j /F ) = slope(F/F j−1 ), the quotient F δ(j) /F is the maximal subbundle of E/F which is a direct sum of stable bundles all having maximal slope among the nonzero subbundles of E/F . The following example shows that even this cannot be achieved in a canonical way.
Example 7.1. Let E 1 and E 2 be semistable bundles over Σ such that slope(E 1 ) equals slope(E 2 ), with maximal Jordan-Hölder filtrations where D 1 , D 2 , E 1 /D 1 and E 2 /D 2 are nonisomorphic stable bundles all of the same slope as E 1 and E 2 , and let E = E 1 ⊕ E 2 . We observed in Remark 4.7 that the maximal Jordan-Hölder filtration of a direct sum of semistable bundles of the same slope is the direct sum of their maximal Jordan-Hölder filtrations (with the shorter one extended trivially at the top if they are not of the same length). Thus the maximal Jordan-Hölder filtration of E is By Lemma 3.2 there are precisely two proper subbundles F of D 1 ⊕ D 2 with slope(F ) = slope(D 1 ⊕ D 2 /F ) = slope(D 1 ⊕ D 2 ), namely D 1 and D 2 . The maximal Jordan-Hölder filtration of E/D 1 = ( so we can refine the filtration (7.5) of E to get is the maximal subbundle of E/F j−1 which is a direct sum of stable subbundles all having maximal slope among the nonzero subbundles of E/F j−1 . Moreover this is trivially still true if we replace F j−1 by any subbundle F of E satisfying F j−1 ⊆ F ⊂ F j and slope(F j /F ) = slope(F j /F j−1 ), since the only such F is F j−1 itself. We can of course reverse the rôles of E 1 and E 2 in this construction, to get another refinement (7.4). Thus there are precisely two refinements of the maximal Jordan-Hölder filtration of E 1 ⊕ E 2 with the required propertes, and if E 1 has the same rank as E 2 and D 1 has the same rank as D 2 then by symmetry there can be no canonical choice. Notice that if rank(D 1 )/rank(E 1 ) = rank(D 2 )/rank(E 2 ) then neither of the δfiltrations (7.5) and (7.6) is balanced since the inequalities (6.1) are not strict; however the δ-filtration (7.4) is balanced and has maximal triviality, in the sense of (6.9), among balanced δ-filtrations of E 1 ⊕E 2 . If on the other hand rank(D 1 )/rank(E 1 ) = rank(D 2 )/rank(E 2 ) then precisely one of the δ-filtrations (7.5) and (7.6) is balanced and it has maximal triviality, in the sense of (6.9), among balanced δ-filtrations of E 1 ⊕ E 2 . This filtration then determines the stratum Σ γ to which E belongs, and in this case E represents an element of the open subset Σ s β,l of Σ γ . Lemma 7.2. Let E be a bundle over Σ with a filtration Proof: This lemma follows immediately from the well known bijective correspondence between holomorphic extensions of a holomorphic bundle D 1 over Σ by another holomorphic bundle D 2 and elements of H 1 (Σ, D 1 ⊗ D 2 ). The extension −1 → 0 induced by the given filtration gives us an element of and the given trivialisation of the extension (7.7) gives us a decomposition of this as . Projection onto the first summand gives us an extension where k 2 > δ(k 1 ) then (e i − e j ).β ≥ ||β|| 2 , and equality occurs if and only if there exists (h, m) ∈ J with (h, m + 1) ∈ J such that k 1 = φ(h, m) and k 2 = φ(h, m + 1). Thus, retaining the notation of the proof of Proposition 6.3, we observe that if E is represented by an element of q i1,i2=1 t k1=1 t k2=δ(k1)+1 then the limit in C as t → ∞ of exp(−itβ)E is the bundle and so the limit p β (E) of exp(−itβ)E in the blow-up of C along G c Z ss R l is an element of the fibre of the exceptional divisor over grE. Indeed p β (E) is the element of this fibre represented by the sum in of the elements of Proposition 7.4. Let β be as in Proposition 5.1 and let E be a semistable bundle representing an element of Σ β,l with a balanced δ-filtration 3. Suppose also that k 1 ∈ {1, ..., t} is such that β.e j < 0 whenever j ∈ ∆ k2 with k 2 > δ(k 1 ).
Then whenever F is a subbundle of E with slope(F ) = slope(E) and such that E k1−1 ⊆ F ⊂ E k1 , the subquotient E δ(k1) /F is the maximal subbundle of E/F which is a direct sum of stable bundles all having the same slope as E/F .
where U i is a linear subspace of C m k 1 i for 1 ≤ i ≤ q, and so is a sum of stable bundles all having the same slope (which is equal to slope(E) and slope(E/F )). Let us suppose for a contradiction that E/F has a subbundle E ′ /F which is not contained in E δ(k1) /F and which is of the required form. Then we can choose k 2 > δ(k 1 ) such that E ′ ⊆ E k2 but E ′ is not contained in E k2−1 , and then the inclusion of E ′ in E k2 induces a nonzero map Since nonzero bundle maps between stable bundles of the same slope are always isomorphisms, by replacing E ′ by a suitable subbundle we can assume that E ′ /F ∼ = D i0 for some i 0 ∈ {1, ..., q}, and that we can decompose C m k 2 i 0 as C⊕C m k 2 i 0 −1 in such a way that the projection θ 0 : E ′ /F → D i0 of θ onto the corresponding component gives us a trivialisation of the extension of E k2−1 /E k1 by this component D i0 of E k2 /E k2−1 . By the definition of Σ β,l the limit p β (E) ∈ P(N l,grE ) of exp(−itβ)E as t → ∞ is semistable for the induced action of Stab(β)/T c β where Stab(β) is the stabiliser of β under the coadjoint action of R l and T c β is the complex subtorus generated by β (see [28]), and by Remark 7.3 p β (E) is represented by the sum over h ∈ {1, ..., L} and m ∈ {l 1 (h), ..., l 2 (h)} of the elements of corresponding to the extensions Let S 0 be the set of ordered pairs (i, j) with i, j ∈ {1, ..., M } such that the component of p β (E) in the weight space corresponding to the weight e i − e j for the action of the maximal torus T l of is nonzero. Since p β (E) is semistable for the action of Stab(β)/T c β , it follows that β is the closest point to 0 in the convex hull of {e i − e j : (i, j) ∈ S 0 }. We may assume that T l acts diagonally with respect to the decomposition of C m k 2 let e j0 be the weight of the action of T l on the component C of C m k 2 i 0 with respect to this decomposition. Since k 2 > δ(k 1 ) and the extension of ⊗ D i corresponding to the weight e j0 is trivial, it follows that if (i, j) ∈ S 0 then j = j 0 . Since β lies in the convex hull of {e i − e j : (i, j) ∈ S 0 } and e 1 , ..., e M are mutually orthogonal, this means that β.e j0 ≥ 0, and as j 0 ∈ ∆ k2 and k 2 > δ(k 1 ), this gives us the required contradiction.
Then whenever F is a subbundle of E with slope(F ) = slope(E) and such that E k1−1 ⊂ F ⊆ E k1 , the subbundle E δ ′ (k1)−1 is the minimal subbundle of F such that F/E δ ′ (k1)−1 is a direct sum of stable bundles all with the same slope as F . Remark 7.7. It follows from the definition of ∆ 1 , ..., ∆ t (Definition 5.5) that if j 1 ∈ ∆ k1 and j 2 ∈ ∆ k2 then β.e j1 < β.e j2 if and only if k 1 > k 2 , so we can choose k − and k + such that β.e j < 0 (respectively β.e j > 0) if and only if j ∈ ∆ k with k > k − (respectively k < k + ). Then k − = k + or k − = k + − 1, depending on whether there exists j with β.e j = 0. Propositions 7.4 and 7.6 tell us that if E is a semistable bundle representing an element of Σ β with filtration as in Proposition 6.3, then There is a converse to Propositions 7.4 and 7.6.
Proposition 7.8. Let β be as in Proposition 5.1 and let E be a semistable bundle with δ-filtration Proposition 6.3. Suppose that every subbundle F of E with the same slope as E satisfies the following two properties: .., t} such that β.e j < 0 whenever j ∈ ∆ k2 with k 2 > δ(k 1 ), then the subquotient E δ(k1)/F is the maximal subbundle of E/F which is a direct sum of stable bundles all with the same slope as E/F ; (ii) if E k1−1 ⊂ F ⊆ E k1 for some k 1 ∈ {1, ..., t} such that β.e j > 0 whenever j ∈ ∆ k2 with k 2 < δ ′ (k 1 ), then the subbundle E δ ′ (k1)−1 is the minimal subbundle of F such that F/E δ ′ (k1)−1 is a direct sum of stable bundles all with the same slope as F . Then E represents an element of the stratum Σ β,l .
Proof: Suppose for a contradiction that E does not represent an element of Σ β,l . Then (cf. [28]) after applying a change of coordinates to C m k i for 1 ≤ i ≤ q and 1 ≤ k ≤ t, we can assume that β is not equal to the closest point to 0 in the convex hull of {e i −e j : (i, j) ∈ S 0 } where S 0 is as in the proof of Proposition 8.4. Moreover the hypothesis (i) on subbundles F of E tells us that if E k1−1 ⊆ F ⊂ E k1 where k 1 ≥ δ(k − ), then the subquotient E δ(k1)/F is the maximal subbundle of E/F which is a direct sum of stable bundles all with the same slope as E/F . This implies that if (h, m) and (h, m + 1) both lie in J and φ(h, m) ≥ k − then every pair (i, j) with i ∈ ∆ h,m and j ∈ ∆ h,m+1 lies in S 0 . Similarly the hypothesis (ii) tells us that if (h, m) and (h, m − 1) both lie in J and φ(h, m) ≤ k + then every pair (i, j) with i ∈ ∆ h,m and j ∈ ∆ h,m−1 lies in S 0 . Since k − ≤ k + this means that This contradicts the fact that β is the closest point to 0 in the convex hull of {e i − e j : (i, j) ∈ S} but does not lie in the convex hull of {e i − e j : (i, j) ∈ S 0 ∩ S}, and thus completes the proof. φ(h 0 , l 1 (h 0 )) > δ(1) + 1, then E δ(1)+1 may not be the maximal subbundle of E which is a direct sum of stable bundles all with the same slope as E; there may be a subbundle of E φ(h0,l1(h0)) /E φ(h0,l1(h0)−1) which provides a trivial extension of E δ(1)+1 by a direct sum of stable bundles all with the same slope as E (see Example 8.1 below). However, even in this case a careful analysis of the proof of Proposition 4.5 reveals that it can be modified to show that the intersection Σ β ∩ S maxJH [n,m] is a locally closed complex submanifold of C ss of finite codimension for each β ∈ Γ and [n, m] ∈ I ss .
Definition 7.10. We shall call a filtration 0 ⊂ P 1 ⊂ ... ⊂ P τ ⊂ E of a semistable bundle E a pivot if each subbundle P j has the same slope as E and P 1 is the minimal subbundle of P τ such that P τ /P 1 is a direct sum of stable bundles of the same slope, while P τ /P 1 is the maximal subbundle of E/P 1 which is a direct sum of stable bundles of the same slope. Any pivot determines a filtration ⊆ ... ⊆ E of E where 0 ⊂ P +1 j /P j ⊂ P +2 j /P j ⊂ ... ⊂ E/P j is the maximal Jordan-Hölder filtration of E/P j , and 0 ⊂ ... ⊂ P −2 j ⊂ P −1 j ⊂ P j is the minimal Jordan-Hölder filtration of P j . A filtration 0 = E 0 ⊂ E 1 ⊂ ... ⊂ E t = E of this form (once repetitions have been omitted) for some pivot 0 ⊂ P 1 ⊂ ... ⊂ P τ ⊂ E will be called a pivotal filtration. It will be called a strongly pivotal filtration if every subbundle F of E such that for some m ≤ 0 has 0 ⊂ ... ⊂ P m−1 j ⊂ P m j ⊂ F as its minimal Jordan-Hölder filtration.
Note that a pivotal filtration is a δ-filtration where δ(k 1 ) is the number of k 2 ∈ {1, ..., t} for which it is not the case that E k1 = P m1 j1 and E k2 = P m2 j2 with m 1 ≥ m 2 or m 1 = m 2 − 1 and j 1 ≥ j 2 ; if the associated δ-filtration is balanced then we will call the pivotal filtration balanced.
Proof: This is an immediate consequence of Propositions 7.4 and 7.6 as in Remark 7.7.

Refinements of the Yang-Mills stratification
We thus have three refinements of the Harder-Narasimhan filtration of a holomorphic bundle E over Σ: the maximal Jordan-Hölder filtration, the minimal Jordan-Hölder filtration and the balanced δ-filtration of maximal triviality obtained by applying Proposition 6.3 to the subquotients of the Harder-Narasimhan filtration. Associated to these we have refinements of the Yang-Mills stratification of C, each of which is a stratification of C by locally closed complex submanifolds of finite codimension, and has the set C s of stable holomorphic structures on E as its open stratum. The first of these refined stratifications is the stratification defined dually using minimal Jordan-Hölder filtrations as in Remark 4.7 and Definition 4.8. A third refinement is the stratification obtained by applying the stratification {Σ γ : γ ∈ Γ} of C ss , whose indexing set was determined in §5 and whose strata were described in terms of balanced δ-filtrations in §6, to the C(n ′ , d ′ ) ss which appear inductively in the description of the Yang-Mills stratification.
Example 8.1. Recall from Remark 4.7 that the maximal Jordan-Hölder filtration of the direct sum E ⊕ F of two semistable bundles of the same slope is the direct sum of the maximal Jordan-Hölder filtrations of E and F with the shorter one extended trivially at the top, while the minimal Jordan-Hölder filtration of E ⊕ F is the direct sum of their minimal Jordan-Hölder filtrations with the shorter one extended trivially at the bottom. Suppose now that E and F have balanced δ-filtrations of maximal triviality given by (1) lie in the interval [−1/2, 1/2). To simplify the notation let us assume that l 1 (1) ≤ l 1 (2) ≤ l 2 (2) ≤ l 2 (1). If ǫ(1) > ǫ(2) then E ⊕ F has a balanced δ-filtration given by (2)+1 ⊕ F l2(2) ⊂ ... ⊂ E l2(1) ⊕ F l2(2) = E ⊕ F. If we assume that E i /E i−1 and F j /F j−1 are stable for l 1 (1) ≤ i ≤ l 2 (1) and l 1 (2) ≤ j ≤ l 2 (2), then this filtration has no proper refinements with subquotients of the same slope as E ⊕ F , so by Corollary 6.4 it is a balanced δ-filtration of E ⊕ F with maximal triviality (and in fact it is not hard to check that this is still true without the simplifying assumption). If ǫ(2) > ǫ(1) then we replace the filtration above with the balanced δ-filtration Thus we see that the the maximal Jordan-Hölder filtration, the minimal Jordan-Hölder filtration and the balanced δ-filtration of maximal triviality of a bundle E can all be different from one another, and that none of them is necessarily a refinement of the other two. Nonetheless, the concepts of maximal Jordan-Hölder filtration, minimal Jordan-Hölder filtration and balanced δ-filtration of maximal triviality on a bundle E are related by Theorem 7.11 via the notion of a pivotal filtration (see also Remark 6.8, Propositions 7.4, 7.6 and 7.8 and Remark 7.9).