Quot-scheme limit of Fubini–Study metrics and Donaldson’s functional for vector bundles

. For a holomorphic vector bundle E over a polarised Kähler manifold, we establish a direct link between the slope stability of E and the asymptotic behaviour of Donaldson’s functional, by deﬁning the Quot-scheme limit of Fubini–Study metrics. In particular, we provide an explicit estimate which proves that Donaldson’s functional is coercive on the set of Fubini–Study metrics if E is slope stable, and give a new proof of Hermitian–Einstein metrics implying slope stability.


Notation
We shall consistently write (X, L) for a polarised smooth projective variety over C of complex dimension n, where we further assume that L is very ample. We work with a fixed Kähler metric ω ∈ c 1 (L) on X defined by a Hermitian metric h L on L. We often write O X (k) for L ⊗k .
We write O X for the sheaf of rings of holomorphic functions on X, C ∞ X for the one of C-valued C ∞ -functions on X.
Throughout in this paper, coherent sheaves of O X -modules will be denoted by calligraphic letters (e.g. E). Block letters (e.g. E) will be reserved for complex C ∞ -vector bundles. This leads to an unavoidable clash of notation for a holomorphic vector bundle, but we use the following convention: both E and E will be used to denote a holomorphic vector bundle, but E will be treated as a C ∞ -object whereas E is meant to be a

Background
A large part of the materials presented in this section can be found in the references [HL10, Kob87, Siu87, LT95, MM07, PS03, Tho06, Wan02].

Slope stability
Let (X, L) be a polarised smooth projective variety of complex dimension n, and E be a holomorphic vector bundle over X of rank r.
Definition 2.1. Let F be a coherent sheaf on X. Its slope µ(F ) = µ L (F ) ∈ Q is defined as where rk(F ) ∈ N is the rank of F (assumed to be non-zero) where it is locally free (cf. [HL10,p.11]), and deg(F ) ∈ Z is defined as X c 1 (det F )c 1 (L) n−1 /(n − 1)! (where det F is a line bundle defined in terms of a locally free resolution of F , see [KM76], [Kob87,V.6] for details).
In general we should define the rank and degree in terms of the coefficients of the Hilbert polynomial [HL10, Definitions 1.2.2 and 1.2.11] and hence they are a priori rational numbers. For us, however, they are defined as integers and the above definition suffices since X is smooth.
The following stability notion was first introduced by Mumford for Riemann surfaces, and was generalised to higher dimensional varieties by Takemoto by choosing a polarisation L.
Definition 2.2 (Slope stability). A holomorphic vector bundle E is said to be slope stable (or Mumford-Takemoto stable) if for any coherent subsheaf F ⊂ E with 0 < rk(F ) < rk(E) we have µ(E) > µ(F ). E is said to be slope semistable if the same condition holds with non-strict inequality, and slope polystable if it is a direct sum of slope stable bundles with the same slope.

Definition 2.3. A holomorphic vector bundle E is said to be
(1) reducible if it can be written as a direct sum of two or more nontrivial holomorphic subbundles as E = j E j ; (2) irreducible if it is not reducible; (3) simple if End O X (E) = C.
It follows immediately that simple vector bundles are irreducible. The following fact is well known.
Definition 2.5. A coherent sheaf F is called reflexive if it is isomorphic to its reflexive hull F ∨∨ .
Definition 2.6. A subsheaf F of E is called saturated if E/F is torsion-free. Any subsheaf F of E has a minimal saturated subsheaf containing F , called the saturation of F in E, defined as the kernel of the surjection E → E/F → (E/F )/T (E/F ), where T (E/F ) is the maximal torsion subsheaf of E/F [HL10, Section 1.1].
It is well known that saturated subsheaves of a locally free sheaf E are reflexive (cf. [Kob87,Proposition 5.5.22]). An important fact is the following.
We have the following lemma, which follows from the well-known fact that, in order to test slope stability of a vector bundle E, it is sufficient to consider its saturated subsheaves [Kob87, Proposition 5.7.6].

Filtrations and Quot-scheme limit
We recall some basic definitions and results that concern Quot-schemes. We refer to the reference book of Huybrechts and Lehn [HL10] for more details. By Serre's vanishing theorem, any coherent sheaf is k-regular for k large enough and thus its regularity is well defined. Recall [HL10,Lemma 1.7.2] that if F is k-regular, then F is k -regular for all k ≥ k and F (k) is globally generated. Moreover, the natural homomorphism is surjective for l ≥ 0.
In particular, if F is k-regular, F can be written as a quotient where the vector space V := H 0 (F (k)) has dimension P F (k), the Hilbert polynomial of F . We now consider the holomorphic vector bundle E. We take k ≥ reg(E) so that E is given as a quotient ρ : V ⊗ O X (−k) → E, where V = H 0 (E(k)), dim V = P E (k) = h 0 (E(k)).
In other words, ρ : V ⊗ O X (−k) → E defines a point in the Quot-scheme Quot(Q, P E ) where we have set Q := V ⊗ O X (−k). The reader is referred to [HL10, Chapter 2] for more details on Quot-schemes.
Suppose that we are given a 1-parameter subgroup (1-PS for short) σ T : C * T → σ T ∈ SL(V ). This defines a C * -orbit in Quot(Q, P E ) defined by {ρ • σ T } T ∈C * . We can give an explicit description of the limit of this orbit as T → 0, following [HL10,Section 4.4].
Recall that giving a 1-PS σ T : C * → SL(V ) is equivalent to giving a weight space decomposition V = i∈Z V i of V , where σ T acts by σ T · v = T i v for v ∈ V i . Note that V i 0 for only finitely many i's. This defines a filtration V ≤i := j≤i V j which in turn induces a subsheaf E ≤i of E by defining a filtration {E ≤i } i∈Z of E by subsheaves. Then ρ induces surjections ρ i : V i ⊗ O X (−k) → E i (with E i := E ≤i /E ≤i−1 ), and summing up these graded pieces we get a point in Quot(Q, P E ) represented by the sheaf The importance of this sheaf is that it is, up to equivalence, the limit of the 1-PS σ T in Quot(Q, P E ); recall that two quotient sheaves ρ i : Q → E i , i = 1, 2 are equivalent if and only if there exists an isomorphism ψ : E 1 → E 2 of sheaves such that ρ 2 = ψ • ρ 1 . Writing [ρ] for the equivalence class of ρ, the above result can be stated as follows. We shall review an outline of the proof of this result, following [HL10], for the reader's convenience.
Sketch of the proof. We construct a family of sheaves •σ T for the fibre θ α over α ∈ C * and look at its behaviour under the limit α → 0; α ∈ C * corresponds to the maximal ideal (T − α) ∈ Spec(C[T ]), and the limit α → 0 corresponds to T → 0. Throughout in the proof, the tensor product is meant to be over C.
Given the decomposition V = i∈Z V i , we define a map Note that γ is an isomorphism of vector spaces which coincides with the action of σ T . In other words, the action of σ T on V defines an isomorphism where we used the same symbol γ by abuse of notation. Note also that ρ induces the following surjection defining a familyẼ of sheaves over C. Thus the desired family θ can be given byρ • γ; restricting to the central fibre at T = 0, we haveẼ while the noncentral fibre is equivalent to ρ • σ T since γ coincides with the action of σ T .
We shall later develop a "differential-geometric" version of this construction, in terms of Fubini-Study metrics. For this purpose, we shall rephrase the above lemma as follows. Suppose that we writê ≤i−1 for the limit in the Quot-scheme by σ T as in (2.2). By abuse of notation we write ρ( and recalling (2.3), we find that ρ(σ T · V i ) must be contained in the sheaf E ≤i . Using the notation above, we may thus write Suppose that we write ρ j→i (V j ) for the elements of V j that are mapped to E i . With this notation, we collect the above for all i's by using V = i∈Z V i , to get Thus, we may write the limit of ρ • σ T as T → 0 as where o(T ) stands for the terms that go to zero as T → 0, and T i is the weight of σ T on E i .
Remark 2.11. It is important to note that E i above is a priori just a coherent sheaf and in general not locally free; it may even have torsion. However, it is still possible to define a Zariski open subset of X over which all E ≤i and E i are locally free (torsion sheaves are treated as zero), recalling that X is smooth (by Lemma 2.7 or [HL10, p.11]).
Remark 2.12. The Castelnuovo-Mumford regularity of E does not play a significant role in what follows, contrary to the important role it plays in the construction of the Quot-scheme as above. In the rest of this paper k only needs to be large enough so that E(k) is globally generated, since all we need is the quotient map (2.1), but there will be little harm in consistently assuming k ≥ reg(E).

Fubini-Study metrics on vector bundles
Suppose that we fix k such that E(k) is globally generated, and hence there exists a holomorphic map Φ : X → Gr(r, H 0 (E(k)) ∨ ) to the Grassmannian such that the pullback under Φ of the universal bundle is isomorphic to E(k). More precisely, writing U for the tautological bundle over the Grassmannian Gr(r, H 0 (E(k)) ∨ ) of r-planes (rather than quotients), we have Φ * U ∨ E(k).
Recall that a positive definite Hermitian form on H 0 (E(k)) defines a Hermitian metric on (the complex vector bundle) U ∨ on Gr(r, H 0 (E(k)) ∨ ), called the Fubini-Study metric on the Grassmannian (see e.g. [MM07, Section 5.1.1]). By pulling them back by Φ, we have Hermitian metrics on E that are also called Fubini-Study metrics. We recall the details of this construction. Noting that the sections of U ∨ can be naturally identified with H 0 (E(k)), Hermitian forms on U ∨ are the elements of H 0 (E(k)) ∨ ⊗ H 0 (E(k)) ∨ , where the bar stands for the complex conjugate.
We We also fix a reference Hermitian form I S on H 0 (E(k)) to be the L 2 -Hermitian form defined by h ref ⊗ h ⊗k L and ω n /n! (see also Remark 2.14); the notation here is meant to imply that the basis S := {s 1 , . . . , s N k }, N k := dim H 0 (E(k)), for H 0 (E(k)) is a fixed I S -orthonormal basis. We may identify I S with S up to unitary equivalence, since fixing a positive Hermitian form is equivalent to fixing its (unitary equivalence class of) orthonormal basis.
We can also define the metric duals of the above. Fixing S as above, we have a basis {s * 1 , . . . , s * N k } for H 0 (E(k)) ∨ that is I S -metric dual of S, and a local frame We also write s i (x) (resp. s * i (x)) for the evaluation of s i (resp. s * i ) at x ∈ X with respect to the trivialisation given by {e 1 (x), . . . , e r (x)}.
We first define C ∞ -maps , with respect to S and {e 1 (x), . . . , e r (x)} as fixed above and hence defined globally over X. The bar, again, denotes the complex conjugate. A well-known result is the following, whose proof can be found in [Wan02,Remark 3.5] where the metric dual * is defined with respect to h ref ⊗ h ⊗k L and I S . The Hermitian metric on E(k) defined by agrees with the pullback of the Fubini-Study metric on U ∨ with respect to I S over the Grassmannian Gr(r, H 0 (E(k)) ∨ ) by the map Φ : X → Gr(r, H 0 (E(k)) ∨ ).
Remark 2.14. In the above, the L 2 -Hermitian form I S (defined by h ref ⊗ h ⊗k L and ω) in fact involves a scaling by dim H 0 (E(k)); see e.g. [MM07, Section 5.2.3].
Suppose that we now change the basis S to σ · S with σ ∈ SL(H 0 (E(k))). This defines another Hermitian metric on the universal bundle U ∨ over the Grassmannian, which we can pull back by Φ to get a Hermitian metric on E(k), exactly as before. Then, noting that (σ −1 ) * • I S • σ −1 has σ · S as its orthonormal basis, we may write the Hermitian metric which is also well known in the literature (see e.g. [PS03, Section 2]). The appearance of the inverse in σ −1 can also be seen from the fact that I S is an element of H 0 (E(k)) ∨ ⊗ H 0 (E(k)) ∨ and hence SL(H 0 (E(k))) acts by the dual action.
In what follows, we shall adopt the following convention to simplify the notation. The Hermitian metrics h ref and h L will be fixed as above, which defines an L 2 -hermitian inner product I S (with the associated orthonormal basis S). Moreover, choosing a local orthonormal frame e L (x) of L with respect to h L , we identify {e 1 (x), . . . , e r (x)} with a local orthonormal frame of E (rather than E(k)), which is associated to the Hermitian metric h S,ref on E in Lemma 2.13. Recalling Q S,ref : In what follows we shall simply write Q * Q for h S,ref and omit I S . Also, in order not to be bothered by the inverse in σ −1 , we shall consistently write We summarise the above argument to define the Fubini-Study metric on the bundle E.
agrees with the pullback of the Fubini-Study metric on the universal bundle over the Grassmannian defined by the Hermitian form I S on H 0 (E(k)). h is called the (reference) Fubini-Study metric on E defined by I S .
Moreover, given σ ∈ SL(H 0 (E(k)) ∨ ), the Hermitian metric agrees with the one defined by the Hermitian form I σ −1 ·S on H 0 (E(k)). h σ is called the Fubini-Study metric on E defined by the positive Hermitian form σ * σ = σ * • I S • σ on H 0 (E(k)).
When it is necessary to make the exponent k more explicit, we also write h k for h, for example. The construction above gives us a way of associating a Hermitian metric FS(H) on E to a positive definite Hermitian form H on H 0 (E(k)); we simply choose σ ∈ GL(H 0 (E(k)) ∨ ) so that H = σ * • I S • σ = σ * σ , and define FS(H) = h σ as above.
Recall that there is an alternative definition ( [Wan02], see also [MM07,  Writing N k = dim H 0 (E(k)) and fixing an isomorphism H 0 (E(k)) C N k , recall that the homogeneous space GL(N k , C)/U (N k ) can be identified with the set B k of all positive definitive Hermitian forms on the vector space H 0 (E(k)), by means of the isomorphism GL( These classical definitions will be important for us later, and are summarised as follows. Definition 2.16. The Bergman space B k at level k is defined to be the set of all positive definite Hermitian forms on the vector space H 0 (E(k)), which can be identified with the homogeneous space More precisely, we have that for any h ∈ H ∞ and p ∈ N there exists a sequence {h k,p } k∈N of Fubini-Study metrics h k,p ∈ H k such that h k,p → h as k → +∞ in the C p -norm. The result (2.9) follows from the so-called Tian-Yau-Zelditch expansion for the Bergman kernel, which is essentially a theorem in analysis. Several proofs have been written especially when E has rank one. For higher ranks, we refer to [Cat99,Wan05]; see also the book [MM07] and references therein. An elementary proof can be found in [BBS08]. Finally, recall that the homogeneous space B k = GL(N k , C)/U (N k ) has a distinguished Riemannian metric called the bi-invariant metric, with respect to which geodesics can be written as t → e ζt for some ζ ∈ gl(N k , C). Recalling the isomorphism GL(N k , C)/U (N k ) σ → σ * σ ∈ B k , we thus have a 1-PS in B k defined for each ζ ∈ gl(H 0 (E(k)) ∨ ) by exp(ζ * t + ζt) (note that we may choose the isomorphism H 0 (E(k)) C N k so that the standard basis for C N k is I S -orthonormal, and hence ζ * is the I S -Hermitian conjugate of ζ). This defines an important 1-PS of Fubini-Study metrics as follows.
The Bergman 1-PS's can be regarded as an analogue of geodesics in B k = GL(N k , C)/U (N k ), and play an important role later.
Remark 2.18. Throughout in what follows, we shall only consider ζ's that have rational eigenvalues, and hence any Bergman 1-PS will be meant to be rational (see e.g. Sections 3 and 5).
We also recall that, in fact, H ∞ can be regarded as an "infinite dimensional homogeneous space", with a well-defined notion of geodesics, as explained in [Kob87, Section 6.2].
. While we do not use this result in this paper, it seems interesting to point out an analogy to the case of varieties, in which test configurations define subgeodesics (see e.g. [BHJ19, Section 3]).

Q * as a C ∞ -quotient map
We give here an interpretation of Q * as a "C ∞ -analogue" of the quotient map ρ : H 0 (E(k)) ⊗ O X (−k) → E as defined in (2.1). While the argument in this section is elementary, it will provide a geometric intuition for the later arguments.
We start with some observations. Suppose that E(k) is a globally generated holomorphic vector bundle with the quotient map Forgetting the holomorphic structure, we get a map Given the above definitions, the following is immediate.
Lemma 2.21. Writing V := H 0 (E(k)), we have the following commutative diagram: Choosing Hermitian metrics for H 0 (E(k)) and E(k), we can take the "metric dual" of (2.12). More precisely, recalling I S and h ref that appeared in the previous section, we have the following C ∞ -maps by means of the metric dual. Composing with these maps, we get which looks similar to the definition of Q * that we had before. We now state the relationship between ρ * C ∞ and Q * .
Proof. Since both Q * and ρ * C ∞ can be written locally as an N k × r matrix of fibrewise full-rank which depends smoothly on x (cf. Lemma 2.13), it is immediate that there exists a fibrewise invertible smooth linear map f S,ref : E ∨ −→ E ∨ so that (2.14) holds.
Remark 2.23. f S,ref above should be regarded as performing a "fibrewise Gram-Schmidt process" with respect to I S and h ref , so that {s * 1 , . . . , s * N k } are mapped to the frame {e * 1 (x), . . . , e * r (x)} as in Lemma 2.13.
While the above lemma looks rather trivial, we observe the following consequence that can be obtained from it combined with Lemma 2.21; when we consider the Quot-scheme limitÊ of E generated by a C * -action given by ζ ∈ sl(H 0 (E(k)) ∨ ), the limit of the Bergman 1-PS Q * e ζ * t e ζt Q as t → ∞ must be in a certain sense "compatible" with this Quot-scheme limit. Indeed, this will be the main topic to be discussed in the next section.

Donaldson's functional and the Hermitian-Einstein equation
As before, (X, L) stands for a polarised smooth complex projective variety and E a holomorphic vector bundle of rank r. We write Vol L := X c 1 (L) n /n!. The following functional plays a central role in our paper.
Definition 2.24. Given two Hermitian metrics h 0 and h 1 on E, the Donaldson functional M Don : In the above, {h t } 0≤t≤1 ⊂ H ∞ is a smooth path of Hermitian metrics between h 0 to h 1 and F t denotes ( √ −1/2π) times the Chern curvature of h t , with respect to the fixed holomorphic structure of E (such that the sheaf of germs of holomorphic sections of E is identified with E). We recall some well-known properties of this functional as established in [Don85]; the reader is also referred to [Kob87, Section 6.3] for more details. First of all it is well-defined, i.e. does not depend on the path {h t } 0≤t≤1 chosen to connect h 0 and h 1 (cf. [Kob87, Lemma 6.3.6]). This easily implies the following cocycle property

Definition 2.27. A Hermitian metric h ∈ H ∞ is called a Hermitian-Einstein metric if it satisfies
where Λ ω is the contraction with respect to the Kähler metric ω on the manifold (i.e. the dual Lefschetz operator).
Critical points of M Don being Hermitian-Einstein metrics is a consequence of the following lemma.
The final important point is that M Don is convex along geodesics in H ∞ , as defined in (2.10). Recalling the important fact that H ∞ is geodesically complete, these properties together imply that a critical point of M Don is necessarily the global minimum. We summarise the above in the following proposition.

Filtrations and metric duals
A central role in this paper is played by 1-parameter subgroups in SL(H 0 (E(k)) ∨ ) and the weight decomposition of H 0 (E(k)) associated to them. This is of course related to the Bergman 1-PS's in Definition 2.17, and for the compatibility with the differential-geometric situation therein we need to assume that 1-parameter subgroups that we consider are the complexifications of the fixed unitary action; more precisely, we fix a Hermitian form I S for H 0 (E(k)) which fixes Q * as in Section 2.3, or alternatively the unitary equivalence class of its orthonormal basis S, and decree that the unitary group U (N k ) acting on H 0 (E(k)) ∨ is always with respect to I S .
It is well known that the set of all 1-parameter subgroups in a torus (in the category of algebraic groups) forms a lattice, and we can take its tensor product over Q. We cannot naively repeat this operation for SL(H 0 (E(k)) ∨ ) as it is nonabelian, but the following is still well defined and plays a very important role in the rest of the paper.
Definition 3.1. We write X Q (k) (or simply X Q ) for the set of 1-parameter subgroups σ : C * → SL(H 0 (E(k)) ∨ ) in the category of complex Lie groups with the following property: for each σ ∈ X Q (k) there exists an integer m ≥ 1 such that σ m : C * → SL(H 0 (E(k)) ∨ ) is a 1-parameter subgroup in the category of algebraic groups and that σ m is the complexification of a unitary 1-parameter subgroup U (1) → U (N k ) where the unitarity is defined with respect to I S on H 0 (E(k)).
In a more down-to-earth terminology, X Q can be defined as matrices as follows. An element σ ∈ X Q corresponds one-to-one with a Hermitian element ζ ∈ sl(H 0 (E(k)) ∨ ) with rational eigenvalues by σ t := e ζt ; it is always understood that ζ is Hermitian with respect to I S . We may change S (without affecting Q * Q) to a unitarily equivalent basis τ · S (τ ∈ U (N k )), say, so that it is compatible with the weight space decomposition Note that ν depends on ζ. Henceforth we shall assume the ordering Remark 3.2. We shall assume without loss of generality that the operator norm (i.e. the modulus of the maximum eigenvalue) of ζ is at most 1; this can be achieved by an overall constant multiple of ζ, which just results in a different scaling of the parameter t. Note also that a Bergman 1-PS {h σ t } t≥0 is rational in the sense of Definition 2.17 if and only if σ ∈ X Q .
We shall introduce an auxiliary variable so that the limit t → +∞ corresponds to T → 0, which makes it easier for us to compare the limit t → +∞ to the description of the Quot-scheme limit as a central fibre, as presented in Lemma 2.10. In this description, for each σ ∈ X Q we may write for some ζ ∈ sl(H 0 (E(k)) ∨ ) and the weight space decomposition (3.1) defines a filtration where we note that the minus sign appears because (3.3) gives the ordering of the weights We now consider the filtration that ζ ∈ sl(H 0 (E(k)) ∨ ) induces on H 0 (E(k)). It turns out that the natural dual action is not the one relevant for our purpose. Recall that Lemma 2.22 implies that we have the following diagram where the left vertical arrow stands for the metric duality isomorphism given by I S and h L , and the right vertical arrow for the one given by h ref and the C ∞ -gauge transformation (written f S,ref in Lemma 2.22). The fact that the right vertical arrow involves taking the dual of the bundle E will have nontrivial consequences concerning the sign (cf. Remark 4.5). We decree that the I S -metric dual gives a weight-preserving isomorphism between H 0 (E(k)) and H 0 (E(k)) ∨ , and hence defines a weight decomposition Recall, as in Lemma 2.10, that the Quot-scheme limit associated to the above filtration is given bŷ The components ofÊ with nontrivial rank are the objects that we need later.
The above definition, together with the invariance of the rank under the flat limit in Quot-schemes, impliesν Remark 3.4. Since E ≤−w 1 is a non-zero torsion-free subsheaf of E, we find1 = 1 in the above notation. In particular, note that We shall mainly work over an open dense subset of X where each E ≤−w i is a holomorphic subbundle of E. We make the following definition, by recalling the standard definition of the singular locus.
We define the regular locus to be the complement of in X, and denote it by X reg (ζ) or X reg .
Thus each E ≤−w i is a holomorphic subbundle of E over X reg , by recalling that a subsheaf F of a locally free sheaf E is a holomorphic subbundle of E if and only if both F and E/F are locally free. Observe that X \ X reg has codimension at least 1 in X by Lemma 2.7.
We define a C ∞ complex vector bundle E over X reg as where E ≤−w α is the C ∞ complex vector bundle over X reg defined by E ≤−w α , and the quotient is the one of the C ∞ complex vector bundles over X reg . Note that we have an isomorphism E It turns out that what we need later is the modification of the filtration 0 E ≤−w 1 ⊂ · · · ⊂ E ≤−w ν = E on a Zariski closed subset of X, given by the following lemma.
Since the above construction does not change the rank of each E ≤−w i for all i = 1, . . . , ν, and hence does not change E ≤−w i where it is locally free, we see that E ≤−w i and E ≤−w i agree on the (Zariski open) locus where they are locally free.

Renormalised Quot-scheme limit of Fubini-Study metrics
Our aim in this section is to find an expansion for h σ t := Q * σ * t σ t Q as t → +∞. Although this limit is divergent, we can find an appropriate rescaling (in terms of the constant gauge transformation) to get a 1-PS of Hermitian metrics {ĥ σ t } t≥0 which is convergent. The limitĥ, which we call the renormalised Quot-scheme limit of h σ t , is well defined only on the Zariski open subset X reg of X, and it may tend to a degenerate metric as it approaches the singular locus X \ X reg . In particular, its curvature may blow up. In spite of this problem, this limit will play an important role later. A combinatorial argument making use of the ordering w 1 > · · · > w ν (3.2) is of crucial importance for the definition ofĥ.
bundles over X reg defined by the reference metric as in (3.8). Let pr α : E ∨ − E −w α be the surjection to each factor.
(1) A C ∞ -mapQ * α : V ∨ ⊗ C ∞ X reg (k) −→ E −w α of locally free sheaves of C ∞ X reg -modules is defined by the composition of Q * and the projection pr α : We have thus a diagram (2.4) and (3.5)). Remark 3.8. In what follows, we assume a convention in which Q * j→α is defined on the whole We defineQ α and Q α→j analogously, taking the fibrewise Hermitian conjugate ofQ * α and Q * j→α , which is equivalent to taking the Hermitian metric dual with respect to the reference metrics defined in Section 2.3.
The main technical result that we establish in this section is the following.
over X reg , in terms of the decomposition ν α=1 E −w α ;Ī σ t is a fibrewise Hermitian form of rank r, whose (α, β) th block is given by which decays exponentially as t → +∞ (since w α − w j > 0 for j > α by (3.2) ).
Thus, we can "separate" the partĥ σ t of h σ t which converges as t → +∞; moreover, we have an explicit description of the rate of convergence. Since they play an important role in what follows, we introduce the following definition.
Definition 3.10. The path of Hermitian metrics {ĥ σ t } t≥0 , whereĥ σ t is a Hermitian metric defined on the regular locus X reg byĥ , is called the renormalised Bergman 1-parameter subgroup associated to σ t , where the C ∞ -isomorphism E ∨ ∼ −→ ν α=1 E −w α and the notation (3.10) are understood. We also call its limit in the renormalised Quot-scheme limit of h σ t , and each component ofĥ is written Thus, "renormalised" in the above means that we apply the constant gauge transform over X reg to h σ t , by (−1 times) the weight of ζ that acts on E = ν α=1 E −w α ; we then get a componentĥ σ t of h σ t which converges as t → +∞.
It turns out that the renormalised Quot-scheme limitĥ defines a Hermitian metric on the vector bundle E over a Zariski open subset X reg = X reg (ζ) of X. In general, however,ĥ fails to be positive definite on the Zariski closed subset X \ X reg of X, which fundamentally comes from the singularity of the sheaves E ≤−w i and E/E ≤−w i .
We now start the proof of Proposition 3.9 by proving the following result, which can be regarded as a C ∞ -analogue of the expansion (2.4).
Lemma 3.11. We have the following expansion over X reg Recalling that e ζt = T −ζ acts on V −w j as e w j t = T −w j , and hence Q * j→α • e ζt = e w j t Q * j→α , we get the claim.
Proof of Proposition 3.9. We start by rewriting Lemma 3.11 in a block-matrix notation, according to the decompositions ν α=1 E −w α and ν j=1 V −w j . We can write Q * in the following (block) row echelon form where we recall Q * j→α = 0 for j < α, and also1 = 1 (as in Remark 3.4). Thus, recalling that ζ is Hermitian and acts on V −w j as T −w j with w j ∈ Q, we compute By taking the conjugate transpose, we obtain a similar formula for e ζt Q. Since1 = 1 (cf. Remark 3.4) and T −w j = e w j t , we thus get where the (α, β) th block ofĪ σ t in the above, with α, β ∈ {1, . . . ,ν}, can be written as Recalling again T = e −t , we get the result as claimed.
Lemma 3.12. The metricĥ is strictly positive definite over X reg .
Proof. It suffices to show that the map Q * α→α : V ∨ −w α ⊗ C ∞ X reg (k) → E −w α defined in Definition 3.7 is surjective, since it implies that Q * α→α Q α→α is strictly nondegenerate.
Lemma 2.22 and the diagram (3.6) imply that the surjectivity of Q * α→α follows from that of ρ α→α . Whileĥ may degenerate as it approaches X \ X reg and hence has no uniform lower bound, it has a uniform upper bound given by Q * Q, stated as follows.

Estimates for Donaldson's functional
We establish several technical estimates, involving the renormalised Fubini-Study metrics and the renormalised Quot-scheme limit. Much of their proofs is elementary, but we shall also need to use the resolution of singularities for saturated subsheaves, as explained in [Jac14,Sib15].
Proposition 4.1. Let {h σ t } t≥0 be a path of Fubini-Study metrics emanating from h k generated by ζ ∈ sl(H 0 (E(k)) ∨ ). Then there exists a constant c 1 (h k ) > 0 depending only on h k (and k), and a constant c 2 (ζ, h k ) > 0 depending on h k and ζ (and k), such that holds for all t ≥ 0, where the graded pieces E −w i := E ≤−w i /E ≤−w i−1 are defined as in Lemma 3.6 by using the filtration 0 E ≤−w 1 ⊂ · · · ⊂ E ≤−w ν = E.
The entire section is devoted to the proof of the above proposition. As the proof is rather involved, the overall strategy is summarised in Section 4.2, which is preceded by some preliminary results in Section 4.1.

Preliminaries
We start with some preliminary results that are necessary for the proof of Proposition 4.1.
Proof. We can write h σ t = e wtĥ σ t e wt by using the renormalised Fubini-Study metricĥ σ t , and note d dt h σ t = e wt wĥ σ t + d dtĥ σ t +ĥ σ t w e wt , which immediately gives us the formula that we claimed.
Note that the connection 1-form of h σ t , which is h −1 σ t ∂h σ t , can be evaluated as e −wt (ĥ −1 σ t ∂ĥ σ t )e wt . This immediately implies that, writing F(ĥ σ t ) for ( √ −1/2π times) the curvature ofĥ σ t , we have the following pointwise equality over X reg : We thus get the following result.

Lemma 4.3. Writingĥ σ t for the renormalised Fubini-Study metric, we have
Remark 4.4. Note that by the cocycle property (2.15), the same result holds if we replace the metric h k by any Hermitian metric on E.
Remark 4.5. An important remark about the sign is in order. We considered the renormalised Bergman 1-PS as a Hermitian metric on E by means of the C ∞ -isomorphism E ∨ ∼ −→ ν α=1 E −w α which involves the metric dual. In other words, the (renormalised) Bergman 1-PS is a priori a Hermitian metric on E ∨ , which is identified with the one on E by means of the metric dual. Although h σ t is a Hermitian metric on E, when we write it as h σ t = e wtĥ σ t e wt as we do here, it is tacitly assumed that E is identified with E ∨ with the metric duality isomorphism as in the diagram (3.6).
With this metric duality isomorphism understood, we should consider h σ t = e wtĥ σ t e wt as a Hermitian metric on E ∨ , and hence we should consider the Donaldson functional for E ∨ , which is nothing but −1 times the usual definition. Thus, given h σ t ∈ H ∞ (E), we apply the metric duality isomorphism H ∞ (E) is meant to be the curvature of E, defined by composing h ∨ σ t = e wtĥ σ t e wt with the metric duality isomorphism. We shall not make distinctions between h σ t and h ∨ σ t = e wtĥ σ t e wt in what follows to avoid further complication in the notational convention, but it is important to note that the sign in Lemma 4.3 is consistent with the metric duality isomorphism that is implicit in this convention. Finally, it is perhaps worth pointing out that E admits a Hermitian-Einstein metric if and only if E ∨ does, and that subbundles of E correspond to the quotient bundles of E ∨ . Lemma 4.3 indicates an interesting role played by the renormalised Fubini-Study metricĥ σ t . Indeed, Proposition 4.1 will be proved by bounding each terms in Lemma 4.3 individually as we discuss in Section 4.2. Before we start evaluating these terms, we recall the following important theorem that will also be used in the proof of Proposition 4.1.
Theorem 4.6 (Chern-Weil formula [Sib15,Sim88]). Let S be a saturated subsheaf of a holomorphic vector bundle E endowed with a Hermitian metric h. Then we have where II(h) is the second fundamental form defined by h associated to S ⊂ E.
In the above, F(h)| S is meant to be the composition of F(h) with the orthogonal projection π : E −→ S defined by h. We shall use the following observation concerning this theorem. Suppose that S is a subsheaf of E that is not necessarily saturated, and let S be its saturation in E. Since S and S agree on a Zariski open subset in X, we find Thus, we get where the second fundamental form that appears on the right hand side is defined with respect to h and the saturation S of S.

Strategy of the proof of Proposition 4.1
We decomposed M Don (h σ t , h k ) in two terms as in Lemma 4.3, and we bound each term individually to get Proposition 4.1. The first term can be evaluated as follows.
Proposition 4.7. There exists a constant c 3 (h k ) > 0 which depends on the reference metric h k (and k) such that uniformly for all t ≥ 0 and ζ ∈ sl Lemma 3.6 ).
The proof of this proposition will be given in Section 4.3. Together with the following result that we prove in Section 4.4, which bounds the second term in Lemma 4.3, we get the desired proof of Proposition 4.1. for all t ≥ 0.
The rest of the section is devoted to the proof of the above results.

Proof of Proposition 4.7
The proof of Proposition 4.7 relies on the resolution of singularities, which we recall below. Recall that we have a sequence of saturated subsheaves E ≤−w1 ⊂ · · · ⊂ E ≤−wν of E. We apply the regularisation (or the resolution of singularities) for the saturated sheaves, as in Jacob [Jac14] and Sibley [Sib15], to reduce to the case of holomorphic subbundles. We closely follow the argument by Jacob [Jac14, Sections 3 and 4]. After a finite sequence of blowups π :X → X, we have a holomorphic subbundleẼ ≤−w α of π * E (i.e.Ẽ ≤−w α and π * E/Ẽ ≤−w α are both locally free) overX, such that π * Ẽ ≤−w α = E ≤−w α (see [Jac14, Proposition 3] and [Sib15, Proposition 4.3]). We thus get a filtration of π * E by holomorphic subbundles overX as 0 Ẽ ≤−w1 ⊂ · · · ⊂Ẽ ≤−wν = π * E, and hence a C ∞ -isomorphism of C ∞ -complex vector bundles overX. The point of this isomorphism is that π * E admits a constant gauge transformation by e wt in the notation of (3.10), globally overX.
We now consider the pullback π * h σ t of the Hermitian metric h σ t . This defines a metric oñ as in [Jac14, Section 2]. As in [Jac14, Proposition 4], we construct Hermitian metricsh σ t |Ẽ ≤−w α onẼ ≤−w α and h σ t | π * E/Ẽ ≤−w α on π * E/Ẽ ≤−w α , by removing the factors that vanish (or blow up) on the exceptional divisor of π. Now we consider the constant gauge transformation by e wt on π * E, and hence on π * h σ t , given by the C ∞ -isomorphism (4.2). Since this gauge transformation is an overall constant scaling on each summand in (4.2), the Hermitian metrics are both well defined globally overX. We now repeat the same operation for the renormalised metric h ren σ t :=ĥ σ t = e −wt h σ t e −wt , which is only well defined on X reg , to get Hermitian metrics defined over π −1 (X reg ), which is Zariski open inX; following the notation as aboveh ren σ t perhaps should be written ash σ t , but we prefer not to use it as it is harder to read. Although these metrics are defined only over π −1 (X reg ), they clearly agree with e −wth σ t e −wt |Ẽ ≤−w α or e −wth σ t e −wt | π * E/Ẽ ≤−w α over π −1 (X reg ). We summarise the conclusion of the above discussion in the following lemma. Lemma 4.9. With the notation as above, the Hermitian metrich ren σ t extends as a smooth Hermitian metric on π * E, globally overX.
We prove in Lemma 4.11 that an analogue of the Chern-Weil formula applies to the above metrich ren σ t . We start by recalling that the L 2 -norm of the second fundamental form remains unchanged under the regularisation process.
Lemma 4.10 ( Jacob [Jac14]). Suppose that we write II α (h ren σ t ) for the second fundamental form defined by the above metrich ren σ t on π * E associated toẼ ≤−w α ⊂ π * E. Then we have Sketch of the proof. We only provide a sketch of the proof as the details can be found in [Jac14, Proofs of Propositions 1 and 4, Lemma 1]. Let us write p (respectivelyp) for the orthogonal projection to E ≤−w α with respect toĥ σ t (respectively tõ E ≤−w α and with respect toh ren σ t ). Arguing as in [Jac14, Proof of Proposition 1, Equation (2.6)], we find by expressing the second fundamental form in terms of the curvatures of E and E ≤−w α , If z defines local holomorphic coordinates and {z = 0} corresponds locally to the exceptional divisor associated to a blow up of the regularisation process, one can express locally the degeneracy of the involved metrics on the bundles. This shows two facts. Firstly, one can identify the two projections p andp via the pull-back map. Secondly, one can express the curvature of the metrics and obtain the following equality in the sense of currents for some non-negative integers a i . Note that the term X i a i √ −1∂∂ log |z| 2 ∧ π * ω n−1 vanishes. Consequently we get from (4.4), By Lemma 4.9h ren σ t extends as a smooth Hermitian metric globally overX which is compact, we get the desired upper bound for the inequality in (4.3).
Lemma 4.11. With the notation as above, we have The point of the above equation is that the analogue of the Chern-Weil formula (4.1) holds for the metriĉ h σ t that is well defined a priori only over X reg .
Proof. This is a consequence of Lemma 4.9: we apply the Chern-Weil formula (Theorem 4.6) forh ren σ t on π * E, which is globally defined overX, and get the claimed result by applying also Lemma 4.10 and [Jac14, Lemma 2] (which states that the degree remains unchanged under the regularisation, by using a degenerate Kähler metric π * ω onX).
Proof of Proposition 4.7. Recall that we have the C ∞ -isomorphism E ν α=1 E −w α over X reg . We compose this with another C ∞ -isomorphism so that the E −w α are pairwise orthogonal with respect toĥ σ t . Thus, Recall now that each sheaf E ≤−w α agrees with its saturation E ≤−w α over X reg , as they are both holomorphic subbundles over X reg . Lemma 4.11 implies where II α (ĥ σ t ) is the second fundamental form associated to the saturated subsheaf E ≤−w α defined with respect toĥ σ t . Thus, In Lemma 4.12 that follows, we prove the existence of a positive constant c 3 (h k ) which depends only on the reference metric h k (and k) such that uniformly for all t > 0 and ζ ∈ sl(H 0 (E(k)) ∨ ). Similarly, we find by noting that saturation does not change the rank. Finally, since ν α=1 E −w α is assumed to be orthogonal with respect toĥ σ t , by our choice of C ∞isomorphism, w must commute withĥ σ t , i.e. [w,ĥ σ t ] = 0. Recalling (4.5), we thus get Recall from Lemma 3.6 that each E −w i is torsion-free or zero for each i = 1, . . . , ν, and that rk(E −w i ) > 0 if and only if i ∈ {1, . . . ,ν}. This means that E −w i is simply zero for all i {1, . . . ,ν}, and hence deg( as required. Thus, granted (4.5) that we prove below, we complete the proof of Proposition 4.7.
Lemma 4.12. Using the notation above, there exists a constant c 3 (h k ) > 0 which depends only the reference metric h k (and k) such that uniformly for all t ≥ 0 and ζ ∈ sl(H 0 (E(k)) ∨ ) (with ζ op ≤ 1, as in Remark 3.2 ).
Step 1: the case when all the E ≤−w i 's are holomorphic subbundles of E. In this case, X reg = X, and it is easy to see that |II α−1 (ĥ σ t )| 2 decays exponentially as t → +∞ with decay rate at least e −|w α −w α−1 |t , by Proposition 3.9. Recalling Lemmas 3.12 and 3.13, this immediately establishes (4.6).
Step 2: the case when all the E <−w i 's are saturated, i.e. E ≤−w i = E ≤−w i . We apply the regularisation (or the resolution of singularities) for the saturated sheaves to reduce to the case of holomorphic subbundles, just as we did at the beginning of this section. After a finite sequence of blowups π :X → X, we have a locally free subsheafẼ ≤−w α of π * E with a locally free quotient π * E/Ẽ ≤−w α overX, such that π * Ẽ≤−w α = E ≤−w α (see [Jac14, Proposition 3] and [Sib15, Proposition 4.3]). Now, the pullback π * h ren σ t of the renormalised metric h ren σ t :=ĥ σ t can be regarded as defining the induced metric oñ (as C ∞ -vector bundles), as in [Jac14, Section 2]. As we did at the beginning of this section and as in [Jac14, Proposition 4], we construct Hermitian metrics by removing the factors that vanish (or blow up) on the exceptional divisor of π.
Recalling Lemma 4.10, we find the following: during the regularisation process for saturated sheaves, we obtain the metrich ren without altering the L 2 -norm of the second fundamental form. Note that we use a degenerate Kähler metric onX in (4.3) but this does not matter for our main purpose. We now consider the limit where t tends to +∞. Since the renormalised metricĥ as in Definition 3.10 may be degenerate on X \ X reg ,h ren may develop further degeneracy on the exceptional divisor of π, as t → +∞. Recall also that Lemma 2.22 implies that the degeneracy ofĥ is exactly of the form as described in [Jac14, equation (3.8), see also Proposition 4], i.e. the metric can be written locally as a smooth Hermitian matrix multiplied by a matrix-valued holomorphic function which can have a zero. Now after pulling back by π, we may further remove these factors that vanish on the exceptional divisors, to get a well-defined metrich, say, on π * E Ẽ ≤−w α ⊕ π * E/Ẽ ≤−w α . The L 2 -norm of the second fundamental form of the metrich agrees with the one defined byĥ by the same reasoning as in the proof of Lemma 4.10. Butĥ has no off-diagonal block (cf. Proposition 3.9 and Definition 3.10). Thus, the L 2 -norm of the second fundamental form of the metrich is zero.
We thus get Inequality (4.7) by recalling that the decay (as t → +∞) of |II α−1 (ĥ σ t )| at each x ∈ X reg is exponential, as we saw in Step 1. Hence, summarising the argument above, after a finite sequence of blowups π :X → X, we may reduce to the case in Step 1. This establishes (4.6) when all E ≤−w i 's are all saturated.
Step 3: we now consider the general case when the E ≤−w i 's are (not necessarily saturated) subsheaves of E. We first apply a sequence of blowups π so that ν α=1 Sing(E ≤−w α ) ∪ Sing(E/E ≤−w α ), whereĥ σ t may be degenerate as t → +∞, is contained in the union of normal crossing divisors (see [Sib15,Theorem 4.4]). As in Step 2, we may remove the factors ofĥ that vanish on the exceptional divisors of π, without changing the L 2 -norm of the second fundamental form at t = +∞ (as in the proof of Lemma 4.10, which holds for an arbitrary blowup). Note that we do not claim that, by such a sequence of blowups, the subsheaves E ≤−w α can be pulled back to holomorphic subbundles; we only remove the factors that vanish on Sing(E ≤−w α ) ∪ Sing(E/E ≤−w α ) that makesĥ σ t degenerate as t → +∞. We then apply the argument in Step 2 to the saturation E ≤−w α of E ≤−w α , by composing π with a further sequence of blowups, where we note that E ≤−w α and its saturation E ≤−w α differ only on a Zariski closed subset and hence this does not affect the value of the integral (4.7), and that the second fundamental form in Step 2 is defined with respect to the saturation E ≤−w α . Thus, we can reduce to the case in Step 2, to establish (4.6) in general.
We finally note that the map defined by is continuous, where we recall that |II α−1 (ĥ σ t )| 2 decays exponentially as t → +∞ at each p ∈ X, with decay rate at least (and in fact faster than) e −|w α −w α−1 |t by Proposition 3.9. (Note in particular that, when we consider the limit of w α−1 tending to w α , the integral over t of (w α − w α−1 )e −|w α −w α−1 |t remains bounded.) By recalling ζ op ≤ 1 (cf. Remark 3.2), we obtain that the bound that does not depend on ζ, as claimed.

Proof of Proposition 4.8
We now prove Proposition 4.8, which establishes the required bound for the second term in Lemma 4.3. We start with the lower bound.
Remark 4.14. The sign convention concerning the metric duality isomorphism, as mentioned in Remark 4.5, still applies, soĥ σ t is meant to be a Hermitian metric on E ∨ .
Proof. We first consider the following general situation. Suppose that we have a geodesic segment γ(s) := e sv (0 ≤ s ≤ 1), with v := log hh −1 0 , which connects h 0 and h, where h 0 is a fixed reference metric. The convexity of the Donaldson functional (Proposition 2.29) implies so that v −v has average 0. Sincev is a constant multiple of the identity, we have Thus, by using Cauchy-Schwarz, we have Defining a constant by recalling (4.8).
Let us consider {ĥ σ t } t≥0 , a renormalised Bergman 1-PS associated to ζ ∈ sl(H 0 (E(k)) ∨ ) emanating from h k . We apply the above argument to h =ĥ σ t and h 0 = h k , in which we replace X by X reg . We note that the convexity of the Donaldson functional (Proposition 2.29) holds for the geodesic segment γ t (s) := e sv t connecting h k andĥ σ t , with v t := logĥ σ t h −1 k which is only well defined on X reg , since the only nontrivial part in proving the convexity is the integration by parts (see [Don85, Proof of Proposition 8]) which we are allowed to do sinceĥ σ t ,ĥ −1 σ t , and F(ĥ σ t ) are all bounded over X reg for any fixed t (we can also prove the convexity by using the regularisation, as we do later in the proof of Lemma 4.15).
Thus (4.10) implies that we have To be more precise, the curvature F(ĥ σ t ) should be defined with respect toĥ σ t composed with the metric duality isomorphism (as in Remark 4.5), but in any case this does not affect the above argument. We prove that v t −v t L 2 can be bounded by a constant depending only on ζ ∈ sl(H 0 (E(k)) ∨ ) and k, irrespectively of t.
We thus evaluate the integral of log(λ α,1 (x) + e −c α t λ α,2 (x)) 2 over the subset of X reg where λ α,1 (x) ≤ 1/2 and 0 < λ α,2 (x) ≤ 1. With this assumption we have log(λ α,1 (x) + e −c α t λ α,2 (x)) 2 ≤ 2 log(λ α,1 (x)) 2 for all t ≥ 0. We may assume that the singular locus is contained in a finite union of normal crossing divisors D := ∪ j D j , by e.g. pulling back to the resolution of X \ X reg (see [Sib15,Theorem 4.4]). We first consider the case where we have a neighbourhood U x of x such that U x intersects with D only at the smooth locus of D. Then, writing z for the local holomorphic coordinate which defines D as z = 0 in such a neighbourhood, we find that λ α,1 (x) tends to zero along the singular locus at a polynomial order in |z|, by recalling Lemma 2.22 and that the quotient map ρ (defined in (2.1)) is algebraic. Thus we get, for all 1 ≤ α ≤ r, where r := |z|, m α ∈ N, and c is some constant. The case when U x contains a (normal crossing) singularity of D can be treated by the same argument. Thus, the conclusion is that there exists a constant c 4 (ζ, h k ) > 0 such that Moreover, the above argument shows that the integrand tr(v 2 t ) ω n n! extends continuously to the whole of X. Thus, henceforth in the proof we may replace X reg by X.
We now prove that the constant c 4 (ζ, h k ) can be bounded uniformly in ζ. To simplify the notation, we write v(ζt) for v t = logĥ σ t h −1 k , by noting that ζ always appears in the form ζt. Suppose by contradiction that there exists a sequence {ζ j } j∈N ⊂ sl(H 0 (E(k)) ∨ ), ||ζ j || op ≤ 1 (cf. Remark 3.2) such that sup t X tr(v(ζ j t) 2 ) ω n n! → +∞ as j → +∞. We may assume that there exists a sequence {t j } j∈N ⊂ R ≥0 such that (4.15) X tr(v(ζ j t j ) 2 ) ω n n! → +∞ as j → +∞, and also assume that {ζ j } j contains a subsequence which converges to ζ ∞ ∈ sl(H 0 (E(k)) ∨ ), by ||ζ j || op ≤ 1 (cf. Remark 3.2). By arguing as in the proof of (4.13), we find If {t j } j is bounded in R ≥0 , we see that (4.15) contradicts (4.16) since is continuous for each fixed t. Thus, by taking a further subsequence in j, we may assume that {t j } j monotonically increases to +∞. Recalling Definition 3.10, the Lebesgue integration theorem implies (4.17) lim j→∞ X tr(v(ζ j t j ) 2 ) ω n n! = X tr (logĥh −1 k ) 2 ω n n! where we wroteĥ := lim j→+∞ e −w j t j h σ j,t j e −w j t j , w j being the eigenvalues of ζ j (as in (3.10)) and σ j,t j := exp(ζ j t j ). However, the right hand side of (4.17) is finite by the same argument as in the proof of (4.13) (the limit depends on the choice of the subsequence in {ζ j } j and {t j } j , but the point is that it is finite for any choice of subsequence). This is a contradiction, and hence there exists a constant c 4 (h k ) > 0 that depends only on k and the reference metric h k , and not on ζ, such that as claimed. Combining (4.10) and (4.14), we establish all the claims stated in the proposition.
We prove the following upper bound, which establishes Proposition 4.8.
Proof. We argue in three steps, as in the proof of Lemma 4.12: when each E <w i is locally free, saturated, and the general case.
Step 1: when each E ≤−w i is locally free, and hence X reg = X,ĥ σ t converges exponentially to the product metric on the C ∞ -vector bundle ν α=1 E −w α over X. The claimed estimate immediately follows from Lemmas 3.12 and 3.13.
Step 2: when each E ≤−w i is saturated, we again argue as in [Jac14, Sections 3 and 4], as we did in Lemma 4.12. We pull back by a finite sequence of blowups, and remove the factors that vanish (or blow up) on the exceptional divisors of π (as in the proof of Lemma 4.12) without altering the value of the Donaldson functional [Jac14, Proof of Propositions 1 and 5]. We thus reduce to the case in Step 1.
Step 3: the general case, where E ≤−w i 's are not necessarily saturated, can be treated by a further sequence of blowups, as in the proof of Lemma 4.12. By removing the factors that vanish on the exceptional divisors of π, we reduce to the case in Step 2.
in place of Lemma 4.12, for all t ≥ 0 and ζ ∈ sl(H 0 (E(k)) ∨ ). In the paper [HK19], we prove that the above uniform bounds lead to an alternative proof of the theorem of Donaldson [Don83,Don85,Don87].

The non-Archimedean Donaldson functional
Recalling Definition 3.1, we find that for any σ ∈ X Q there exists m ∈ N such that σ m is a 1-parameter subgroup in the category of algebraic groups, which in particular has integral weights. This allows us to clear up the denominators in the materials in Section 3, so that the filtration 0 E ≤−w 1 ⊂ · · · ⊂ E ≤−w ν = E in (3.7) is graded by integers; more precisely, writing w 1 , . . . , w ν for the weights of σ and defining mw i =:w i ∈ Z, where m ∈ N is the least integer such that σ m is algebraic, we define the filtration in an obvious way.
Recall also that the 1-PS's X Q in Definition 3.1 correspond bijectively with Hermitian elements ζ ∈ sl(H 0 (E(k)) ∨ ) with rational eigenvalues, which provides a more down-to-earth description of the above. In particular, the integer m that appeared above is exactly the following.
Given the integrally graded filtration (5.1), we also have the filtration 0 E ≤−w 1 ⊂ · · · ⊂ E ≤−w ν = E of E by saturated subsheaves as in Lemma 3.6. With this convention understood, we prove the following result.
Proof. We simply write down Propositions 4.1 for the specific filtration defined by ζ F . Direct computation immediately yields the claimed results.
Finally, we end this section by proving the following rather technical result which we need later. Recall that the distance dist(h 1 , h 2 ) between two Hermitian metrics h 1 , h 2 ∈ H can be defined by using the geodesic between them, as in Section 2.3. Lemma 6.3. Given a Bergman 1-PS {h σ t } t≥0 emanating from h k , generated by ζ F ∈ sl(H 0 (E(k))) associated to a saturated subsheaf F ⊂ E, we have dist(h σ t , h k ) → +∞ as t → +∞.
Proof. We can explicitly write down the geodesic {γ s (t)} 0≤s≤1 connecting h k and h σ t as e sv(σ t ) h k where v(σ t ) := log(h σ t h −1 k ). We compute its Riemannian length as |∂ s γ s (t)| = X tr(v(σ t ) · v(σ t )) 1/2 ω n n! , where we recall that γ s (t) being a geodesic is equivalent to ∂ s γ s (t) being independent of s. We show that this length segment tends to +∞ as t → +∞.

Summary of the main results
We recall the following terminology (cf. [BHJ19, Definition 5.1]).
Definition 7.1. Let N be a function N : SL(N , C) → R that is continuous with respect to the topology on SL(N , C) defined by a submultiplicative matrix norm || · || (e.g. the operator norm).
(1) N is said to be coercive if there exist two constants a, b ∈ R with a > 0 such that N (g) ≥ a log g − b holds for any g ∈ SL(N , C); (2) N is said to have log norm singularities if it can be written as N (g) = a log g + O(1) for some constant a ∈ R, where O(1) stands for the terms that remain bounded over SL(N , C).
We shall also say that N is coercive (resp. has log norm singularities) along any rational Bergman 1-PS if in the above notation it satisfies N (σ t ) ≥ a log σ t − b (resp. N (σ t ) = a log σ t + O(1)) for any σ ∈ X Q .
Observe that by means of Bergman 1-PS's, we can naturally define M Don as a map from SL(H 0 (E(k)) ∨ ) to R as with the fixed reference metric h k , while we are mostly interested in the case when the domain of M Don (−, h k ) is restricted to a Bergman 1-PS {σ t } t≥0 . Important functionals concerning the constant scalar curvature Kähler metrics have log norm singularities, as proved by Paul [Pau12]; see also [BHJ19,Section 5]. Note also that the property of having log norm singularities is independent of the choice of a matrix norm, up to changing the constants c 0 and c 1 .
A particularly important role is played by coercive functions; they are in particular bounded from below. If N is coercive then it is proper, i.e. the preimage of any compact set is compact, which is equivalent in this setting to saying that N (g) tends to +∞ as g tends to +∞.
We now summarise what we have established so far. Suppose that we take k 0 ∈ N to be large enough for E(k) to be globally generated, and also that we have a sequence {h k } k≥k 0 ⊂ H ∞ , h k ∈ H k , converging to the reference metric h ref ∈ H ∞ in C p for p ≥ 2, afforded by (2.9). Write {h σ t } t≥0 for the Bergman 1-PS emanating from h k associated to the 1-PS σ ∈ X Q . In what follows we assume k ≥ k 0 . this does not matter), and the geodesic distance dist(h σ t , h min ) between them. Then, from Lemma 6.3, we have dist(h σ t , h min ) → +∞ as t → +∞; recall that the sequence {h k } k∈N ⊂ H ∞ , h k ∈ H k , converges to the reference metric h min in C p for p ≥ 2, as in Section 7. Thus, using the strict geodesic convexity of M Don we obtain M Don (h σ t ) → +∞, as t → +∞. But Theorem 7.3 implies that this contradicts the assumption M NA (σ F ) = 0. Consequently, we get M NA (σ F ) > 0, which implies (by recalling Proposition 6.2 or 7.4) µ(E) > µ(F ), and hence the slope stability of E by Lemma 2.8.
Finally, the case when E is reducible can be treated similarly. By applying the previous argument to each irreducible component of E, we find that E is a direct sum of slope stable bundles. The Hermitian-Einstein equation implies that the slopes of these components are equal, and hence E is slope polystable.