Hermitian-Yang-Mills connections on some complete non-compact K\"ahler manifolds

We give an algebraic criterion for the existence of projectively Hermitian-Yang-Mills metrics on a holomorphic vector bundle $E$ over some complete non-compact K\"ahler manifolds $(X,\omega)$, where $X$ is the complement of a divisor in a compact K\"ahler manifold and we impose some conditions on the cohomology class and the asymptotic behaviour of the K\"ahler form $\omega$. We introduce the notion of stability with respect to a pair of $(1,1)$-classes which generalizes the standard slope stability. We prove that this new stability condition is both sufficient and necessary for the existence of projectively Hermitian-Yang-Mills metrics in our setting.


Introduction
The celebrated Donaldson-Uhlenbeck-Yau theorem [8,28] says that on a compact Kähler manifold (X, ω), an irreducible holomorphic vector bundle E admits a Hermitian-Yang-Mills (HYM) metric if and only if it is ω-stable. After this pioneering work, there have been several results aiming to generalize this to non-compact Kähler manifolds [25,1,22,21,15,19]. A key issue is to understand what role stability plays on the existence of projectively Hermitian-Yang-Mills (PHYM) metrics. An interesting special case in the non-compact setting is when (X, E) both can be compactified, i.e. X is the complement of a divisor in a compact Kähler manifold X and E is the restriction of a holomorphic vector bundle E on X, and when the Kähler metric has a known asymptotic behaviour. Under these assumptions, one wants to build a relation between the existence of PHYM metrics on E and some algebraic data on E. In this paper, we prove a result in this setting.
Let X be an n-dimensional (n ≥ 2) compact Kähler manifold, D be a smooth divisor and X = X\D denote the complement of D in X. Let E be a holomorphic vector bundle on X, which we always assume to be irreducible unless otherwise mentioned. Let E, E| D denote its restriction to X and D respectively. Suppose the normal bundle N D of D in X is ample. On X we consider complete Kähler metrics ω = ω 0 + dd c ϕ satisfying Assumption 1 (see Section 2 for a precise definition). Roughly speaking, we assume ω 0 is a smooth closed (1,1)-form on X vanishing when restricted to D, ϕ is a smooth function on X, and ω is asymptotic to some model Kähler metrics given explicitly on the punctured disc bundle of N D . Typical examples satisfying these assumptions are Calabi-Yau metrics on the complement of an anticanonical divisor of a Fano manifold and its generalizations [27,14,13] (see Section 6.2 for a sketch).
To state our theorem, we need two ingredients: the existence of a good initial hermitian metric on E and the definition for stability with respect to a pair of classes. The following lemma is proved in Section 4. Lemma 1.1. If E| D is c 1 (N D )-polystable, then there is a hermitian metric H 0 on E satisfying: (1). there is a hermitian metric H 0 on E and a function f ∈ C ∞ (X) such that H 0 = e f H 0 , Date: June 29, 2022.
1 (2). |Λ ω F H 0 | = O(r −N 0 ), where r denotes the distance function to a fixed point induced by the metric w and N 0 is the number in Assumption 1-(3).
We call H 0 conformal to a smooth extendable metric if it satisfies the first condition in Lemma 1.1. A key feature we use in this paper is that the induced metic on End(E) is conformally invariant with respect to metrics on E. Therefore the two hermitian metrics H 0 and H 0 induce the same metric on End(E) and this is the norm used in Lemma 1.1- (2). Then naturally (following [25]) one wants to find a PHYM metric in the following set Here we use √ −1su(E, H 0 ) to denote the subbundle of End(E) consisting of the trace-free and self-adjoint endomorphisms with respect to H 0 . Though H 0 in general is not unique, we will show that if we fix the induced metric on det E, then the set P H 0 is uniquely determined as long as H 0 satisfies conditions in Lemma 1.1 (see Proposition 4.7).
Next we define stability with respect to a pair of (1,1)-classes, which generalizes the standard slope stability defined for Kähler classes in [16,Chapter 5]. In the following, we use µ α (S) to denote the slope of a torsion-free coherent sheaf with respect to a class α ∈ H 1,1 (M) on a compact Kähler manifold M (see Section 3.2 for a more detailed discussion), i.e. µ α (S) := 1 rank(S)ˆM c 1 (det S) ∧ α n−1 .
The main result of this paper is By the definition of (α, β)-stability in Definition 1.2, we have the following consequence. Now let us give a brief outline for the proof of Theorem 1.3. For the "if" direction, we follow the argument in [25,19] by solving Dirichlet problems on a sequence of domains exhausting X. A key issue here is to prove a uniform C 0 -estimate. For this we rely on a weighted Sobolev inequality in [27,Proposition 2.1] and Lemma 5.4 which builds a relation between weakly holomorphic projection maps over X and coherent subsheaves over X. For the "only if" direction, we use integration by parts to show that the curvature form on E can be used to compute the degree of E with respect to [ω 0 ] (see Lemma 5.3). For both directions, the asymptotic behaviour of the Kähler metric ω plays an essential role.
Then let us compare Theorem 1.3 with some results existing in the literature. In [25] and [19], by assuming some conditions on the base Kähler manifold (X, ω) and an initial hermitian metric on E, it was proved that for an irreducible vector bundle E the existence of a PHYM metric is equivalent to a stability condition called analytic stability. In our case, since we assume that E has a compactification E on X, the existence of good initial metrics is guaranteed by the polystablity assumption of E| D . Moreover the stability we used in Theorem 1.3 is for E which is purely algebraic, i.e. independent of choice of metrics. In [1], for asymptotically conical Kähler metrics on X, it was proved that if E| D is c 1 (N D )polystable, then there exists PHYM metrics on E. No extra stability condition is needed in this case. Therefore the necessity of stability conditions depends on the geometry of (X, ω) at infinity. Another typical example for such a phenomenon is the problem for the existence of bounded solutions of the Poisson equation on noncompact manifolds. See Section 6.1 for a brief discussion.
The paper is organized as follows. In Section 2, we discuss the assumptions on the Kähler manifold (X, ω) and prove a weighted mean value inequality for nonnegative almost subharmonic functions. In Section 3, we give a brief review of some standard results for hermitian holomorphic vector bundles and give a detailed discussion on (α, β)-stability used in Theorem 1.3. In Section 4, Lemma 1.1 is proved and we also show that the assumption in Lemma 1.1 is necessary. In Section 5, we prove Theorem 1.3 and give an example which does not satisfy the stability assumption. In Section 6.1, we discuss some other results on the existence of PHYM metrics. In Section 6.2, we discuss some Calabi-Yau metrics satisfying Assumption 1. In Section 6.3, we prove a counterpart of Theorem 1.3 in a different setting where X is a compact Kähler surface and c 1 (N D ) is trivial. In Section 6.4, we discuss some problems for further study.
Notations and conventions.
• B r (p) denotes the geodesic ball centered at p with radius r and if the basepoint p is clear from the context, we will just write it as B r . • In this paper, we identify a holomorphic vector bundle with the sheaf formed by its holomorphic sections. • When we integrate on a Riemannian manifold (M, g), typically we will omit the volume element dV g . • Let (M, ω) be a Kähler manifold and (E, H) be a hermitian holomorphic vector bundle over M. We use C ∞ (M, E) to denote smooth sections of E; W k,p (M, E; ω, H) (respectively W k,p loc (M, E; ω, H)) to denote sections of E which are W k,p (respectively W k,p loc ) with respect to the metric ω and H. If bundles or metrics are clear from the text, we will omit them.

On the asymptotic behaviour of ω
As mentioned in the introduction, the asymptotic behaviour of the Kähler metric on the base manifold is crucial to make the argument in this paper work. In this section we will discuss these assumptions. Let X be an n-dimensional (n ≥ 2) compact Kähler manifold, D be a smooth divisor and X = X\D denote the complement of D in X. Let L D denote the holomorphic line bundle determined by D.
From now on, we assume the normal bundle of D, i.e. N D = L D | D is ample unless otherwise mentioned. Then we know that c 1 (D) is nef and big. We fix a hermitian metric h D on N D such that We are mainly interested in the region where |ξ| h D is small, which will be viewed as a model of X at infinity. Then we have a well-defined positive smooth function on C where using the obvious projection map p : C → D, we view ω D as a form on C. Then for every smooth function F : (0, ∞) −→ R with F ′ > 0 and F ′′ > 0, defines a Kähler form on C. Let g F denote the corresponding Riemannian metric and r F denote the induced distance function to a fixed point p.
We need a diffeomorphism to identify a neighborhood of D in N D with a neighborhood of D in X. For this, we use the following definition introduced by Conlon and Hein in [6,Definition 4.5].
p . Now we can state the assumptions for the Kähler metric ω on X. We consider a special class of potentials: H := F (t) : F (t) = At a for some constant A > 0 and a ∈ 1, n n − 1 (2.4) Assumption 1. Let ω be a Kähler form on X and g be the corresponding Riemannian metric. We assume that (1) the sectional curvature of g is bounded, (2) ω can be written as ω 0 + √ −1∂∂ϕ, where ϕ a smooth function on X and ω 0 is a smooth (1, 1)-form on X with ω 0 | D = 0, (3) there exists an exponential-type map Φ from a neighborhood of D in N D to a neighborhood of D in X and a potential F ∈ H such that There are (lots of) other potentials F besides those given in (2.4) making the argument in this paper work, but for simplicity of the statement and some computations we only consider potentials in H. The order in (2.5) is not optimal either and again we just choose the number 8 for a neat statement. From now on, unless otherwise mentioned, N 0 denote the number in (2.5).
Here are the main properties we will use for Kähler metrics defined by the potentials in H. For simplicity of notation, we omit the subscript for the dependence on F . Proposition 2.3. For the Kähler metric defined by a potential F = At a ∈ H, we have (1) The metric is complete as |ξ| h D → 0, if θ is a smooth form on D with θ| D = 0, then |θ| g = O(e −δt ) for some δ > 0.
Conditions (1)-(3) follow directly from (2.3) and (2.4). Condition (4) can be proved directly by doing computation in local coordinates on D as in [14,Section 3]. For completeness and later reference, we include some details.
Proof of (4): We choose local holomorphic coordinates z = {z i } n−1 i=1 on the smooth divisor D and fix a local holomorphic trivialization e 0 of N D with |e 0 | h D = e −ψ , where ψ is a smooth function on D satisfying √ −1∂∂ψ = ω D . Then we get local holomorphic coordinates {z 1 , · · · , z n−1 , w} on C by writing a point ξ = we 0 (z). Then in these coordinates we can write (2.3) as Then it is easy to check the following estimates: Then (4) follows directly from (2.7).
Remark 2.4. Actually, from the proof of Proposition 2.3-(4), we can give an effective lower bound for δ. For example, for 2-forms, δ can be chosen to be any positive number sufficiently close to (and less than) 1/2. However δ > 0 is sufficient for our later use.
Remark 2.5. Although not needed in this paper, we mention that following the computation in [27,Section 4] or [3, Section 3], we can show that Rm ≤ Cr 2( 1 a −1) .
In Assumption 1, we only assume the asymptotics of the Kähler forms. To get the asymptotic behaviour of the corresponding Riemannian metrics, we need to show that the complex structure of X and D are sufficiently close under the metric g F . When D is an anticanonical divisor, the following result is proved in [14,Proposition 3.4]. For a general smooth divisor D, the author learned the following proof from Song Sun. Lemma 2.6. Let J D and J X denote the complex structure on D and X respectively. And Φ * J X := dΦ • J X • (dΦ) −1 denote the pullback of J X under an exponential-type map Φ. Then we have Proof. Since dΦ p is complex linear for all p ∈ D, we know Φ * J X − J D is smooth section of End(T D) vanishing on D. But this is not enough to get the bound claimed in (2.8). We will use the integrability of Φ * J X and property (3) in Definition 2.1. In the following, we ignore the pull-back notation. Around a fixed point in D we can choose local holomorphic coordinates {w, z 1 · · · , z n−1 } of the total space of N D so that the zero section is given by w = 0. Then we can write for α = 1, · · · , n − 1 that where P α and Q αβ are linear functions of w and w, i.e. there are smooth functions p α and p α of {z i } such that P α = p α w + p α w and a similar expression for Q αβ . There are no type (1, 0) vectors in the linear term of the right hand side because J 2 is still of type (1, 0) with respect to J X , which coincides with J D when restricted to D. Therefore ∂ w P α = p α = 0. By the property (2) and (3) in Definition 2.1 and the following standard exact sequence of the holomorphic vector bundles on D 0 −→ T 1,0 D −→ T 1,0 X −→ N D −→ 0, we know that on D, the dz α component of∂ J X ∂ w is tangential to D. Note that by definition we have∂ J X ∂ w = L ∂w J X , therefore we know that Since on D, the dz α component of∂ J X ∂ w is tangential to D, we obtain p α = 0. So we have for α = 1, · · · , n − 1 9) Now on D we consider the local basis of holomorphic vector fields (with respect to J C ): e n = w∂ w , e α = ∂ zα , α = 1, · · · , n − 1 and correspondinglyē n ,ē α the conjugate vector fields, and e n , e α the dual frame etc. Then we can write where i, j ranges from 1, · · · , n, 1, · · · ,n. Then (2.9) implies that we have J j i = O (|w|) for all i, j. Then the lemma follows from the explicit expression of the Kähler metric on D, see (2.7).
From the assumption (2.5) on the Kähler form and (2.8) on the complex structure asymptotics, we obtain that for the corresponding Riemannian metric It is also useful to write down the Riemannian metric g F explicitly in real coordinates. Note that the set ξ ∈ N D : Then we can write the Riemannian metric g F as follows where g D is the corresponding Riemannian metric for ω D and θ is a connection 1-form on Y such that dθ = ω D . From the asymptotic of the Riemannian metric tensor (2.11), the explicit expression of the Riemannian metric g F in (2.12) and conditions in Assumption 1, one can directly show the following result.
Lemma 2.7. Suppose (X, ω, g) satisfy Assumption 1, then (1) the volume growth of g is at most 2, i.e. there exists a constant C > 0 such that Vol(B R (p)) ≤ CR 2 for all R sufficiently large.
(2) for large numbers K, α = 2 and β = 4 a − 2, (X, ω) is of (K, α, β)-polynomial growth if θ is a smooth form on X vanishing when restricted to D, then That (M, g) is of (K, α, β)-polynomial growth is important for us since we need the weighted Sobolev inequality in [27, Proposition 2.1] to prove a weighted mean value inequality in the next subsection.

2.1.
A weighted mean value inequality. In this subsection, using a weighted Sobolev inequality in [27], we prove a weighted mean value inequality for nonnegative functions which are almost subharmonic. This is important when we run Simpson's argument to get a uniform C 0 -estimate. As usual, r denotes the distance function to a fixed base point induced by a Riemannian metric.
Lemma 2.8. Let (X, g) be a Riemannian manifold which is of (K, α, β)-polynomial growth as defined in [27]. Let u be a nonnegative compactly supported Lipschitz function satisfying ∆u ≤ f in the weak sense. Suppose that |f | = O(r −N ), for some N ≥ 2 + α + β, then there Proof. The following argument is the standard Moser iteration with the help of the weighted Sobolev inequality in [27, Proposition 2.1].
Let γ = 2n + 1 2n − 1 . Note that we haveˆu p ∆u ≤ˆu p f for any p ≥ 1. Integration by parts and using that |f | = O(r −N ), we havê Let dµ = (1 + r) −N dV g and without loss of generality, we may assume dµ has total mass 1.
Then the weighted Sobolev inequality shows that Applying the triangle inequality and Hölder inequality, we get . Let p i = γ i , i = 0, 1, · · · . We have for any i Either there exists a sequence of p i j → ∞ such thatˆu p i j dµ ≤ 1, which implies that u L ∞ ≤ 1, or there exists a smallest i 0 such that andˆu p i dµ > 1 for i ≥ i 0 . In the second case, we have Iterating gives that The assumption on the degree a. The only reason why we need to assume a ≤ n n − 1 is that the volume growth of the corresponding Riemannian metric is at most 2. In fact we have the following easy but useful degree vanishing property for Rimannian manifolds with at most quadratic volume growth. Lemma 2.9. Let (M, g) be a complete Riemannian manifold with volume growth order at most 2. Let u ∈ C ∞ (M) satisfying |∇u| ∈ L 2 and ∆u ∈ L 1 , then M ∆u dV g = 0.
Proof. By the Cauchy-Schwarz inequality and the assumption on the volume growth, we have Therefore there is a sequence R i → ∞ such thatˆ∂ ∆u dV g for any sequence R i going to infinity. Using Stokes' theorem, 2.3. Assumption on Φ and ω. By Proposition 2.3 and the assumption on the decomposi- (2.14) and integrating this exact 2-form, we can show the following result, whose proof is similar to that given in [14,Lemma 3.7].
Lemma 2.10. There exists a real 1-form η outside a compact set of C with And it suffices to write θ = dη with |η| . We identify C with R + × Y in such a way that the Riemannian metric g F can be written as dr 2 + g r , where r is the coordinate function on R + and g r is a metric on {r} × Y 2n−1 that depends on r. Then θ is supported on the region {r > r 0 } for some r 0 > 0. Then there exist a 1-form α and a 2-form β supported on the region {r > r 0 } such that ∂ r α = 0 and θ is closed, therefore dα + ∂ r β = 0 and then one can directly check that θ = dη. Since dr ∧ α is perpendicular to β and we assumed |θ| Fix a smooth background Riemannian metricḡ on Y . Then from (2.12) and (2.11), we obtain the following estimate Then the estimate for |η| g F follows from a direct computation.
Remark 2.11. A similar argument can be applied to dd c ϕ directly on X (using Assumption 1 ) and we obtain that there exists a cut-off function χ supported on a compact set and a smooth real 1-form ψ supported outside a compact set satisfying |ψ| = O(r 1+ 1 a ) such that dd c ϕ = dd c (χϕ) + dψ This is quite useful when we want to integrate by parts on X.
We assumed that ω 0 is a closed (1,1)-form on X and vanishes when restricted to D. In particular,ˆX c 1 (D) ∧ ω n−1 0 = 0. Then by the Lelong-Poincaré formula, we obtain the following.
Lemma 2.12. Let S ∈ H 0 (X, L D ) be a defining section of D and h be any smooth hermitian

Hermitian holomorphic vector bundles
Firstly let us recall the definition of projectively Hermitian-Yang-Mills metrics. Given a hermitian metric H on a holomorphic vector bundle E, there is a unique connection compatible with these two structures and it is called the Chern connection of (E, H). Let F H denote the curvature of the Chern connection and we call it the Chern curvature of (E, H). Let E be a holomorphic vector bundle on a Kähler manifold (X, ω). A hermitian metric H is called an ω-projectively Hermitian-Yang-Mills metric (ω-PHYM) if for some constant λ. We also use the notation F ⊥ H to denote the trace-free part of the curvature form, i.e. F ⊥ 3.1. Basic differential inequalities. Let E be a holomorphic vector bundle and H, K be two hermitian metrics on E, then we have an endomorphism h defined by We will write this as H = Kh and h = K −1 H interchangeably. Note that h is positive and self-adjoint with respect to both H and K. Let ∂ H and ∂ K denote the (1, 0) part of the Chern connection determined by H and K respectively. By abuse of notation, we use the same notation to denote the induced connection on End(E). Simpson showed that (2) and (3), if det(h) = 1 then the curvatures can be replaced by the trace-free curvatures F ⊥ .

Slope stability.
If |tr(Λ ω F H )| ∈ L 1 , the ω-degree of (E, H) and ω-slope of (E, H) are defined to be Now let us assume M is compact. Integration by parts shows that the degree defined above is independent of the metric H and only depends on the cohomology class of [ω] ∈ H 2 (X, R), i.e. by the Chern-Weil theory, And for any coherent subsheaf S of E, one can define its ω-degree as follows (see [16,Chapter 5]). It is shown that det S := (∧ r S) * * is a line bundle, where r is the rank of S and define And as before we define µ ω (S), the ω-slope of S, to be deg ω (S) rank(S) . Note that for the definition of ω-degree and ω-slope, we do not need ω to be a Kähler form at all, and a real closed (1, 1)-form is enough. That is for every real closed (1, 1)-form α, we can define The slope µ α (S) is defined similarly as before and we will use the notation µ(S, α) and µ α (S) interchangeably.
We have the following definition which generalizes the standard slope stability defined for Kähler classes in [16,Chapter 5]. ( From the definition, we know that if β = 0, then E is (α, β)-stable if and only if it is α-stable; if E is α-stable, then it is (α, β)-stable for any class β. In applications, typically the first class α has some positivity. For example, in our Theorem 1.3, α = c 1 (D) is nef and big.
Remark 3.4. For every coherent subsheaf S of a holomorphic vector bundle E, we have an exact sequence of sheaves: 0 → S → S * * → S * * /S → 0, where S * * /S is a torsion sheaf and supports on an analytic set with codimension at least 2. Then by [16,Section 5.6], we know det S = det(S * * ). In particular, we know that E is α-stable (respectively (α, β)-stable) if and only if the conditions in (1) (respectively (2)) hold for every coherent subsheaf of E.
3.3. Coherent subsheaves and weakly holomorphic projection maps. Let (E, H) be a hermitian holomorphic vector bundle over a Kähler manifold (M, ω). Suppose S is a coherent subsheaf of E, since E is torsion-free, then S is torsion free and hence locally free outside Σ which is a closed analytic set of codimension at least 2. Moreover on X\Σ we have an induced orthogonal projection map π = π H S satisfying Outside the singular set Σ, the Chern curvature of (S, H| S ) is related to the Chern curvature (E, H) by (3.5) Let us mention a result in current theory: ). Let Σ be a closed analytic subset of codimension at least 2 in a Kähler manifold (M, ω). Assume T is a closed positive current on M\Σ of bidegree (1, 1), i.e a (1, 1)form with distribution coefficients, then the mass of T is locally finite in a neighborhood of Σ. More precisely, every p ∈ Σ has a neighborhood U ⊆ M such that U T ∧ ω n−1 < ∞.
Applying the above theorem to tr √ −1∂π ∧∂π , one gets And in general we call π ∈ W 1,2 loc (M, End(E); ω, H) is a weakly holomorphic projection map if it satisfies (3.4) almost everywhere. By the discussion above, we know that for a coherent subsheaf S of E, π H S is a weakly holomorphic projection map. A highly nontrivial result due to Uhlenbeck and Yau [28] is that the converse is also true (see also [24]).
Theorem 3.6 ( [28]). Suppose there is a weakly holomorphic projection map π, then there exists a coherent subsheaf S of E such that π = π H S almost everywhere. If X is compact, deg ω (S) defined in (3.3) can be computed using the curvature form F S,H . The following result is well-known, see [16,Section 5.8]. We include a simple proof using Theorem 3.5.
Proof. Let r denote the rank of S. Since S is a subsheaf of E, there is a natural sheaf homomorphism Φ : Note that Φ is only injective on M\Σ in general. Let ∧ r H denote the metric on ∧ r E induced from H, then Φ * (∧ r H) defines a singular hermitian metric on (∧ r S) * * which is smooth outside Σ and whose curvature form is equal to tr(F S,H ). Since Φ is a holomorphic bundle map, by choosing a local holomorphic basis of (∧ r S) * * and ∧ r E, it is easy to show that Φ * (∧ r H) = f K, where K is a smooth hermitian on (∧ r S) * * , the function f is positive smooth outside Σ and converge to 0 polynomially along Σ. Then by Theorem 3.5, it suffices to prove the following: for every smooth positive function Note that | log f | ∈ L 2 , then (3.8) follows from the Cauchy-Schwarz inequality and existence of good cut-off functions. More precisely, since Σ has real codimension at least 4, it is wellknown that there exists a sequence of cut-off functions χ ǫ such that 1 − χ ǫ is supported in the ǫ-neighborhood of Σ and we have uniform L 2 bound on ∆χ ǫ . We briefly describe a construction of these cut-off functions. Let s be a regularized distance function to Σ in the sense that s : M → R ≥0 is smooth and satisfies that there exist positive constants C k such that The existence of such a regularized distance function can be derived from [26, Theorem 2 on page 171]. After a rescaling, we may assume s < 1 on M. For every ǫ > 0, let ρ ǫ be a smooth function which is equal to one on the interval (−∞, ε −1 ) and zero on (2 + ε −1 , ∞). Moreover we can have |ρ ′ ǫ | + |ρ ′′ ǫ | ≤ 10. Then we define χ ǫ = ρ ǫ (log(− log s)), and we can directly check they satisfy the desired properties.
Motivated by the above result, Simpson [25] uses the right hand side of (3.7) to define an analytic ω-degree of a coherent subsheaf on a noncompact Kähler manifold. Typically one needs to assume |Λ ω F H | ∈ L 1 to ensure the first term of (3.5) is integrable. Then the degree of a coherent subsheaf is either a finite number or −∞ depending on whether |∂π| ∈ L 2 . In general, this analytic degree depends on the choice of the background metric H. And a key observation in this paper is that when E has a compactification and H is conformal to a smooth extendable hermitian metric, this analytic degree does have an algebraic interpretation, see Lemma 5.3 and Lemma 5.4.

Dirichlet problem.
We have the following important theorem of Donaldson: . Given a hermitian holomorphic vector bundle (E, H 0 ) over (Z, ω) which is a compact Kähler manifold with boundary, there is a unique hermitian metric H on E such that (1) As observed in [19], one can do conformal changes to H to fix the induced metric on det E and still have it to be a projectively Hermitian-Yang-Mills metric. Proposition 3.9 ( [19]). Given a hermitian holomorphic vector bundle (E, H 0 ) over (Z, ω) which is a compact Kähler manifold with boundary, there is a unique hermitian metric H on E such that (1) Donaldson functional for manifolds with boundary. Next we recall Simpson's construction [25] for Donaldson functional. We follow the exposition in [19,Subsection 2.5] and focus on compact Kähler manifolds with boundary.
Let (Z, ω) be a compact Kähler manifold with boundary, (E, H 0 ) be a hermitian holomorphic vector bundle. Let b be a smooth section of End(E) which is self-adjoint with respect to H 0 . Then for any smooth function f : as follows: at each point p ∈ Z, choose an orthonormal basis Let Ψ(λ 1 , λ 2 ) denote the smooth function (3.9) Then put Mochizuki proved the following important result 3.6. Bando-Siu's interior estimate. The following result shows that to get a local uniform bound for a sequence of PHYM metrics, it suffices to have a uniform C 0 -bound.

Existence of a good initial metric
In this section, we continue to use notations in Section 2 and always assume the Kähler metric on X satisfies the Assumption 1. We begin by working on the model space (C, ω C ), where ω C = dd c F (t) for some potential F (t) ∈ H. Using the explicit expression of ω C in (2.3), it is easily to show that Lemma 4.1. Let E be a holomorphic vector bundle on D and p * (E) be its pull back to C. Let E be a holomorphic vector bundle on D. We still use p to denote the projection map from D to D. Then we can compare the holomorphic structure on E and p * (E| D ) as follows. In a neighborhood of D, which we may assume to be C, we fix a bundle map Φ : E → p * (E| D ) such that Φ| D is the canonical identity map and Φ is an isomorphism as maps between complex vector bundles. Then Φ pulls back the holomorphic structure on p * (E| D ) to E. Now we have two holomorphic structures on E and denote them by∂ 1 and ∂ 2 . Then the difference β =∂ 2 −∂ 1 is a smooth section of A 0,1 (End(E)) and is 0 when restricted to D. Locally near a point in D, choose holomorphic coordinates {z 1 , · · · , z n−1 , w} such that D = {w = 0}. Using these coordinates, β can be written as f i dz i + hdw where f i and h are smooth sections of End(E) and f i | w=0 = 0. Now suppose we have a Hermitian metric H on p * (E| D )| C , then via Φ we view it as a metric on E| C . Let ∂ i denote the (1, 0) part of the Chern connection determined by∂ i and H. Then one can check that where β * H denote the smooth section of A 1,0 (End(E)) obtained from β by taking the metric adjoint for the End(E) part and taking conjugate for the 1-form part. Locally Extend H D smoothly to get a hermitian metric H 0 on E. Using the diffeomorphism Φ given in the Assumption 1 -(3) we get a positive smooth function on X by abuse of notations stilled denoted by t, which is equal to (Φ −1 ) * t outside a compact set on X.
Define a hermitian metric on E using Then we claim that From the construction, Recall that for a 2 form θ, Since we assume that |Φ * (ω) − ω C | = O(r −N 0 ), then (4.4) will follow from the following estimate on C: there exist a δ > 0 such that Using the same argument as we did before Proposition 4.3, we can show that there exists a δ > 0 such that .  Remark 4.5. From the above discussion, we also obtain that |F H 0 | = O(r 1−a ). In general, we can not expect a higher decay order for the full curvature tensor F H 0 since it has nonvanishing component along the directions tangential to D, but if n = 2 we actually proved that From the proof given above, the assumption that E| D is c 1 (N D )-polystable is used crucially to have a good initial metric H 0 satisfying (1) and (2) in Lemma 1.1, which both are important for the proof. We show the assumption that E| D is c 1 (N D )-polystable is also necessary subject to the conditions in Lemma 1.1. More precisely, we have that Proof. By these two assumptions, we have is a smooth bundle valued (1,1)-form on X, its pull-back under Φ to D is a smooth bundle valued 2-form satisfying that its restriction to D is of type (1,1) and . From the explicit expression of the Kähler form ω F in (2.3) and the assumption on the potential F in Assumption 1, we know that Next we can show that the set P H defined in (1.1) is unique if we fix the induced metric on the determinant line bundle. More precisely, we have In particular, there exists a constant C > 0 such that C −1 H 0 ≤ H 1 ≤ CH 0 .
Proof. By condition (1) in Lemma 1.1, we know that there are smooth hermitian metrics H 0 and H 1 and smooth functions f 0 and f 1 on X such that for i = 0, 1 where ∇ denotes the induced connection on End(E) from the Chern connection on (E, H 0 ). From this and noting that H 0 is conformal to an extendable metric, we can check directly that |∇h| ∈ L 2 (X; ω, H 0 ) Then from the definition of P H 0 , we obtain that P H 0 = P H 1 .

Proof of the main theorem
We first prove a lemma on the degree vanishing property. Proof. Fix a base point p ∈ X and let ρ R be a smooth cut-off function which is 1 on B R (p), 0 outside B 2R (p) and |∇ρ R | ≤ C R where C is a constant independent of R. Integrating by parts, we have the following which is bounded by CˆB |β| ω n . And this term tends to 0 as R → ∞ since |β| ∈ L 1 .
From now on, we assume ω = ω 0 + dd c ϕ is a Kähler form satisfying the Assumption 1 in Section 2. Note that we only proved that dd c ϕ = dψ for a smooth form ψ with |ψ| = O(r 1+ 1 a ) (see Remark 2.11). Therefore we can not apply Lemma 5.1 directly. Typically we have a definite decay order for |β|, so we can still use integration by parts to show some degree vanishing properties. More precisely, we have Proof. By a similar integration by part argument as in the proof of Lemma 5.1, it suffices to show that lim This follows from the facts that |β| = O(r −N 0 ), |ψ| = O(r 1+ 1 a ) and the volume growth order of ω is at most 2.
The following two lemmas are crucial for us since they relate information on X and that on X. Lemma 5.3. Let H 0 be the metric constructed in Lemma 1.1. One has the following equality: Proof. Firstly, recall that By the construction in Lemma 1.1, we know that |Λ ω tr(F H 0 )| = O(r −N 0 ). Since the volume growth order of ω is at most 2, we know that Λ ω tr(F H 0 ) is absolutely integrable. Therefore the left hand side of (5.3) is well-defined. By the Chern-Weil theory, for any smooth hermitian metric H 0 on E we havê By the construction (4.3), H 0 = e Ct H 0 for some constant C and t defined in Section 4. Moreover by Remark 4.4, t = − log |S| 2 h for some smooth hermitian metric on L D . By Lemma 2.12, we obtain thatˆX dd c t ∧ ω n−1 Using (5.2) and ω = ω 0 +dd c ϕ, to prove (5.1), it suffices to show that for any k = 1, · · · , n−1, Since ω 0 vanishes when restricted to D, by Lemma 2.7, we know that |ω 0 | = O(r −N 0 ). Combining this with Remark 4.5, we know that tr(F H 0 ) ∧ ω n−1−k 0 is a closed (n − k, n − k)-form with decay order at least r −N 0 . Therefore Lemma 5.2 implies that its integral is 0. Case 2. k = n − 1. If n = 2, then by (4.8) we can still apply Lemma 5.2. If n ≥ 3, note that though |Λ ω tr(F H 0 )| = O(r −N 0 ), | tr(F H 0 )| is not in L 1 in general. So we can not apply Lemma 5.2 directly. Instead we shall use the asymptotic behaviour of tr(F H 0 ) obtained from the construction. Integrating by parts and pulling back via Φ, we know that Then by (2.5), Lemma 2.10 and the assumption N 0 > 8, we obtain that the right hand side of the above equality equals By (4.6) and (4.7), we know that it equals Note that when restricted to the level set of t, (1). Let S be a coherent reflexive subsheaf of E. If S| D is locally free and a splitting factor of E| D , then∂π H 0 S ∈ L 2 (X; ω, H 0 ). (2). Let π ∈ W 1,2 loc (X, E * ⊗ E; ω, H 0 ) be a weakly holomorphic projection map. If∂π ∈ L 2 (X; ω, H 0 ), then there exists a coherent reflexive subsheaf S of E such that π = π H 0 S a.e. and S| D is a splitting factor of E| D .
Proof. A crucial point here is that H 0 is conformal to a smooth extendable metric H 0 . In particular, for a coherent subsheaf S of E, the projections induced by H 0 and H 0 are the same. Note that by [4, Lemma 3.23 and Remark 3.25], for every coherent reflexive subsheaf S of E, S| D is torsion free and can be naturally viewed as a subsheaf of E| D .
(1) Let π = π H 0 S . Then π is smooth in a neighborhood of D and∂π| D = 0 by assumption. Note that π H 0 S = π| X , so it suffices to show∂π ∈ L 2 (X, ω, H 0 ). Fix small balls U i of X covering D such that there are holomorphic coordinates {z 1 , · · · , z n−1 , w} on each U i with D ∩U i = {w = 0} and E is trivial on each ball U i . Under these coordinates and trivializations we can write∂ π =∂ z πdz +∂ w πdw, where we view∂ z π and∂ w π as matrices of smooth functions and∂ z π| w=0 = 0. So we have |∂ z π| ≤ C|w| and |∂ w π| ≤ C. Then the result follows from the explicit estimate given in (2.7).
Since |π| H 0 ≤ 1 and by [7,Lemma 7.3], it suffices to show∂π ∈ L 2 (X; ω X , H 0 ). By (2.8) and (2.11), we may assume in local coordinates around D the Kähler metric ω is exactly given by the model space. We choose local holomorphic coordinates z = {z i } n−1 i=1 on the smooth divisor D and fix a local holomorphic trivialization e 0 of N D with |e 0 | h D = e −ψ , where ψ is a smooth function on D satisfying √ −1∂∂ψ = ω D . Then we get local holomorphic coordinatea {z 1 , · · · , z n−1 , w} on C by writing a point ξ = we 0 (z). Choose a basis of (0, 1)forms dz 1 · · · dz n−1 , dw w −∂ z i ψdz i . Then we can writē where f i and h are sections of End(E). Notice that dz i is perpendicular to the dw Since∂π is in L 2 with respect to ω, by (2.7) we know that Then we know that f i −h∂ z i ψ, h w are all L 2 -integrable with respect to the Lebesgue measure. Therefore the claim is proved: Then Uhlenbeck-Yau's result (Theorem 3.6) implies that there exists a coherent subsheaf S of E such that π = π H 0 S outside the singular set of S. Taking the double dual, we may assume S is reflexive. By the integrability condition (5.5),∂π H 0 S | D = 0, which means that S| D is a splitting factor of E| D since E| D is polystable. Now we are ready to prove the main theorem. We decompose it into two propositions.
(When E is a vector bundle, this follows from the fact that the first Chern class c 1 (D) is the Poincaré dual of the homology class defined by the divisor D. For a general reflexive sheaf, the key point is to show that c 1 (E)| D = c 1 (E| D ) using the fact that E is locally free outside an analytic set of (complex) codimension at least 3.) Therefore we have µ(S| D , c 1 (N D )) ≥ µ(E| D , c 1 (N D )). (5.6) By assumption, E| D is c 1 (N D )-polystable, so (5.6) implies that S| D is locally free and is a splitting factor of E| D . Then by Lemma 5.4, we havē ω, H). For simplicity of notation, in the following we omit the dependence on S. By the definition of P H 0 and H ∈ P H 0 , we know that H = H 0 e s with s L ∞ + ∂ s L 2 < ∞. The claim follows directly from the following pointwise inequality (outside the singular set Σ of S) where C is a constant independent of points and all the norms are with respect to H 0 . Let r 0 , r denote the rank of S and E respectively. Near any given point p ∈ X\Σ, we can find a local holomorphic basis {e 1 , · · · , e r 0 , e r 0 +1 , · · · , e r } of E such that S = Span{e 1 , · · · , e r 0 }, e i , e j H 0 (p) = δ ij , ∂ e i , e j H 0 (p) = 0 for 1 ≤ i, j ≤ r 0 and r 0 + 1 ≤ i, j ≤ r .
In the following we use Einstein summation convention and use i, j to denote numbers from 1 to r, α, β to denote numbers from 1 to r 0 . Under this basis π H 0 can be written as Similarly, π H can be written as e ∨ α ⊗ e α + H iβ H βα e ∨ i ⊗ e α . Note that as a matrix H = H 0 h, where h is the matrix representation of e s under the basis which gives (5.7). Let π = π H S . Using the Chern-Weil formula and the fact that H ∈ P H 0 is PHYM, we have and consequently is L 1 .
Assume this for a moment, then by Lemma 5.3, we know that and equality holds if and only if∂π = 0. Suppose∂π = 0. Since Again by [7,Lemma 7.3], there is a global holomorphic section of End(E), which is still denoted by π, such that π = π H S a.e. and π 2 = π. Note that since rank(π) = tr(π) is real valued and holomorphic, it follows that rank π is a constant. Thus E holomorphically splits as the direct sum of ker π and Im π, which contradict with our assumption that E is irreducible. Therefore we prove that Proof of the claim: since H ∈ P H 0 , we have tr(F S,H ) − tr(F S,H 0 ) = ∂∂u, for a bounded real valued smooth function u with |∇u| ∈ L 2 . By Lemma 2.9, tr(F S,H ) ∧ ω n−1 =ˆtr(F S,H 0 ) ∧ ω n−1 .
By the same argument in Lemma 5.3, we can shoŵ Hence we complete the proof of the claim. Proof. Uniqueness is obvious. Suppose we have two ω-PHYM metrics H 1 , By the definition of P H 0 , we know that det h = 1 and h is both bounded from above and below and |∂h| ∈ L 2 . Then by taking the trace of the differential equality in Therefore∂h = 0 and since h is self-adjoint with respect to H i , it is parallel with respect to the Chern connection determined by (∂, H i ). Then its eigenspaces give a holomorphic decomposition of E which contradicts the assumption that E is irrducible unless h is a multiple of identity map. Since det h = 1, it must be that h is the identity map, i.e. H 1 = H 2 .
For the existence part, we follow Simpson and Mochizuki's argument [25,19]. For completeness, we include some details. Let {X i } be an exhaustion of X by compact domains with smooth boundary and we solve Dirichlet problems on every X i using Donaldson's theorem (Theorem 3.8). Then we have a sequence of PHYM metrics H i on E| X i such that H i | ∂X i = H 0 | ∂X i and det H i = det H 0 . Let s i be the endomorphism determined by H i = H 0 h i = H 0 e s i . Then we have s i | ∂X i = 0 and tr(s i ) = 0.
We argue by contradiction to prove a uniform C 0 -estimate for s i . First note that by Lemma 3.2, e s i satisfies the elliptic differential inequality ∆ log(tr(e s i )) ≤ |ΛF ⊥ H 0 | (5.8) therefore tr(e s i ) satisfies the weighted mean value inequality in Lemma 2.8. Since tr(e s i ) and |s i | are mutually bounded, we know that |s i | also satisfies the weighted mean value inequality (2.13). Lemma 2.8 plays an essential role since it ensures that after normalization we can have a nontrivial limit in W 1,2 loc . Suppose there is a sequence s i such that sup Then by Lemma 2.8, we obtain Let u i = l −1 i s i . Then by Lemma 2.8 again we obtain there is a constant C independent of i such thatˆX i |u i |(1 + r) −N 0 = 1 and |u i | ≤ C, (5.9) where the norms are with respect to the back ground metric H 0 . Then following Simpson's argument, we can show that Lemma 5.7. After passing to a subsequence, u i converge weakly in W 1,2 loc to a nonzero limit u ∞ . The limit u ∞ satisfies the following property: if Φ : R × R → R is a positive smooth function such that Φ (λ 1 , By the definition of Ψ in (3.9), we know that as l → ∞, lΨ (lλ 1 , lλ 2 ) increases monotonically Fix a Φ as in the statement of the lemma. We know that for all A > 0 there exists l A such that if |λ i | ≤ A and l > l A , then we have Since sup |u i | are bounded, its eigenvalues are also bounded. Then by (5.11) and (5.12), we obtain that for i sufficiently large Again since sup |u i | is bounded we can find Φ satisfying the assumption in the lemma and Φ(u i ) = c 0 for all i, where c 0 a fixed small positive number. Then by (5.13) and the construction of H 0 , there exists a positive constant C such that Therefore by a diagonal sequence argument and after passing to a subsequence we may assume u i converge weakly in W 1,2 loc to a limits u ∞ withˆX |∂u ∞ | 2 ≤ C. We claim that u ∞ = 0. Indeed by (5.9), there exists a compact set K ⊆ X independent of i such that Since on compact sets the embedding from W 1,2 to L 1 is compact, after taking the limit, we In particular u ∞ = 0.
Next we prove (5.10). By the uniform boundedness of u i , the O(r −N 0 ) decay property of |ΛF H 0 | and the nonnegativity of the second term of the left hand side in (5.13), we know that there exists ǫ i → 0 such that for any j ≥ i, we have Note that Φ (u j ) ∂ u j ,∂u j H 0 = |Φ 1 2 (u j )(∂u j )| 2 H 0 . By [25, Proposition 4.1], we know that on each fixed X i , Φ 1 2 (u j ) → Φ 1 2 (u ∞ ) in Hom L 2 , L q for any q < 2. Since∂u j converge weakly in L 2 (X i ) to∂u ∞ , we obtain that Φ 1 2 (u j )(∂u j ) converge weakly to Φ 1 2 (u ∞ )(∂u ∞ ) in L q (X i ) for any q < 2. Then we know that for any q < 2, Letting i → ∞, the inequality (5.10) is proved.
Simpson's argument in [25, Lemma 5.5 and Lemma 5.6] can be applied verbatim to the infinite volume case, so we have Lemma 5.8 ([25]). Let u ∞ be a limit obtained in the previous lemma. Then we have (1) The eigenvalues of u ∞ are constant and not all equal.
Moreover using (5.10), Simpson proved that Lemma 5.9 ([25]). There exists at least one γ such that By Lemma 5.4, we get a filtration of E by coherent reflexive subsheaves S i whose restrictions to D are splitting factors of E| D . Since we assume that E| D is c 1 (N D )-polystable, we know that for every i µ(S i | D , c 1 (N D )) = µ(E| D , c 1 (N D )).
Then by Lemma 5.3, (5.15) which contradicts with the (c 1 (D), [ω 0 ])-stability assumption. Therefore we do have a uniform C 0 -estimate for s i .
Bando-Siu's interior regularity result Theorem 3.11 can be applied to get local uniform estimate for all derivatives of s i . Then we can take limits to get a smooth section s ∈ End(E), which is self-adjoint with respect to H 0 and tr(s) = 0 and more importantly s L ∞ < ∞ and H = H 0 e s is a PHYM metric.
Indeed taking the trace of the equality in Lemma 3.2-(2) and noting that where ν i denotes the outward unit normal vector of ∂X i . Integrating (5.16) over X i and using Stoke's theorem in the left hand side, we obtainˆX Since we have uniform C 0 -estimate for s i = log h i , there exist constants C 1 and C 2 independent of i such thatˆX On the stability condition. Note that global semistability is known [18], if we assume the restriction to D is semistable. There do exist irreducible holomorphic vector bundles which are polystable when restricted to D but not globally stable, even under more restrictive assumptions that X is Fano and D ∈ |K −1 X |.  ) corresponding to a non-splitting exact sequence of holomorphic vector bundles whose restriction to D splits as a direct sum of two line bundles with the same degree. Therefore E itself is not c 1 (D)-stable but E| D is c 1 (N D )-polystable. Such an E is irreducible, because if E = L 1 ⊕ L 2 , then deg(L i , c 1 (D)) = deg(L i | D ) = 0 since E| D is polystable of degree 0, which implies that S has to be one of the L i and Q is the other one. This contradicts with the construction of E.
6. Discussion 6.1. More results on the existence of PHYM metrics. By Donaldson's theorem on the solvability of Dirichlet problem (Theorem 3.8), the elliptic differential inequality (Lemma 3.2-(3)), the maximal principle and Bando-Siu's interior estimate (Theorem 3.11), we get the following well-known existence result. There are many examples for which (6.1) has a positive solution and even bounded solutions [1,21,23].
for some ǫ > 0, then (6.1) admits a bounded solution. In particular, if (M, g) has nonnegative Ricci curvature, volume growth order greater than 2, |ΛF ⊥ H 0 | = O(r −2 ) and |ΛF ⊥ H 0 | ∈ L 1 , then (6.1) admits a bounded solution. Theorem 6.1 can not be applied to (X, ω, g) satisfying Assumption 1 since we do not know whether (6.1) admits a positive solution (for this volume growth order at most 2 is a key issue). And actually Theorem 1.3 tells us that there are some obstructions for the existence of ω-PHYM metrics which are mutually bounded with the initial metric.
Such a phenomenon also appears when we seek a bounded solution for the Poisson equation on a complete noncompact Riemannian manifold (M, g) with nonnegative Ricci curvature. Suppose f is compactly supported for simplicity, then we know that (1) if the volume growth order is greater than 2, i.e. there is a constant c > 0 such that Vol(B r ) ≥ cr 2+ǫ for some ǫ > 0, then (6.2) admits a bounded solution. (Since by Li-Yau [17], (M, g) admits a positive Green's function which is O(r −ǫ ) at infinity, a bounded solution of (6.2) is obtained by the convolution with the Green's function.) (2) if the volume growth order does not exceed 2, i.e. there is a constant C > 0 such that Vol(B r ) ≤ C(r + 1) 2 , then (6.2) admits a bounded solution if and only ifˆM f = 0.
(For the "if" direction, see [11,Theorem 1.5]. For the "only if" direction, suppose we have a bounded function u and a compactly supported function f such that ∆u = f . Then by Cheng-Yau's gradient estimate [5], we obtain |∇u| ≤ C r for some C > 0 independent of r. Multiplying u both sides in (6.2) and integrating by parts, we obtain that |∇u| ∈ L 2 . Then Lemma 2.9 impliesˆM f = 0.) Next we discuss another result whose proof is similar to the proof of Theorem 1.3. Let (X, ω) be an n-dimensional (n ≥ 2) compact Kähler manifold, D be a smooth divisor. Let ω D = ω| D denote the restriction of ω to D and X = X\D denote the complement of D in X. Let L D be the line bundle determined by D and S ∈ H 0 (X, L D ) be a defining section of D. Fix a hermitian metric h on L D . Then after scaling h, the function t = − log |S| 2 h is smooth and positive on X. For any smooth function F : (0, ∞) −→ R with |F ′ (t)| → 0 as t → ∞ and F ′′ (t) ≥ 0 there exists a large constant A such that is a Kähler form on X. By scaling ω we may assume A = 1. One can easily check that ω is complete is and only ifˆ∞ 1 √ F ′′ = ∞ and it always has finite volume. In the following, we always assume the function F satisfies |F ′ (t)| → 0 as t → ∞ and F ′′ (t) ≥ 0. Then we can state assumptions on ω.

Assumption 2.
Let ω be the Kähler form defined by (6.3) and g be the corresponding Riemannian metric. We assume that (1) the sectional curvature of g is bounded.
A consequence of these assumptions is that (X, g) is complete and of (K, α, β)-polynomial growth defined in [27, Definition 1.1], so we can use the weighted Sobolev inequality as we did for the proof of Lemma 2.8.
Let E be an irreducible holomorphic vector bundle on X such that E| D is ω D -polystable. Then by Donaldson-Uhlenbeck-Yau theorem, there exists a hermitian metric H D on E| D such that Extend H D smoothly to get a smooth hermitian metric H 0 on E. Then by (6.4) and Assumption 2 -(2), one can easily show that . Then we have the following result Theorem 6.3. Suppose (X, ω) satisfies Assumption 2 and E| D is ω D -polystable. Let H 0 be a hermitian metric as above and P H 0 be defined by (1.1). Then there exists an ω-PHYM metric in P H 0 if and only if E is ω-stable.
Using the argument in Proposition 5.5, the "only if" direction follows from Lemma 2.9 and the following lemma .
Lemma 6.4. For every smooth closed (1,1)-form θ on X, we havê X θ ∧ ω n−1 =ˆX θ ∧ ω n−1 (6.5) Proof. Firstly note that since there exists a positive number c > 0 such that ω > cω and ω n < ∞, the left hand side of (6.5) is well-defined. Therefore it suffices to show that for Let S ǫ denote the level set {|S| h = ǫ}. By integration by part, it suffices to show that Case 1. k = 1. Note that with respect to the smooth back ground metric ω, Vol(S ǫ ) = O(ǫ) and |d c F | ≤ C|F ′ (t)|ǫ −1 on S ǫ . Then (6.6) follows from the assumption that |F ′ | → 0 as t → ∞. Case 2. 2 ≤ k ≤ n − 1. Then (6.6) follows from the fact that |F ′ (t)| → 0 as t → ∞ and For the "if" direction, the argument in Proposition 5.6 applies. We will not give the details and just point out the following two observations which make the argument work in this setting. The key points are (1) Assumption 2 and Lemma 6.2 ensure that we can apply the weighted mean value inequality proved in Lemma 2.8. (2) We have L 2 (X, ω) ⊂ L 2 (X, ω) since ω ≥ cω for some c > 0, therefore by Uhlenbeck-Yau's theorem (Theorem 3.6) a weakly projection map π of E over X with |∂π| ∈ L 2 (X, ω) defines a coherent torsion free sheaf S of E.

6.2.
Calabi-Yau metrics satisfying Assumption 1 . As mentioned in the Introduction, there do exist interesting Kähler metrics satisfying the Assumption 1, which contain Calabi-Yau metrics on the complement of an anticanonical divisor of a Fano manifold and its generalizations [27,14,13]. We will call them Tian-Yau metrics. Here we give a sketch for the construction of these Calabi-Yau metrics and refer to [13]-Section 3 for more details. Let X be an n-dimensional (n ≥ 2) projective manifold, D ∈ |K −1 X | be a smooth divisor and X = X\D be the complement of D in X. Suppose that the normal bundle of D in X, N D = K −1 X | D is ample. Fixing a defining section S ∈ H 0 (X, K −1 X ) of the divisor D whose inverse can be viewed as a holomorphic volume form Ω X on X with a simple pole along D.
Let Ω D be the holomorphic volume form on D given by the residue of Ω X along D. Using Yau's theorem [29] , there is a hermitian metric h D on K −1 X | D such that its curvature form is a Ricci-flat Kähler metric ω D with by rescaling S if necessary. One can show that the hermitian metric h D extends to a global hermitian metric h X on K −1 X such that its curvature form is nonnegative and positive in a neighborhood of D.
By glueing a smooth positive constant on a compact set, we get a global positive smooth function z which is equal to (− log |S| 2 h X ) 1 n outside a compact set. For any A ∈ R, we denote h A = h X e −A and v A = n n + 1 n , which is viewed as a smooth function defined outside a compact set on X. We denote by H 2 c,+ (X) the subset of Im(H 2 c (X, R) → H 2 (X, R)) consisting of classes k such thatˆY k p > 0 for any compact analytic subset Y of X of pure dimension p > 0. Then Hein-Sun-Viaclovsky-Zhang proved the following result Theorem 6.5. [13] For every class k ∈ H 2 c,+ (X), there is a unique Kähler metric ω ∈ k such that (1) ω n = ( √ −1) n 2 Ω X ∧ Ω X , and for some δ, A > 0 and all l ≥ 0.
And from the construction in [13]-Section 3, we have the decomposition ω = ω 0 + dd c ϕ, where ω 0 is a smooth (1,1)-form on X vanishing when restricted to D. And by Theorem 6.5 and the estimate in [14,Proposition 3.4], one can directly check that these Kähler metrics satisfy Assumption 1.
Remark 6.6. It was proved in [14] that Tian-Yau metrics ω T Y can be realized as the rescaled pointed Gromov-Hausdorff limits of a sequence of Calabi-Yau metrics ω k on a K3 surface. We expect that ω T Y -PHYM connections we obtained in this paper give models for the limits of ω k -HYM connections on the K3 surface.
6.3. On the ampleness assumption of the normal bundle N D . In this subsection, we first explain why we assume the normal bundle of D is ample and then discuss the case where the normal bundle is trivial on compact Kähler surfaces.
In order to have the above equality, a natural (possibly the only reasonable) choice is that α = c 1 (D).
To make the argument in this paper work, we also need the following property: if a vector bundle F on D is polystable with respect to α| D and S is a coherent subsheaf of F with the same α| D -degree as F , then S is a vector bundle and is a splitting factor of F . (Note that this does not follow from the definition since α| D may not be a Kähler class. For example if α| D is 0, then definitely it does not satisfy this property.) In general in order to have this property, we need α| D to be a Kähler class. This is one of the reasons why we assume that the normal bundle of D is ample, i.e. c 1 (D)| D is a Kähler class. Another reason is that by assuming N D is ample, on the punctured disc bundle C we have explicit exact Kähler forms, which give models of the Kähler forms on X.
However if X is a compact complex surface, in which case the divisor D now is a smooth Riemann surface, then the property mentioned above always holds. Note that on a Riemann surface D, the slope of a vector bundle is canonically defined and independent of the choice of cohomology classes on D.
Lemma 6.7. Let X be a compact Kähler surface and D be a smooth divisor. Suppose E| D is polystable. Let S be a coherent reflexive subsheaf of E. Then µ(S, c 1 (D)) = µ(E, c 1 (D)) if and only if S| D is a splitting factor of E| D .
Using this, most of the arguments in Section 5 can be modified to work for divisors D with c 1 (N D ) = 0 in complex dimension 2. In the following, we assume c 1 (N D ) = 0 in H 2 (D, R). Then it is easy to see that c 1 (D) is nef and by the global ∂∂-lemma on D, we know that there exists a hermitian metric h D on N D with vanishing curvature. Let L D be the line bundle determined by D and S ∈ H 0 (X, L D ) be a defining section of D. Then we can extend h D smoothly to get a smooth hermitian metric h on L D and after a rescaling, we may assume that t = − log |S| 2 h is positive on X. In this case, we can consider (at least) all monomials potentials with degree greater than 1 H := {F (t) = At a : A > 0 is a constant and a > 1} . (6.8) Assumption 3. Let ω be a Kähler form on X and g be the corresponding Riemannian metric. We assume that (1) the sectional curvature of g is bounded.
Suppose (X, ω, g) satisfies Assumption 3, then we have the following consequences: • the Riemannian metric g is complete and has volume growth order at most 2, • (X, g) is of (K, 2, β)-polynomial growth as defined in [27, Definition 1.1] for some positive constants K and β. Let E be an irreducible holomorphic vector bundle over X such that E| D is polystable with degree 0. Then by Donaldson-Uhlenbeck-Yau theorem (for Riemann surfaces this was first proved by Narasimhan and Seshadri [20]), there exists a hermitian metric H D on E| D such that Λ ω D F H D = 0.
Since D is a Riemann surface, this is equivalent to say that H D gives a flat metric on E| D , i.e. F H D = 0. (6.9) Extending H D smoothly to get a hermitian metric H 0 on E then by (6.9) and the proof of Lemma 1.1, we know that H 0 is already a good initial metric in the following sense: |F H 0 | = O(e −δt ). (6.10) Then we have the following result, whose proof is essentially the same as that for Theorem 1.3. We just point out the difference. The argument in Section 5 can be applied if Lemma 5.3 still holds. The analog of Lemma 5.3 in this case is the following lemma, for which we need to assume E| D is flat. Lemma 6.9. Suppose (X, ω) satisfies Assumption 3 and E| D is flat. Let H 0 be a hermitian metric as above. Then we have the following equality: Proof. By Chern-Weil theory, it suffices to show that X √ −1 2π tr(F H 0 ) ∧ dd c ϕ = 0. (6.11) The argument in Lemma 2.10 can be used again to show that there exists a cut-off function χ supported on a compact set and a smooth 1-form ψ supported outside a compact set such that dd c ϕ = dd c (χϕ) + dψ. Moreover |ψ| grows at most in a polynomial rate of r. Then (6.11) follows from integration by parts and (6.10).
Example 6.10. Let X = CP 1 × D, where D is a compact Riemann surface. Then D = {∞} × D is a smooth divisor with trivial normal bundle. Fix a Kähler form ω D on D and also view it as a form on CP 1 × D via the pull-back of the obvious projection map. Note that up to a scaling [ω D ] ∈ c 1 (CP 1 ) in H 1,1 (X). We can consider asymptotically cylindrical metrics on X = C × D given by the Kähler forms where z denotes the coordinate function on C and ϕ = Φ ′′ is a positive smooth function defined on R such that ϕ(t) = e t when t is sufficiently negative and ϕ(t) = 1 for t sufficiently positive. Then one can easily check that (X, ω) satisfies Assumption 3 with F (t) = t 2 . Let E be an irreducible holomorphic vector bundle on CP 1 × D such that E| D is flat. Then by Theorem 6.8, we know that Similar examples as in Example 5.10 show that the condition c 1 (D), c 1 (CP 1 ) -stability is non-trivial. More specifically, let D be a Riemann surface with genus g ≥ 1 and k ≥ 2 be an integer. Then similar argument as in Example 5.10 shows that there exists a non-splitting extension 0 −→ O −→ E −→ p * 1 (O P 1 (−k)) −→ 0. whose restriction to D splits. Then one can easily check that E is irreducible and not c 1 (D), c 1 (CP 1 ) -stable.
6.4. Some problems for further study. Let (X, ω) satisfy the Assumption 1. As illustrated by Theorem 6.5 it is more natural to assume a stronger condition on the background Kähler metric ω. More precisely, we assume that in (2.5) the right hand side is replaced by O(e −δ 0 r α 0 ) for some δ 0 , α 0 > 0 and we also have the same bound for higher order derivatives. Under these assumptions and motivated by the result of Hein [12] for solutions of complex Monge-Ampère equations, we make the following conjecture. Note that the key issue is to prove that |s| decays exponentially, since all of the higher order estimates will follows form standard elliptic estimates.