Kobayashi-Hitchin correspondence for twisted vector bundles

We prove the Kobayashi-Hitchin correspondence and the approximate Kobayashi-Hitchin correspondence for twisted holomorphic vector bundles on compact K\"ahler manifolds. More precisely, if $X$ is a compact manifold and $g$ is a Gauduchon metric on $X$, a twisted holomorphic vector bundle on $X$ is $g-$polystable if and only if it is $g-$Hermite-Einstein, and if $X$ is a compact K\"ahler manifold and $g$ is a K\"ahler metric on $X$, then a twisted holomorphic vector bundle on $X$ is $g-$semistable if and only if it is approximate $g-$Hermite-Einstein.


Introduction
The Kobayashi-Hitchin correspondence for vector bundles is a nowadays well-established result in complex geometry, saying that a holomorphic vector bundle on a compact complex manifold X is polystable if and only if it admits a Hermite-Einstein metric. Here a holomorphic vector bundle is polystable if it is the direct sum of stable holomorphic vector bundles (where stability is the slope-stability, or Mumford-Takemoto stability) with the same slope, and a Hermite-Einstein metric is a Hermitian metric whose mean curvature is a constant multiple of the identity.
This result was proved in an increasing order of generalization by several authors. First, in 1980 Kobayashi introduced in [15] the notion of Hermite-Einstein metric on a holomorphic vector bundle over a complex manifold. In [16] he showed that an irreducible Hermite-Einstein vector bundle on a compact Kähler manifold is polystable with respect to a Kähler metric. A different proof of this was given by Lübke in [20].
Shortly after [16], Donaldson proved in [6] that on a Riemann surface even the opposite is true, i. e. that a stable holomorphic vector bundle carries a Hermite-Einstein metric. This gave a new proof of the Narasimhan-Seshadri theorem (see [23]) saying that a holomorphic vector bundle on a Riemann surface X is stable if and only if it has an irreducible projective unitary representation of the fundamental group of X.
Donaldson's result motivated Kobayashi and Hitchin, indipendently, to conjecture that this result holds for every holomorphic vector bundle on a compact Kähler manifold: it is this correspondence which is usually referred to as Kobayashi-Hitchin correspondence. The first proof of this correspondence for higher dimensional manifolds was given by Donaldson in [7] for algebraic surfaces, and then in [8] for algebraic manifolds.
Uhlenbeck and Yau proved in [27] and [28] that the Kobayashi-Hitchin correspondence holds on arbitrary compact Kähler manifolds, completing the proof of the original conjecture of Kobayashi and Hitchin. A few years after that, Buchdahl proved in [1] that the Kobayashi-Hitchin correspondence holds on any compact complex surface, and in [18] Li and Yau proved that it holds on every compact complex manifold, i. e. that if X is a compact complex manifold and g is a Gauduchon metric on X, then a holomorphic vector bundle E is g−Hermite-Einstein if and only if it is g−polystable.
Instead of Hermite-Einstein holomorphic vector bundle, i. e. a holomorphic vector bundle admitting a Hermite-Einstein metric, one can consider the weaker notion of approximate Hermite-Einstein holomorphic vector bundle, i. e. a holomorphic vector bundle E such that for each > 0 there is a Hermitian metric h on E whose mean curvature K g (E, h ) verifies where c ∈ R depends only on X, g, c 1 (E) and the rank of E.
It was shown by Kobayashi in [17] that an approximate Hermite-Einstein holomorphic vector bundle on a compact Kähler manifold X is semistable (with respect to a Kähler metric g), and that if X is a projective manifold, then even the converse holds. This equivalence is often referred to as approximate Kobayashi-Hitchin correspondence. In [13] Jacob proved that the approximate Kobayashi-Hitchin correspondence holds for every holomorphic vector bundle on a compact Kähler manifold.
A natural generalization of holomorphic vector bundles on a compact complex manifold X is given by twisted holomorphic vector bundles, where the twist is a 2−cocyle representing an element in the Brauer group Br(X) of X (i. e. the torsion of H 2 (X, O * X )). Twisted sheaves were introduced by Giraud in [10], and can be defined in several equivalent ways: as family of sheaves on an open covering of X together with a twisted gluing, as sheaves of modules over an Azumaya algebra on X (see [3]), as sheaves on a gerb on X (see [10], [11], [5]), as sheaves on a O * X −gerbe (see [19]) or as sheaves on a projective bundle over X (see [30]).
Stability for coherent twisted sheaves was introduced first by Lieblich in [19] in the language of sheaves on O * X −gerbes, and by Yoshioka in [30] in the language of sheaves on a projective bundle over X. In [24] stability of coherent twisted sheaves is discussed in the language of twisted gluing of coherent sheaves and in the language of modules over an Azumaya algebra. In all of these categories in order to define stability one needs a definition of Chern classes of coherent twisted sheaves.
The notion of connection on a twisted holomorphic vector bundles appears in [11], [5] (see even [22]), and was used in [29] to prove the Kobayashi-Hitchin correspondence for twisted holomorphic vector bundles on a compact Kähler manifold. All the definitions there are in the category of twisted holomorphic vector bundles as holomorphic bundles over a gerb.
In the present paper we will consider twisted vector bundles following Cȃldȃraru's point of view, i. e. local vector bundles on a open covering together with a twisted gluing. Connections on such vector bundles may be found in [14], and in the present paper we present a definition of Hermitian metric. The twist α will be a 2−cocycle (given once an open covering U = {U i } i∈I of X is fixed) whose cohomology class lies in Br(X), and which is associated a B−field, i. e. a family of closed (1, 1)−forms B i ∈ A 1,1 (U i ) such that B i − B j is an exact form.
The aim of the paper is to prove the following: Theorem 1.1. Let X be a compact Kähler manifold with a Kähler metric g, α the twist associated to a B−field and E an α−twisted holomorphic vector bundle on X.
(1) E is g−polystable if and only if it is g−Hermite-Einstein.
(2) E is g−semistable if and only if it is approximate g−Hermite-Einstein .
As already mentioned, the first item of the statement of Theorem 1.1 was already proved by Wang in [29]. Here we present a proof in the language of local vector bundles with twisted gluing, and we provide a generalization of Wang's result to compact complex manifolds with a Gauduchon metric g (see Theorem 5.17). The second item is the twisted version of the approximate Kobayashi-Hitchin correspondence, and is new. The proof is an adaptation to twisted holomorphic vector bundles of the original proofs of Kobayashi and Jacob.
The structure of the paper is the following: section 2 contains the main basic facts about twisted vector bundles and sheaves, and about connections and metrics on them. As this was not available in the literature, we gave a global overview of this.
We prove in particular that every twisted holomorphic vector bundle E over which we fix a Hermitian metric h, carries a unique connection which is compatible with the holomorphic structure of E and the metric h. In analogy with the untwisted case, we will call this connection the Chern connection of the pair (E, h).
The B−field fixed at the beginning will allow us to define the curvature of every connection, which is a global 2−form with values in the (untwisted) vector bundle End(E). The curvature of the Chern connection will be called Chern curvature, and it will be a global (1, 1)−form with values in End(E). It is the fact that End(E) is a true (i. e. untwisted) holomorphic vector bundle on X that will allow us to provide a proof of Theorem 1.1.
We will moreover discuss how connections, curvatures and Hermitian metrics behave under various operations of twisted bundles, like direct sum, tensor product, dual, pull-back, sub-bundle and quotient bundles, proving in particular a twisted version of the Gauss-Codazzi formulas.
Section 3 is devoted to introduce the notion of g−Hermite-Einstein and approximate g−Hermite-Einstein twisted bundles. To do so, we need to introduce Chern forms and Chern classes for twisted sheaves: as already done by Wang in [29], we define Chern forms and Chern classes by means of the curvature of a connection, similarly to what happens for holomorphic vector bundles.
Once the Chern forms and classes are introduced, we define the mean curvature of a pair (E, h) of a twisted holomorphic vector bundle E and a Hermitian metric h: exactly as in the untwisted case, this will be a smooth endomorphism of E (which is Hermitian with respect to h). The notion of (weak) g−Hermite-Einstein metric is then as in the untwisted case, and we prove that in the conformal class of a weak g−Hermite-Einstein metric there is always a g−Hermite-Einstein metric.
We define g−Hermite-Einstein and approximate g−Hermite-Einstein vector bundles as in the untwisted case, and we will provide several properties of (approximate) g−Hermite-Einstein bundles following closely the analogous properties for untwisted bundles.
Section 4 is devoted to the notion of g−semistable and g−stable twisted holomorphic vector bundles, proving several properties of these bundles, and in particular that g−Hermite-Einstein bundles are g−polystable, and that approximate g−Hermite-Einstein bundles are g−semistable. The proof is essentially the same as in the untwisted case, and we follow closely Lübke's argument in [20]. This proves half of Theorem 1.1.
In section 5 we prove that a g−stable twisted holomorphic vector bundle is g−Hermite-Einstein: this is the content of Theorem 5.1, which completes the proof of point 1 of Theorem 1.1. The proof we present is identical to the one given by Uhlenbeck and Yau in [27], and its adaptation to Gauduchon metrics on compact complex manifold as presented in [18] and in section 3 of [21].
Wang's approach in [29] was to adapt to twisted bundles the original argument of Donaldson, adapted by Simpson in [25]. Since [18] and [21] work more generally if g is a Gauduchon metric on a compact complex manifold X, we will finally prove that the Kobayashi-Hitchin correspondence for twisted holomorphic vector bundles holds on every compact complex manifold (with respect to a chosen Gauduchon metric on it), thus generalizing [29].
The remaining part of the paper is devoted to the proof of the approximate Kobayashi-Hitchin correspondence, namely that each g−semistable twisted holomorphic vector bundle is approximate g−Hermite-Einstein: this is the content of Theorem 6.1, which completes the proof of point 2 of Theorem 1.1.
The proof will follow closely the original argument in the untwisted case as presented in [17] and [13]. As in [17] we first define the Donaldson Lagrangian for Hermitian metrics on a twisted holomorphic vector bundle, and prove that if it is bounded below, then the bundle is approximate g−Hermite-Einstein. We will then prove that the Donaldson Lagrangian of a g−semistable twisted holomorphic vector bundle is bounded below. The proof is based on the one proposed by Jacob in [13] for untwisted vector bundles, and we adapt it to the twisted case.

Notation
All along the paper we consider a C ∞ −differentiable manifold M . If J is a complex structure on M , we will let X = (M, J) be the induced complex manifold. We moreover fix a sufficiently fine open covering U = {U i } i∈I of M , where I is a set of indexes. We write U ij := U i ∩ U j and U ijk : Once the open covering U is fixed, we choose a B−field B on X with respect to U: for further reference and more details on B−fields, see [11], [5], [29] and [4].
A B−field on X with respect to U is a family B = {B i } i∈I where B i is a 2−form on U i , such that there are 1−forms ω ij on U ij with the property that We notice that dB i − dB j = d 2 ω ij = 0, hence the 3−forms dB i glue together to give a closed 3−form dB on X, whose cohomology class is an element in H 3 (X, Z).
Notice that d(ω ij + ω jk + ω ki ) = 0, so if the covering U is sufficiently fine we may find U (1)−valued functions α ijk on U ijk such that Then α B = {α ijk } is a 2−cocycle whose cohomology class lies in H 2 (X, O * X ). The 2−cocyle α B will be called twist induced by B, and all along the paper we will use the notation α instead of α B . The natural morphism H 2 (X, O * X ) −→ H 2 (X, Z) from the exponential sequence sends the cohomology class [α] to the cohomology class [dB] of dB.
If [dB] is torsion in H 3 (X, Z), then [α] is torsion in H 2 (X, O * X ), i. e. it corresponds to an element in the Brauer group Br(X) of X. In this case [dB] is trivial in H 3 (X, R), and we may and will choose the forms B i to be d−closed for every i ∈ I. We will moreover ask that B i is a purely imaginary (1, 1)−form on U i , and that ω ij is a (1, 0)−form on U ij .
If H 3 (X, Z) is free, then every element in Br(X) may be represented by a twist induced by some B−field.

Connections and metrics
In this section we introduce the definitions of twisted vector bundle and of twisted coherent sheaf that we will use all along the paper. After having reviewed all the basic operations we will use on twisted sheaves, we will introduce the notion of connection, of curvature and of Hermitian metric on a twisted vector bundle, showing that once a holomorphic twisted vector bundle and a Hermitian metric on it are given, then there is a unique connection on it which is compatible with the metric and with the holomorphic structure. In analogy to the untwisted case, this connection will be called the Chern connection of the twisted bundle, whose curvature will be the most important tool in the paper, exactly as in the untwisted case.
2.1. Twisted vector bundles. Let M be a C ∞ −differentiable manifold over which we have a complex structure, and let X be the induced complex manifold. We first recall the definition of twisted vector bundle on X.
(1) for each i ∈ I, E i is a complex C ∞ vector bundle on U i , (2) for each i, j ∈ I, φ ij : E i|U ij −→ E j|U ij is an isomorphism of complex C ∞ vector bundles on U ij , Morphisms of twisted bundles are defined in a natural way: (1) for each i ∈ I, we have that f i : E i −→ F i is a morphism of complex C ∞ vector bundles on U i , (2) for each i, j ∈ I, we have ψ ij • f i = f j • φ ij .
The category of α−twisted complex C ∞ vector bundles on X will be denoted Bun C ∞ (X, α). The objects that will be under investigation in this paper will anyway more precisely be twisted holomorphic vector bundles, defined as follows: 3. An α−twisted complex C ∞ vector bundle E = {E i , φ ij } i,j∈I on X will be called α−twisted holomorphic vector bundle on X if E i is a holomorphic vector bundle on U i and φ ij : E i|U ij −→ E j|U ij is an isomorphism of holomorphic vector bundles.
Morphisms among twisted holomorphic vector bundles are then defined as follows: Definition 2.4. If E and F are two α−twisted holomorphic vector bundles on X, a morphism of α−twisted holomorphic vector bundles from E to F is a morphism f = {f i } i∈I : E −→ F of α−twisted complex C ∞ bundles such that for every i ∈ I we have that f i : E i −→ E j is a morphism of holomorphic vector bundles on U i .
The category of α−twisted holomorphic vector bundles on X will be denoted Bun(X, α). If instead of looking at vector bundles we are willing to look at sheaves, we will talk about twisted sheaves as follows.
Definition 2.5. An α−twisted sheaf on X is a family E = {E i , φ ij } i,j∈I where (1) for each i ∈ I, E i is sheaf of Abelian groups on U i , (2) for each i, j ∈ I, φ ij : E i|U ij −→ E j|U ij is an isomorphism of sheaves of Abelian groups on U ij , If F i is a sheaf of O U i −modules for every i ∈ I, then we say that F is an α−twisted sheaf of O X −modules. If moreover F i is coherent (resp. quasi-coherent), we will say that F is an α−twisted coherent sheaf (resp. an α−twisted quasi-coherent sheaf).
Moreover, we have the notion of morphism between α−twisted sheaves.
Definition 2.7. An α−twisted sheaf E = {E i , φ ij } of O X −modules is said to be locally free (of rank r) if for each i ∈ I we have that E i is locally free (of rank r).
The full-subcategory of Sh O X (X, α) whose objects are α−twisted locally free sheaves is denoted Lf (X, α). It is easy to prove that there is an equivalence of categories between Bun(X, α) and Lf (X, α).
Remark 2.8. If U is a refinement of U, by restriction we see that a B−field relative to U gives a B−field relative to U , whose associated twist is ǎ Cech 2−cocycle α relative to U . We moreover get a canonical equivalence between Bun C ∞ (X, α) and Bun C ∞ (X, α ) (and similarly for the other categories we mentioned before).
If E = {E i , φ ij } is an α−twisted complex C ∞ vector bundle relative to U, we may refine U so that the twisted vector bundle corresponding to E will be {E i , φ ij } i∈I where E i is the trivial vector bundle.

2.2.
Operations with twisted bundles. The usual operations between vector bundles (C ∞ or holomorphic) and sheaves can be defined as well in the twisted setting. We will only consider the case of α−twisted complex C ∞ vector bundles, but the same definitions work for α−twisted holomorpic vector bundles and for α−twisted sheaves (of O X −modules, coherent or quasi-coherent). We refer the reader to [3] for further details.
where E * i is the dual vector bundle of E i on U i , and φ * ij : E * i −→ E * j is the dual of φ ij : more precisely, if η is a local section of E * i , then φ * ij (η) is the local section of E * j mapping a local section ξ of E j to η(φ −1 ij (ξ)). It is easy to see that E * is a complex C ∞ vector bundle twisted by α −1 = {α −1 ijk }.
j is the conjugate of φ ij . Then E is a complex C ∞ vector bundle twisted by α = {α ijk }. If E is holomorphic, then E is holomorphic over X (the complex manifold obtained by putting on M the conjugate complex structure J).
Direct sum. If E = {E i , φ ij } and F = {F i , ψ ij } are two α−twisted complex C ∞ vector bundles, their direct sum is E⊕F := {E i ⊕F i , φ ij ⊕ψ ij }, which is an α−twisted complex C ∞ vector bundle as well.
Tensor product. Consider B and B two B−fields with respect to U, and let α and α be the respective twists. If E = {E i , φ ij } is an α−twisted complex C ∞ vector bundle and F = {F i , ψ ij } an α −twisted complex C ∞ vector bundle, their tensor product is E ⊗ F : In particular E ⊗ E * and E * ⊗ E * are untwisted complex C ∞ vector Wedge product. If E is an α−twisted complex C ∞ vector bundle on X, for every p ≥ 0 we may consider the p−th wedge product p E = { p E i , ∧ p φ ij }. This is a direct summand of E ⊗p , hence it is an α p −twisted complex C ∞ vector bundle. In particular, if E has rank r, then we have that det(E) := r E is an α r −twisted complex C ∞ vector bundle.
The canonical isomorphism E * i ⊗F i Hom(E i , F i ) (coming from the universal property of tensor product) induces a canonical isomorphism between E * ⊗F and Hom(E, F ). In particular we see that End(E) and Hom(E, E * ) are untwisted complex C ∞ vector bundles.
Notice that if E and F are α−twisted holomorphic vector bundles, then End(E) and Hom(E, F ) are holomorphic vector bundles. As such we may consider their global sections as complex C ∞ vector bundles, or as holomorphic vector bundles.
In the first case, the global sections of End(E) (resp. of Hom(E, F )) are the smooth endomorphisms of E (resp. the smooth morphisms from E to F ), and will be denoted A 0 (End(E)) or simply End(E) (resp. A 0 (Hom(E, F )), Hom(E, F )). In the second case, we will use the notation H 0 (End(E)) and H 0 (Hom(E, F )).
Pull-back. Let X and Y be two C ∞ differentiable manifolds and f : , so that we may glue together the traces of the endomorphisms f i 's to get a global smooth (resp. holomorphic) function T r(f ), called trace of f .
Similarily we have If x ∈ U ij , as φ ij and ψ ij are isomorphisms of vector bundles we have It follows that rk x (f i ) does not depend on the choice of i ∈ I: we will write it rk x (f ) and call it the rank of f at x. We now need to make some remarks about eigenvalues of endomorphisms of twisted bundles. Remark 2.9. If E is an α−twisted holomorphic vector bundle and f = {f i } is a smooth endomorphism of E, then it makes sense to consider the eigenvalues of f (which are smooth functions on X).
Indeed, suppose that λ i is an eigenvalue of f i , i. e. λ i is a smooth function on U i for which there is a nowehere vanishing smooth section s of E i with f i (s) = λ i s. Then λ i is an eigenvalue for f j over U ij : indeed φ ij (s) is a nowhere vanishing smooth section of E j over U ij , and we have Hence the eigenvalues of the f i 's glue together to give global smooth functions on X that on each U i rectrict to the eigenvalues of f i , and that will be referred to as eigenvalues of f .
As for morphisms of untwisted sheaves, the trace of a morphism of twisted sheaves is the sum of the eigenvalues, and its determinant is their product. The previous Remark 2. 9 shows moreover that f i is diagonalizable if and only if f j is, hence it makes sense to talk about diagonalizable endomorphisms of α−twisted (holomorphic) vector bundles.
If f is a diagonalizable endomorphism of E whose eigenvalues are λ 1 , · · · , λ r , consider a smooth function ϕ : R −→ R and suppose that the images of λ 1 , · · · , λ r are all contained in the definition domain of ϕ. In particular we see that ϕ • λ i is a smooth function on X.
This allows us to perform the following general construction: for every i ∈ I consider a local frame σ i of E i which diagonalizes f i . With respect to σ i we then have that f i is represented by a diagonal matrix F i whose diagonal entries are the eigenvalues of f i (each one appearing with its respective multiplicity). We then let ϕ(F i ) be the diagonal matrix whose diagonal entries are the ϕ•λ i (each one with the respective multiplicity), and consider the endomorphism ϕ(f i ) of E i corresponding to ϕ(F i ).
Particular cases are exp(f ), the exponential of f (which may be defined for every endomorphism of E), log(f ), the logarithm of f (which may be defined for positive definite endomorphisms) and f σ for every σ ∈ (0, 1] (which may be defined for positive semidefinite endomorphisms).

Connections and curvatures.
We now define connections and curvatures on twisted vector bundles. Before doing this, we recall some very basic facts about connections on vector bundles: we refer the reader to [17] for further details. If V is a complex C ∞ vector bundle on X of rank r, we use the notation A p (V ) for the space of p−forms on X with values in V , and A p (X) for the space of p−forms on X.
A connection on V is a C−linear map D : A 0 (V ) −→ A 1 (V ) such that for every f ∈ A 0 (X) and every s ∈ A 0 (E) we have If s = {s 1 , · · · , s r } is a local frame of V , then the connection form of D relative to s is a matrix Γ of 1−forms on X such that It is easy to see that to give a connection on V is equivalent to give an where a ij is the r × r−matrix of smooth functions on V representing φ ij with respect to s i and s j .
We introduce now the notion of connection on a twisted bundle (see [11], [5] and [29] for connections on gerbs, and [14] for connections on twisted vector bundles). Let E = {E i , φ ij } i,j∈I be an α−twisted complex C ∞ vector bundle of rank r.
for every i, j ∈ I if Γ i is a connection form of D i with respect to a local frame of E i , and if a ij is the matrix of smooth functions representing φ ij with respect to the chosen local frames, we have The motivation of the previous definition comes from the following remark: if V is a complex C ∞ vector bundle and D is a connection on it, take a family of connections form Γ i associated to local frames on an open covering of X, and let a ij be the matrix representing the transition function with respect to the given local frames. We then have If a ki a jk a ij = I r , this last line is Γ i . But in the twisted case we have that a ki a jk a ij = α ijk · I r , hence we get Γ i = Γ i + α −1 ijk dα ijk . In order to avoid this discrepancy we need to add ω ij · I r in the relation between Γ i and Γ j .
The existence of a connection on any α−twisted complex C ∞ vector bundle on M is granted by Example 7.2 of [14]. We present here a more general construction that will be used in what follows.
Proposition 2.13. Let E be an α−twisted complex C ∞ vector bundle on M . Then E admits a connection.
Proof. Write E = {E i , φ ij }, and let p = {p i } i∈I be a partition of the unity with respect to U. Choose a connection D i on E i , and let Γ i be the connection form of D i with respect to a chosen local frame of E i . We write a ij for the matrix of smooth functions representing φ ij with respect to these local frames.
We consider which are two matrices of 1−forms on U i . Notice that Now recall that a jk a ij = α ijk a ik , so Now, recall that a ki a jk a ij = α ijk · I r , so that a ik = α −1 ijk a jk a ij . It follows that We now let Γ i := Γ i + Φ i for every i ∈ I, so that Consider now the family D = { D i } i∈I where D i is the connection whose connection form is Γ i with respect to the given local frame: we then see that D is a connection on E.
Remark 2.14. The set of connections on an α−twisted complex C ∞ vector bundle E is an affine space over the vector space . As a consequence of this, any affine linear combination of connections on an α−twisted complex C ∞ vector bundle E is again a connection on E.
With respect to a local frame of E i we represent R i by a matrix Ω i of 2−forms. For each i ∈ I recall that Ω i = dΓ i + Γ i ∧ Γ i (see as instance section 1 in Chapter I of [17]). But then Definition 2.16. The 2−form R D ∈ A 2 (End(E)) is the curvature of D.

Connections and holomorphic structures.
Let us now fix a holomorphic structure, getting a complex manifold X. Let E = {E i , φ ij } be an α−twisted complex C ∞ vector bundle on X and D = {D i } a connection on E. The holomorphic structure on X gives a direct sum decomposition Composing D i with the two projections we get If Γ i is a connection form for D i with respect to a given frame, we have a natural decomposition Γ i = Γ 1,0 i +Γ 0,1 i since Γ i is a matrix of 1−forms. It follows from the definition of connection and the fact that Suppose now furthermore that E is an α−twisted holomorphic vector bundle, i. e. E i has a holomorphic structure and φ ij is an isomorphism of holomorphic vector bundles. We then represent the connection D i by a matrix Γ i of 1−forms with respect to a holomorphic local frame, and φ ij by a matrix a ij whose entries are holomorphic functions. In this case we get The holomorphic structure of E i corresponds to a semi-connection This is equivalent to asking that Γ 0,1 i = 0 for every i, or even that for every holomorphic section ξ of E i we have D i (ξ) = D 1,0 i (ξ) (see Proposition 3.9 in Chapter I of [17]).
The following shows that each twisted holomorphic vector bundle carries a connection compatible with its holomorphic structure. Lemma 2.18. Let E be an α−twisted holomorphic vector bundle on X. Then E admits a connection D compatible with its holomorphic structure, and if the B−field B is such that B 0,2 i = 0 for every i ∈ I, then R 0,2 where E i is a holomorphic vector bundle and φ ij is holomorphic for every i, j ∈ I. We know that E i admits a connection D i compatible with its holomorphic structure (see Proposition 3.5 in Chapter I of [17]). We let Γ i be its connection form with respect to a holomorphic local frame of E i . Consider moreover a partition of the unity p = {p i } relative to U.
The proof of Proposition 2.13 tells us that if we let D = { D i } whose connection form, with respect to the given holomorphic local frames, is then D is a connection on E. Notice that since Γ 0,1 j = 0 by the fact that D is compatible with the holomorphic structure of E, and ∂a ij = 0 since a ij is a matrix of holomorphic functions. It follows that D is compatible with the holomorphic structure of E.
To conclude, notice that R 0,2 D = 0 if and only if its restriction to U i is 0 for every i ∈ I, i. e. if and only if ( Now, recall that as D 0,1 i = ∂ i we have R 0,2 i = 0 (see Proposition 3.5 in Chapter I of [17]), and we are done.
A converse of the previous Lemma holds too. Lemma 2.19. Let E be an α−twisted complex C ∞ vector bundle on X and D a connection on E. Suppose that the B−field B = {B i } is such that B 0,2 i = 0 for every i ∈ I, and that R 0,2 D = 0. Then there is a unique holomorphic structure on E with which D is compatible.
Proof. As in the proof of Lemma 2.18, the fact that R 0,2 D = 0 and that B 0,2 i = 0 imply that R 0,2 i = 0 for every i ∈ I. Proposition 3.7 in Chapter I of [17] then implies the existence of a unique holomorphic structure on E i with which D i is compatible.
We now need to prove that φ ij is holomorphic with respect to the holomorphic structures of E i and E j . Let Γ i be the connection form of D i with respect to a holomorphic local frame of E i , and a ij the matrix of smooth functions representing φ ij with respect to the chosen local frames of E i and E j . Then Γ 0,1 i = 0, and since Γ i = a −1 ij Γ j a ij + a −1 ij da ij + ω ij · id E i , by multiplying by a ij on both sides we then get But as ω ij is a (1, 0)−form, we see that da ij is a matrix of (1, 0)−forms: hence a ij is a matrix of holomorphic functions, and φ ij is holomorphic.
2.5. Hermitian metrics and connections. We now introduce the notion of Hermitian metric on a twisted bundle. We recall that if V is a complex C ∞ vector bundle on X, a Hermitian metric on V is a C ∞ field of positive definite Hermitian products on the fibers of V .
It follows that there is no discrepancy on U ijk , and the definition makes sense.

Remark 2.22.
If H i is the matrix of smooth functions representing h i with respect to a given local frame of E i , and a ij is the matrix of smooth functions representing φ ij with respect to the chosen local frames of E i and E j , then H i is a Hermitian matrix and H i = T a ij H j a ij .
We first show that Hermitian metrics exist on every α−twisted C ∞ vector bundle: Lemma 2.23. Let E be an α−twisted C ∞ vector bundle. Then E admits a Hermitian metric.
Proof. Let h i be a Hermitian metric on E i , and p = {p i } i∈I a partition of the unity with respect to U. Let But since φ jk • φ ij = α ijk · φ ik and |α ijk | = 1, we see that As for untwisted vector bundles, we look for relations between Hermitian metrics and connections. More precisely, let E = {E i , φ ij } be an α−twisted complex C ∞ vector bundle, D = {D i } a connection on E and h = {h i } a Hermitian metric on E.
Definition 2.24. We say that D and h are compatible (or that D is a h−connection) if for every i ∈ I we have that D i is a h i −connection, i. e. for every sections ξ and η of E i we have Representing h i and D i by matrices H i (of smooth functions) and Γ i (of 1−forms) with respect to a chosen local frame of E i , this reads as Using this we prove the following: (1) There is a unique connection D on E which is compatible both with the holomorphic structure of E and the Hermitian metric h. (2) If the B−field B is such that B i is a (1, 1)−form for every i ∈ I, then R D ∈ A 1,1 (End(E)).
Proof. As E i is a holomorphic vector bundle on U i and h i is a Hermitian connection on E i , there is a unique connection D i on E i which is compatible with the holomorphic structure of E i and with the Hermitian metric h i . If Γ i is a connection form of D i and H i is a matrix representing h i with respect to a given local frame of E i , we know that We will moreover let a ij be the matrix of smooth functions representing φ ij with respect to the chose local frames. Let p = {p i } be a partition of the unity with respect to U. The proof of Lemma 2.18 tell us that if we let D be the connection on E whose connection form (with respect to the local frame given above) is then D is compatible with the holomorphic structure of E.
It only remains to show that D and h are compatible. Notice that Using the fact that which proves that D is compatible with Hermitian metric h. Let now R D be the curvature of D: as D is compatible with the holomorphic structure of E, we know from Lemma 2.18 that R 0,2 D = 0. Moreover, for every i ∈ I we have where R i is the curvature of D i . As D i is compatible with h i we know that also R 2,0 i = 0 (see section 4 in Chapter I of [17]). But since B i is a (1, 1)−form by hypothesis, it follows that R D ∈ A 1,1 (End(E)). Now, if E is an α−twisted holomorphic vector bundle and h is a Hermitian metric on E, the previous Lemma allows us to give the following: Definition 2.26. The unique connection D on E which is compatible with h and with the holomorphic structure of E is called the Chern connection of the pair (E, h), and its curvature will be called Chern curvature of the pair (E, h). We will sometimes use the notation D h and R h for them.
Notice that by definition we have that D = {D i } is the Chern connection of (E, h) if and only if D i is the Chern connection of (E i , h i ). As an immediate consequence of Lemma 2.19 we get the following converse of Lemma 2.25: Lemma 2.27. Let E be an α−twisted complex C ∞ vector bundle on X, h a Hermitian metric on E and D a connection on E compatible with h. If the B−field B is such that B i is a (1, 1)−form for every i ∈ I, then there is a unique holomorphic structure on E so that D is the Chern connection of (E, h).

2.6.
Connections and metrics on associated bundles. We resume here the basic facts about how a connection (or a Hermitian metric) on a twisted vector bundle E induces a connection (or a Hermitian metric) on twisted vector bundles that may be constructed from E.
In particular D i is a connection on E i , so we may use it to produce a connection D * i on E * i : for every local section ξ of E * i , we need to define a 1−form D * i (ξ) with coefficients in E * i . We then define D * i (ξ) by expressing the 1−form (D * i (ξ), η) obtained by evaluating the coefficients of D * i (ξ) on η. We then let ). If Γ i is the connection form of D i with respect to a local frame of E i , the connection form of D * i is −Γ i with respect to the dual local frame (see section 5 in Chapter I of [17]).
Proof. For every local sections ξ (of E * i ) and η (of ij be the matrix representing φ * ij , and Γ * i the connection form of D * i with respect to the dual local frame. In conclusion, we get i is the curvature of the dual connection D * i (here we use the fact that R * i = −R i , see section 5 in Chapter I of [17]).
The connection D * is called the dual connection of D, or equivalently connection induced by D on E * .
Let now h = {h i } be a Hermitian metric on E. Then h i is a Hermitian metric on E i , i. e. an isomorphism h i : i is the Chern connection of (E * i , h * i ), so that D * is the Chern connection of (E * , h * ) (see Lemma 2.25).
The Hermitian metric h * is called the dual Hermitian metric of h, or equivalently Hermitian metric induced by h on E * .
vector bundles, and consider a connection D = {D i } on E and a connection D = {D i } on F . In particular D i is a connection on E i and D i is a connection on F i , so we may use them to produce a connection D i ⊕ D i on E i ⊕ F i : a local section of E i ⊕ F i is of the form ξ ⊕ ξ for a local section ξ of E i and a local section ξ of F i , so we let If Γ i is the connection form of D i with respect to a local frame of E i and Γ i is the connection form of D i with respect to a local frame of F i , then Proof. It is easy to see that in Chapter I of [17]). We then get R D⊕D = R D ⊕ R D , and we are done.
The connection D ⊕ D is called the direct sum connection of D and D , or equivalently connection induced by D and D on E ⊕ F . Let now h = {h i } be a Hermitian metric on E and h = {h i } a Hermitian metric on F . Then h i is a Hermitian metric on E i and h i is a Hermitian metric on F i , and we define the sum Hermitian metric for every local sections ξ, ξ of E i and η, η of F i . Lemma 2.31. Let E and F be two α−twisted holomorphic vector bundles on X, h a Hermitian metric on E and h a Hermitian metric on F . The family h⊕h = {h i ⊕h i } i∈I is a Hermitian metric on E ⊕F , and D h⊕h = D h ⊕D h .
Proof. If H i and H i represent h i and h i with respect to local frames of E i and F i , then The Hermitian metric h⊕h is called the direct sum Hermitian metric of h and h , or equivalently Hermitian metric induced by h and h on E ⊕ F .
Tensor product. Let E = {E i , φ ij } be an α−twisted complex C ∞ vector bundle and F = {F i , ψ ij } be an α −twisted complex C ∞ vector bundle. Consider a connection D = {D i } on E and a connection D = {D i } on F . In particular D i is a connection on E i and D i is a connection on F i , so we may use them to produce a connection D i ⊗ D i on E i ⊗ F i : we let If Γ i is the connection form of D i with respect to a local frame of E i and Γ i is the connection form of D i with respect to a local frame of F i , then Γ i ⊗ I s + I r ⊗ Γ i is the connection form of D i ⊗ D i with respect to the corresponding local frame of E i ⊗F i (where A⊗B is the Kronecker product).
Proof. We know that D i ⊗ D i is a connection on E i ⊗ F i , and an easy calculation shows that where a ij and b ij are matrices of smooth functions representing φ ij and ψ ij respectively with respect to local frames of E i and F i . Hence D ⊗ D is a connection on E ⊗ F . Now, let B a B−field inducing the twist α and B a B−field inducing the twist α . Then B + B = {B i + B i } i∈I is a B−field inducing the twist αα , and we have [17]), which implies the statement.
The connection D ⊗ D is called the tensor product connection of D and D , or equivalently connection induced by D and D on E ⊗ F .
Let now h = {h i } be a Hermitian metric on E and h = {h i } a Hermitian metric on F . Then h i is a Hermitian metric on E i and h i is a Hermitian metric on F i , and we define the product Hermitian metric so h ⊗ h is a Hermitian metric on E. The remaining part of the proof is straightforward.
The Hermitian metric h ⊗ h is called the tensor product Hermitian metric of h and h , or equivalently Hermitian metric induced by h and h on E ⊗ F .
As a particular case, if E is an α−twisted complex C ∞ vector bundle E and two integers p, q ≥ 0, we let which is then an α p−q −twisted complex C ∞ vector bundle. If D is a connection on E, it induces a connection D p,q on E p,q , and if h is a Hermitian metric on E, it induces a Hermitian metric h p,q on E p,q .
The most important case to consider is when p = q = 1, in which case E 1,1 = E ⊗ E * = End(E): this is a usual complex C ∞ vector bundle. If D is a connection on E, the connection D 1,1 is a usual connection on a vector bundle, and if h is a Hermitian metric on E, then h 1,1 is a usual Hermitian metric on a vector bundle. The Chern connection of (E 1,1 , h 1,1 ) is the Chern connection of (End(E), h 1,1 ).
Wedge product. Let E be an α−twisted complex C ∞ vector bundle on X and p a strictly positive integer. Consider a connection D = {D i } on E. The product connection D ⊗p on E ⊗p is easily seen to verify the following: if ξ is a section of p E i , then D ⊗p i (ξ) ∈ A 1 (∧ p E i ). It follows that D ⊗p |∧ p E is a connection on p E, denoted D p and called wedge connection on E, or equivalently connection induced by D on ∧ p E.
Let now h = {h i } be a Hermitian metric on E. The induced Hermitian metric h ⊗p restricted to ∧ p induces a Hermitian metric, denoted h p , and called wedge Hermitian metric on E, or equivalently Hermitian metric induced by h on ∧ p E. The following is immediate: Lemma 2.34. Let E be an α−twisted holomorphic vector bundle on X and h a Hermitian metric on E. Then D h p = D p h . Particular case is when p is the rank r of E, in which case we have which is an α r −twisted complex C ∞ line bundle on X. If D is a connection on E, it induces a connection det(D) on det(E), called determinant connection, and if h is a Hermitian metric on E, it induces a Hermitian metric det(h) on det(E), called determinant Hermitian metric. We moreover have D det(h) = det(D h ).
Pull-back. Let now X and Y be two complex manifolds and f : X −→ Y be a holomorphic map between them.
In particular D i is a connection on E i , so we may use it to produce a connection f * D i on f * E i : to define it, we notice that if ξ is a local section of f * E i , then there is a unique local section ξ of E i such that ξ , where on the right we have the pull-back under f of the 1−form D i (ξ ) with coefficients in E i . If Γ i is the connection form of D i with respect to a local frame of E i , then f * Γ i is the connection form of f * D i with respect to the pull-back local frame.
The connection f * D is called the pull-back connection of D, or equivalently connection induced by D on f * E.
Let now h = {h i } be a Hermitian metric on E. Then h i is a Hermitian metric on E i , i. e. an isomorphism of complex C ∞ vector bundles h i : i is an isomorphism of complex C ∞ vector bundles, getting a Hermitian metric f * h i . The following is immediate. The Hermitian metric f * h = {f * h i } is called pull-back Hermitian metric or Hermitian metric induced by h on f * E.

2.7.
Subbundles and quotients. Let E = {E i , φ ij } be an α−twisted holomorphic vector bundle on a complex manifold X, and let r be its rank.
Definition 2.37. A twisted holomorphic subbundle of E is an α−twisted holomorphic vector bundle S = {S i , ψ ij } on X such that for every i ∈ I we have an injective morphism of α−twisted holomorphic vector bundles f : Let now S be a twisted holomorphic sub-bundle of E, and let f : S −→ E be the inclusion. For every i ∈ I we then may consider the quotient vector bundle Q i := E i /S i , and we let It is easy to verify that ϕ ij is a well-defined isomorphism of holomorphic vector bundles on U ij , and that Q = {Q i , ϕ ij } is an α−twisted holomorphic vector bundle, called quotient of E by S.
Moreover, for every i ∈ I we have a natural projection p i : E i −→ Q i , and we have ϕ ij • p i = p j • φ ij . The family p = {p i } i∈I is then a morphism p : E −→ Q of α−twisted holomorphic vector bundles, called projection.
We notice that we have an exact sequence of α−twisted holomorphic vector bundles 2.7.1. Hermitian metrics and orthogonals. Let now h = {h i } be a Hermitian metric on E. As f i : where we used the fact that f is a morphism of α−twisted holomorphic vector bundles. As a consequence, the family h S is a Hermitian metric on S.
For every i ∈ I and every x ∈ U i we define As h is a Hermitian metric on E, this holds if and only if h i,x (s, f i,x (t)) = 0, and this last holds as s ∈ S ⊥ i,x .
As a consequence we see that ψ ⊥ ij : S ⊥ i|U ij −→ S ⊥ j|U ij . By definition, this map is the restriction of a biholomorphism to a complex C ∞ sub-bundle, hence ψ ⊥ ij is injective and C ∞ . We need to show that it is surjective.
In any case, as for every we have an isomorphism of α−twisted complex C ∞ vector bundles between Q and S ⊥ . We then have an exact sequence The morphism ϕ is then an injective morphism of α−twisted complex C ∞ vector bundles, i. e. Q is an α−twisted complex C ∞ subbundle of E. We then may use π to define a Hermitian metric h Q on Q. The Hermitian metrics h S and h Q are called Hermitian metrics induced by h on S and Q.

Connections and orthogonals.
Let now E, S and Q as before, consider a Hermitian metric h on E, and h S and h Q the induced Hermitian metrics. Let D be the Chern connection of (E, h), and consider the exact sequence of α−twisted holomorphic vector bundles, and the exact sequence ). We then get two maps i } i∈I is the Chern connection of (S, h S ), and the maps A i glue together to form an element A ∈ A 1,0 (Hom(S, Q)).
Proof. It is known that D S i is the Chern connection of (S i , h S i ) (see Proposition 6.4 in Chapter I of [17]). Choose now a local frame , · · · , f i (t s )} may be completed to a local frame t i of E i . Let Γ i be the connection form of D i with respect to t i , and let F i be the matrix representing f i with respect to t i and t i , and Π i be the matrix representing π i with respect to t i and t i .
If Γ S i is the connection form of D S i with respect to t i , we then have where a ij is the matrix representing φ ij with respect to t i and t j , and b ij is the matrix representing ψ ij with respect to t i and t j . But as showing that D S is a connection on S. It then follows that D S is the Chern connection of (S, h S ).
Moreover, we know that A i ∈ A 1,0 (Hom(S i , Q i )) (see Proposition 6.4 in Chapter I of [17]). Complete t i as t i = {t 1 , · · · , t s , t s+1 , · · · , t r }, so that t i = {p i (t s+1 ), · · · , p i (t r )} is a local frame of Q i . Let P i be the matrix representing p i with respect to t i and t i . If we represent A i by a matrix A i with respect to t i and t i , we have In a similar way one produces a connection D Q on Q as D Q = {D Q i }, which turns out to be the Chern connection of (Q, h Q ), and an element C ∈ A 0,1 (Hom(Q, S)). The form A is called second fundamental form of S, and the form Q is called second fundamental form of Q.
Let now E be an α−twisted holomorphic vector bundle on X and D a connection on it.
The following will be used in the proof of the Kobayashi-Hitchin correspondence.
Lemma 2.42. Le E be an α−twisted holomorphic vector bundle on X, h a Hermitian metric on E and D the Chern connection of (E, h). Suppose that E is a D−invariant subbundle of E, and let E := (E ) ⊥ . Then E and E are both α−twisted holomorphic subbundles of E, and the direct sum Proof. This is an immediate consequence of the untwisted analogue, see Proposition 4.18 in Chapter I of [17].
An immediate corollary of this is the following: Corollary 2.43. Let E be an α−twisted holomorphic vector bundle on X, h a Hermitian metric on E and D the Chern connection of (E, h). Suppose that S is an α−twisted holomorphic subbundle, and let A ∈ A 1,0 (Hom(S, S ⊥ )) be as before. If A = 0, then S ⊥ is an α−twisted holomorphic subbundle which is isomorphic, as an α−twisted holomorphic bundle, to Q.
We end this section with the Gauss-Codazzi equations in the twisted setting. Let E be an α−twisted holomorphic vector bundle, h a Hermitian metric on E, D the Chern connection of (E, h), S a twisted holomorphic sub-bundle of E, Q the quotient of E by S and A ∈ A 1,0 (Hom(S, Q)), C ∈ A 0,1 (Hom(Q, S)) as before.
For every i ∈ I we let R i be the curvature of D i , R S i the curvature of D S i and R Q i the curvature of D Q i of (Q i , h Q i ). By Lemma 2.40, the Gauss-Codazzi equations in the untwisted setting give where we let D 1,0 i and D 0,1 i for the (1, 0)−part and the (0, 1)−part of the connection induced by D i on Hom(S i , Q i ) and Hom(Q i , S i ) (see section 6 in Chapter I of [17]).
We notice that the Hom(S i , Q i )'s glue together to give a holomorphic vector bundle Hom(S, Q), and D induces the Chern connection on it: hence the D 1,0 i 's glue together to give the (1, 0)−part of the Chern connection on this bundle (and similarly for the Now all the forms in the formula glue together to give where R S := R D S and R Q = R D Q . These are called the twisted Gauss-Codazzi equations.
2.8. Hermitian forms and Hermitian endomorphisms. We now define the notion of Hermitian form on an α−twisted complex C ∞ vector bundle E.
Hermitian metrics on E are clearly examples of Hermitian forms. Another useful example is the following.
Example 2.45. Take A ⊆ R an interval and consider a differentiable family h = {h t } t∈A of Hermitian metrics on E, i. e. a family of Hermitian metrics such that if we write h t = {h t,i } i∈I and represent h t,i by a matrix H t,i with respect to a local frame, then the entries of H t,i are differentiable in t.
For every t ∈ A let V t,i := ∂ t H i,t , i. e. the matrix whose entries are the derivatives of the entries of H i,t with respect to t. We then let v t,i be the form on E i represented by the matrix V i with respect to the given local frame, i. e. v t,i = ∂ t h i,t .
As h t,i is a Hermitian metric, it is easy to see that v t,i is a Hermitian form We will use the notation is a family of Hermitian forms on E, called derivation of h.

Endomorphism from a Hermitian metric.
Let h be a Hermitian metric in E, and v a Hermitian form on E. For every i ∈ I let us define an . The fact that h i is a Hermitian metric and v i is a Hermitian form imply that For every x ∈ U ij and every a, b ), by using the fact that h is a Hermitian metric, and ). As φ ij is an isomorphism, this implies that for every a ∈ E i,x and every b ∈ E j,x we have ). As this holds for every x ∈ U ij and for every a ∈ E i,x we then finally get 46. The endomorphism f h,v will be called endomorphism associated to h and v.
Choosing a local frame on U i and representing v i by a matrix V i and h i by a matrix H i , we see that f h,v is represented by the matrix H −1 i V i . Remark 2.47. If h and k are two Hermitian metrics, the endomorphism f h,k is an automorphism whose inverse is f k,h . Indeed, for every i ∈ I and sections ξ, η of E i , we have Remark 2.49. If v 1 , v 2 are Hermitian forms on E, λ 1 , λ 2 ∈ R and h is a Hermitian metric on E, then Indeed, for every i ∈ I and every two sections ξ, η of E i we have But as we have we conclude.
Remark 2.50. If h is a Hermitian metric on E and v is a Hermitian form on E, then f h,v is diagonalizable and has the same signature of v. Indeed, consider a local frame σ of E i with respect to which we represent h i and v i by Hermitian matrices As H i is positive definite and V i is diagonalizable, by classical linear algebra (see as instance Theorem 7.6.3 of [12]) their product F h,v i is diagonalisable, has real eigenvalues and same signature of V i , proving the claim.
In particular, if k is a Hermitian metric, then f h,k is diagonalizable and its eigenvalues are all strictly positive smooth function. It then makes sense to consider log(f h,v ) and (f h,v ) σ for every σ ∈ (0, 1].
A particular example of this construction is obtained by taking a differentiable family h = {h t } t∈A . By Example 2.45 we know that h t is a Hermitian form on E for every t ∈ A, and hence we may consider the endomorphism f ht,h t of E: we then get a function If the family h of Hermitian forms is differentiable as well, then f h is differentiable.

Hermitian endomorphisms.
A converse of the previous construction is also possible. Before, recall the following definition: We let End h (E) be the set of h−Hermitian endomorphisms of E, which is easily seen to be a real vector space. We first provide an easy example of h−Hermitian endomorphism: Example 2.52. Let h be a Hermitian metric on E and v a Hermitian form on E. The endomorphism f h,v is h−Hermitian: indeed, by definition for every i ∈ I and every sections ξ and η of E i we have ). If h and k are both Hermitian metrics on E, then f h,k is moreover k−Hermitian. Indeed, for every i ∈ I and ξ, η two sections of E i we have h is a Hermitian metric on E and f ∈ End h (E), then we define a Hermitian form f h on E as follows: for every sections ξ, η of E i we let It is easy to see that as , for every i ∈ I and every sections ξ, η of E i .
Conversely, if f a h−Hermitian endomorphism of E we have f = f h, f h . Indeed, for every i ∈ I and every sections ξ, η of E i we have A useful remark is about the eigenvalues of Hermitian endomorphisms: Indeed if λ is an eigenvalue of f , and if s is an eigenvector of eigenvalue λ over U i , then But as s is nowhere vanishing we have h i (s, s) > 0, so λ = λ.
As a consequence, it makes sense to consider the subset End + h (E) of End h (E) given by h−Hermitian endomorphisms of E whose eigenvalues are all strictly positive: it is easy to see that it is a convex domain in End h (E).
Finally, we remark the following: 56. Let f be a h−Hermitian diagonalizable endomorphism of E whose eigenvalues are λ 1 , · · · , λ r , and ϕ : R −→ R is a smooth function such that the image of λ 1 , · · · , λ r lies in the definition domain of ϕ. Then ϕ(f ) is a h−Hermitian diagonalizable endomorphism.
Proof. By Lemma 2.10 we have that ϕ(f ) is a diagonalizable endomorphism. Let σ i = {ξ 1 , · · · , ξ r } be a local frame of E i diagonalizing f i , and λ j be the eigenvalue corresponding to ξ j . If we take ξ, η two sections of E i , write Notice that as f is h−Hermitian, by Remark 2.55 we get If h i (ξ j , ξ k ) = 0, we then get λ j = λ k , so ϕ(λ j ) = ϕ(λ k ) and hence ϕ(f i ) is h i −Hermitian, concluding the proof.
2.9. Space of Hermitian metrics. Let now E = {E i , φ ij } be an α−twisted complex C ∞ vector bundle of rank r on X. We let Herm(E) be the set of Hermitian forms on E, and Herm + (E) be the set of Hermitian It is easy to see that v + w and λv are Hermitian forms on E, and that under these two operations Herm(E) is a real vector space (of dimension r 2 ). Moreover, the subset Herm + (E) of Herm(E) is a convex domain in Herm(E), since the same holds for Hermitian metrics on vector bundles. As a consequence, we see that if h ∈ Herm + (E), then we may view Herm(E) as the tangent space of Herm + (E) at h, i. e.
We have an action of the group of automorphisms of E on Herm(E). More precisely, we let GL(E) be the group of automorphisms of E, that we will call complex gauge group of E, and define the action We will moreover let gl(E) be the Lie algebra of GL(E), i. e. the Lie algebra of global sections of End(E). If k ∈ Herm + (E), we let i. e. the stabilizer of k under the action of GL(E): this will be called complex gauge group of the pair (E, k), and we let and that This implies that for every two local sections ξ and η of E i we have The previous Lemma allows us to identify Herm + (E) with the quotient GL(E)/U k (E), so we may consider Herm + (E) as a symmetric space with respect to the involution mapping Remark 2.58. We notice that if h is a Hermitian metric on E, the map is an isomorphism of real vector spaces: it is linear by Remark 2.49, and by Example 2.54 its inverse is If k is a Hermitian metric, then by Remark 2.50 we know that f h,k is positive definite, i. e. we have Conversely, the Spectral Theorem implies that if f is a positive definite h−Hermitian endomorphism, we have It follows that f h is a Hermitian metric, and that λ induces an identification between Herm + (E) and End + h (E). The previous Remark 2.58 allows us to provide an action of GL(E) on End h (E) and End + h (E), which is easily described by the following: Remark 2.59. If h ∈ Herm + (E), v ∈ Herm(E) and a ∈ GL(E), then we have ah ∈ Herm + (E) and av ∈ Herm(E). The associated endomorphism Indeed, for every i ∈ I and every two sections ξ, η of 2.9.1. Riemannian metric and geodesics. Now, let h ∈ Herm + (E), and take v, w ∈ Herm(E). As already noticed, we have Herm(E) = T h (Herm + (E)), i. e. we may view v, w as tangent vectors to Herm + (E) at h. We now want to define a metric on T h (Herm + (E)), so to have a metric on the space To do so, suppose that X is a compact complex manifold of dimension n and that g is a Kähler metric on X, whose associated (1, 1)−form is denoted Lemma 2.60. For every h ∈ Herm + (E) the map is a positive definite bilinear symmetric product, which depends smoothly on h and which is GL(E)−invariant.
Proof. Linearity follows from Remark 2.49, while symmetry is a consequence of the properties of the trace.
is the Hilbert-Schmidt inner product, and hence it is positive definite. The GL(E)−invariance is an immediate consequence of Remark 2.59, while the smooth dependence on h is obvious.
As a consequence of Lemma 2.60 we get a GL(E)−invariant Riemannian metric on Herm + (E). For every h ∈ Herm + (E) and for every v ∈ T h Herm + (E) we let Now, consider a < b two real numbers and h, k ∈ Herm + (E). We let and a function We notice that The critical points of this functional correspond then to the geodetics in which is a piecewise differentiable function for every s. Moreover, if s 1 h,k (E): this element is a small deformation of h in the direction of v.
We then have so an easy calculation gives As h is a critical point for the functional E if and only if i with respect to the same local frame: here we let H t,i be the matrix whose entries are the derivatives in t of the entries of H t,i . Hence we have that Proof. For every i ∈ I we have a natural morphism η i : the last being the sheaf of Hermitian forms on the (untwisted) vector bundle E i . The sheaf H E i is known to be locally free of rank r 2 , and we now show that η i is an isomorphism: this will imply the statement. For It is immediate to verify that η i,x is injective. For the surjectivity, let β ∈ Herm(E i,x ), and take an open subset U of U i containing x and a Hermitian form h i on U such that h i,x = β. For every j ∈ I we then let ij , which is a Hermitian form on E j|U , and we see that h := {h j } j∈I is a Hermitian metric on E |U whose germ at x has image β under η i,x .
As H E is a locally free sheaf of C ∞ X −modules of rank r 2 on X, there is a C ∞ real vector bundle H E of rank r 2 corresponding to it, whose space of global sections is Herm(E).
2.9.3. Norms. We now introduce various norms on the space of p−forms with values in a vector bundle: we refer the reader to [17] and to [21], chapter 7, for more details.
Let V be a real vector bundle of rank s on a differentiable compact manifold X of dimension d. Let g be a Riemannian metric on X and h a fiber metric on V . Recall that on A * (V ) we have a pointwise inner product associated to g and h, and defined as follows: which is a smooth function on M . In particular, to every ξ ∈ A * (V ) we associate a smooth function We then have a L p −norm on A * (V ) defined as Remark 2.62. As a particular case, if E is an α−twisted vector bundle on X and ξ ∈ A p (End(E)), if h is a Hermitian metric on E and g is a Hermitian metric on X, we associate to ξ a smooth function |ξ| on X, which has the property that |ξ| 2 σ d g = T r(ξ ∧ * ξ * ), where ξ * is the adjoint of ξ, and * ξ * is the * −Hodge of ξ * (see section 3.2).
Let now ∇ be the Levi-Civita connection induced by g on the tangent bundle T X of X, and consider a connection D on V which is compatible with h. For every k ∈ N, the covariant derivative of the connection induced on Sym k (Ω X ) ⊗ V (where Ω X is the cotangent bundle of X) gives a linear map The composition of these maps gives then a linear map Using the metric on Sym k (Ω X ) induced by h and g we then may define for every ξ ∈ A 0 (V ), and for every p, q ∈ N a L p q −norm as follows: We will let L p q (V ) be the completion of A 0 (V ) with respect to this norm. A similar construction may be done in a relative context. More precisely, let a > 0 be a real number, and consider the projection π a : X ×[0, a] −→ X: the pull-back π * a V is a real smooth vector bundle of rank s on X × [0, a], whose global sections are smooth curves in the space of global sections of Take a smooth family of fiber metrics {h t } t∈[0,a] on V , and let D t be a connection on V compatible with h t , and we suppose that the family {D t } t∈[0,a] is smooth. Consider a global section f of π * a V , and notice that This is a norm on the space of global sections of π * V , and we let where d(z, w) is the distance between z and w, and X a = X × [0, a].
All these definitions will be applied to the real vector bundle H E (of rank r 2 ) on a compact Kähler manifold X of (real) dimension 2n. The metric h on H E is the natural Riemannian metric we defined on Herm(E).

Hermite-Einstein condition
In this section we introduce the notion of Hermite-Einstein and approximate Hermite-Einstein twisted vector bundles: both notions will be identical to the corresponding notions for untwisted bundles.
To do so, we will first need to introduce Chern forms and the Chern classes of twisted holomorphic vector bundles, which will be defined starting from the choice of a connection on a twisted bundle. We start with some preliminary notation.
Let V be a complex vector space and k ∈ N. We let f k : V k −→ C be a symmetric multilinear form of degree k on V , and define the associated degree k homogeneous polynomial. If G is a linear group acting freely on V , we say that f k is G−invariant if for every g ∈ G and for every v 1 , · · · , v k ∈ V we have A particular case is when V = gl r (C), the Lie algebra of G = GL r (C), on which G acts by conjugation, i. e.
We now define r homogeneous polynomials F 1 , · · · , F r on V by letting F k (X) be the homogeneous part of degree k of i. e. we have hence it follows that F 1 , · · · , F r are G−invariants.
3.1. Chern classes from Chern connection. Let E be an α−twisted holomorphic vector bundle of rank r on a complex manifold X, and consider ω ∈ A 2 (End(E)). Choose a local frame s of E i over an open subset U ⊆ U i , with respect to which we represent ω |U by a matrix Ω U of 2−forms. For every k ∈ N we then let If s is a local frame of E j over an open subset U ⊆ U j , then: • if i = j, then s are s are two local frames of E i over U ∩ U , and Ω U and Ω U represent the same 2−form with values in End(E i ) with respect to these two local frames. There is then an invertible matrix M of smooth functions such that Ω U = M Ω U M −1 (see section 1 in Chapter I of [17]). • If i = j and U ∩ U = ∅, let a ij be the matrix representing φ ij with respect to s and s . We then have Ω U = a −1 ij Ω U a ij . In any case, we see that there is an invertible matrix A of smooth functions If now E is an α−twisted holomorphic vector bundle E and D is a connection on it, then R D ∈ A 2 (End(E)), so the previous construction gives us Proof. Let D = {D i }, and choose an open covering U of X as before, i. e. U = {U j } j∈J is such that for every j ∈ J there is i ∈ I such that U j ⊆ U i , and there is a local frame s j of E i over U j . Represent R D|U j with respect to s j by a matrix Ω j of 2−forms, and the curvature R i of D i by a matrix Ω j of 2−forms. We have If Γ j is the connection form of D i with respect to s j , the Bianchi identity for D i (see section 1 in Chapter I of [17]) gives We finally get dγ k (R D ) = 0.
We now analyze how γ k (R D ) varies with D.
Proof. By Remark 2.14, for every t ∈ [0, 1] the affine linear combination be an open covering of X as in the proof of Lemma 3.1, and let Γ j be the connection form of D, Γ j the connection form of D and Γ j,t the connection form of D t with respect to a local frame over U j . We clearly have Γ j,t := (1 − t)Γ j + tΓ j .
If ∆ j := Γ j − Γ j , then Γ j,t = Γ j + t∆ j and by Remark 2.14 we see that the ∆ j represents a global δ ∈ A 1 (End(E)) with respect to the given local frame. As in the proof of Lemma 3.1 we represent R D by a matrix Ω j , R D by a matrix Ω j , and the curvature R t of D t by a matrix Ω j,t . Moreover, we represent the curvature of D i (resp. of D i , of D i,t ) by a matrix Ω j (resp. Ω j , Ω j,t ), so that we have We then have Let us now consider the smooth (2k − 1)−form on U j defined by As the ∆ j 's glue together to form a global 1−form, it is easy to prove that if U j is another open subset, then ϕ k,j|U jj = ϕ k,j |U jj , so there is a unique We now have But this implies that γ k (R D ) − γ k (R D ) = dϕ k , and we are done.
As a consequence, the k−th Chern class of E does not depend on D, so we will write it as c k (E), and call it k−th Chern class of E.
If now E is an α−twisted holomorphic vector bundle and h is a Hermitian metric on E, we may choose the Chern connection D h in order to calculate the Chern forms and the Chern classes. We will use the notation γ k (E, h) instead of γ k (R D h ), and call it the k−th Chern form of (E, h).
As the Chern curvature of (E, h) is a (1, 1)−form, it follows that γ k (E, h) is a (k, k)−form, and hence the k−th Chern class of E is c k (E) ∈ H k,k (X).

Mean curvature.
If M is a C ∞ differentiable manifold, a Hermitian metric g on T M ⊗ C (the complexified tangent bundle of M ) is called Hermitian metric on M . A holomorphic structure on M is a holomorphic structure on T M , so that the pair given by the differentiable manifold M and the given complex structure is a complex manifold X, whose tangent bundle is denoted T X (i. e. it is the tangent bundle of M with the given complex structure).
The Hermitian metric g on X induces a Hermitian metric, still denoted g, on k T X and k Ω X for every k, where Ω X is the cotangent bundle of X. In particular, if ξ, η are two k−forms on X, i. e. two local sections of k Ω X , we may calculate g(ξ, η), which is a smooth function on X. If X has dimension n, then for every smooth k−form ξ on X there is a unique smooth (2n − k)−form on X, denoted * ξ, such that for every smooth k−form η on X we have η ∧ * ξ = g(η, ξ) · σ n g , where σ g is the real (1, 1)−form associated to g and to the complex structure of X, i. e. if z 1 , · · · , z n are local holomorphic coordinate on an open subset U of X and g ij = g(∂/∂z i , ∂/∂z j ), then If ξ is a (p, q)−form, then * ξ is a (n − q, n − p)−form, so we have the Hodge * −operator * : A p,q (X) −→ A n−q,n−p (X), whose conjugate is * : A p,q (X) −→ A n−p,n−q (X).
We then have an inner product Another useful operator is the Lefschetz operator for every ξ ∈ A p,q (X) and η ∈ A p+1,q+1 (X). A formula for the adjoint of the Lefschetz operator is It is easy to extend all these operators to (p, q)−forms with coefficients in any vector bundle on X.
Let now E be an α−twisted holomorphic vector bundle on X and h a Hermitian metric on E. Suppose that the B−field B is given by d−closed, purely imaginary (1, 1)−forms, and let moreover R h be the Chern curvature of (E, h). We define K g (E, h) := iΛ g (R h ).
As R h ∈ A 1,1 (End(E)), we have that K g (E, h) ∈ A 0 (End(E)), i. e. it is an endomorphism of the complex C ∞ vector bundle End(E).
Definition 3.4. The endomorphism K g (E, h) is called g−mean curvature of (E, h).
By the very definition of Λ g we then see that Proof. By the very definition of K g (E, h) we have that is the mean curvature of the vector bundle E i with Hermitian metric h i . The mean curvature of a holomorphic vector bundle with respect to a Hermitian metric is known to be Hermitian with respect to that metric, Notice that if f : U i −→ R is a real smooth function, for every sections ξ, η of E i we have The Hermitian form associated to the Hermitian metric h and to the h−Hermitian endomorphism K g (E, h) is denoted K g (E, h) and called g−mean curvature Hermitian form of (E, h).
Another useful result is the following: Lemma 3.6. Let h ∈ Ω 0,1 h,k (E) be a differentiable family of Hermitian metrics on E. We then have Proof. Let h t := h(t) and D t the Chern connection of (E, h t ). Given a local frame of E i , let Γ i,t be the connection form of D i,t , and represent h i,t by a matrix H i,t . As (see the proof of Proposition 4.9 in Chapter I of [17]), we get that If we let V t,i := ∂ t H i,t , we get Written in another way we then get . Passing from the matrix notation to the usual notation we then get that since the matrices V t,i 's represent the Hermitian form h t with respect to the chosen local frames. Applying D 0,1 i to both sides, and using the fact that D 0,1 i = D 0,1 t,i = ∂ i (since both are Chern connections, and hence compatible with the holomorphic structure) we get then where R i,t is the curvature of the Chern connection D i,t of (E i , h i,t ).
As the B−field B = {B i } i∈I does not depend on t, i. e. we have ∂ t B i = 0. This implies that As both sides glue together to give global elements of A 1,1 (End(E)), we finally get ∂ t R t = ∂D 1,0 t f ht,h t . Applying iΛ g to both sides we conclude.
3.3. Hermite-Einstein metrics. Let now E be an α−twisted holomorphic vector bundle on X, and consider a Hermitian metric h on E.
Definition 3.7. The pair (E, h) verifies the weak g−Hermite-Einstein condition if there is a real function ϕ : X −→ R such that The function ϕ is called Einstein function of (E, h) relative to g. If ϕ is constant, we will say that (E, h) verifies the g−Hermite-Einstein condition, and the constant number ϕ will be called Einstein factor of (E, h) relative to g.
Let us first see some properties.
Proposition 3.8. If L is an α−twisted holomorphic line bundle on X and h is a Hermitian metric on L, then (L, h) verifies the weak g−Hermite-Einstein condition for every Hermitian metric g on X. Proof. By Lemma 2.29 if D is the Chern connection of (E, h), then D * is the Chern connection of (E * , h * ), and we have R D * = −R D . Hence and we are done. Proposition 3.10. If (E 1 , h 1 ) verifies the weak g−Hermite-Einstein condition with Einstein function ϕ 1 , and (E 2 , h 2 ) verifies the weak g−Hermite-Einstein condition with Einstein function ϕ 2 , then (E 1 ⊗E 2 , h 1 ⊗h 2 ) verifies the weak g−Hermite-Einstein condition with Einstein function ϕ 1 + ϕ 2 .
Proof. By Lemma 2.33 if D i is the Chern connection of (E i , h i ), then D 1 ⊗D 2 is the Chern connection of (E 1 ⊗ E 2 , h 1 ⊗ h 2 ), and we have The statement then follows readily.
Proof. By Lemma 2.31 if D i is the Chern connection of (E i , h i ), then D 1 ⊕D 2 is the Chern connection of (E 1 ⊕ E 2 , h 1 ⊕ h 2 ), and we have The statement then follows readily.
The following is an immediate consequence of Propositions 3.9, 3.10 and 3.11: Proposition 3.12. If (E, h) verifies the weak g−Hermite-Einstein condition with Einstein function ϕ, then (1) for every p, q ∈ N, (E p,q , h p,q ) verifies the weak g−Hermite-Einstein condition with Einstein function (p − q)ϕ; (2) for every p ∈ N, (∧ p E, ∧ p h) verifies the weak g−Hermite-Einstein condition with Einstein function pϕ.
Finally, we have the following: Proof. By Lemma 2.36 if D is the Chern connection of (E, h), then f * D is the Chern connection of (f * E, f * h), and we have R f * D = f * R D . It follows that so the statement follows readily.
We now give the following definition: 14. An α−twisted holomorphic vector bundle on a compact, complex manifold with Hermitian metric g is called g−Hermite-Einstein if it admits a Hermitian metric h such that (E, h) verifies the g−Hermite-Einstein condition.
The previous Propositions 3.8 to 3.13 tell us that: • twisted holomorphic line bundles are all g−Hermite-Einstein; • the dual of a g−Hermite-Einstein twisted holomorphic vector bundle is g−Hermite-Einstein; • the tensor product of g−Hermite-Einstein twisted holomorphic vector bundles is g−Hermite-Einstein; • the direct sum of twisted holomorphic vector bundles is g−Hermite- Einstein if and only if the summands are g−Hermite-Einstein with the same Einstein factor; • the pull-back of a g−Hermite-Einstein twisted holomorphic vector bundle is Hermite-Einstein (with respect to the pull-back metric). The following is a key result in the proof of one direction of the Kobayashi-Hitchin correspondence.
Proposition 3.15. Let E 1 and E 2 be two α−twisted holomorphic vector bundles on X, and let h i be a Hermitian metric on E i for i = 1, 2. Let g a Hermitian metric on X and suppose that (E i , h i ) verifies the weak g−Hermite-Einstein condition with Einstein function ϕ i .
Proof. Let us first suppose that ϕ 2 < ϕ 1 , so we prove that if f : Recall that f is a global section of E * 1 ⊗ E 2 (which is an untwisted holomorphic vector bundle). By Lemmas 2.29 and 2.33 the Hermitian metric h 1 and h 2 induce the Hermitian metric h * 1 ⊗ h 2 on E * 1 ⊗ E 2 , and if D i is the Chern connection of (E i , h i ), then D * 1 ⊗ D 2 is the Chern connection of (Hom(E 1 , E 2 ), h * 1 ⊗ h 2 ). By Propositions 3.9 and 3.10 it then follows that (E * 1 ⊗ E 2 , h * 1 ⊗ h 2 ) is a holomorphic vector bundle verifying the weak g−Hermite-Einstein condition with Einstein function ϕ 2 − ϕ 1 . As ϕ 2 < ϕ 1 the mean curvature of E * 1 ⊗ E 2 is negative definite everywhere on X: by Theorem 1.9 in Chapter III of [17] it has then no non-zero global sections, so f = 0.
Suppose now that ϕ 2 ≤ ϕ 1 and let f : E 1 −→ E 2 be a morphism of α−twisted sheaves. The previous part of the proof tells us that f is a global section of the g−Hermite-Einstein holomorphic vector bundle E * 1 ⊗E 2 , whose Einstein function is ϕ 2 −ϕ 1 : the mean curvature is then everywhere negative semi-definite, hence by Theorem 1.9 in Chapter III of [17] f has to be parallel with respect to the Chern connection D * 1 ⊗ D 2 . If f = {f i }, this means that f i is parallel with respect to the Chern connection D * 1,i ⊗ D 2,i , so that the rank of f i has to be constant. It follows that f i is a morphism of holomorphic vector bundles, and hence that f is a morphism of α−twisted holomorphic vector bundles.
As a consequence, ker(f ) is an α−twisted holomorphic subbundle of E 1 and Im(f ) is an α−twisted holomorphic subbundle of E 2 . As f is parallel, they are both invariant with respect to the Chern connections. We then let E 1 := ker(f ) ⊥ and E 2 := Im(f ) ⊥ , where the orthogonality is with respect to h 1 and h 2 respectively. By Lemma 2.42 the statement follows.
3.4. First Chern class and Hermite-Einstein. Let now E be an α−twisted holomorphic vector bundle of rank r on a complex manifold X of dimension n, and let us fix a Hermitian metric h on E and a Hermitian metric g on X. We let σ g be the real (1, 1)−form on X associated to g and to the complex structure of X.
If R h is the Chern connection of (E, h), then by its very definition the first Chern form of (E, h) is Lemma 3.16. If X is compact and (E, h) verifies the weak g−Hermite-Einstein condition with Einstein function ϕ, then As (E, h) verifies the weak g−Hermite-Einstein condition with Einstein function ϕ, ϕσ n g , and the statement follows.
If g is a Kähler metric, then σ g is d−closed, hence the value is ∂∂−closed.

3.5.
Hermite-Einstein and weak Hermite-Einstein. We now show that if a twisted vector bundle verifies the weak g−Hermite-Einstein condition, then it is g−Hermite-Einstein. Let E be an α−twisted holomorphic vector bundle on a complex manifold X, h a Hermitian metric on E and χ : X −→ R a positive smooth function. For every i ∈ I and for every sections ξ, η of E i we let It is easy to see that h χ i is a Hermitian metric on E i , and since The conformal change of Hermitian metric affects the curvature, and we have: If (E, h) verifies the weak g−Hermite-Einstein condition with Einstein function ϕ, then (E, h χ ) verifies the weak g−Hermite-Einstein condition with Einstein function ϕ + iΛ g ∂∂χ.
Proof. Let D be the Chern connection of (E, h) and D χ the Chern connection of (E, h χ ). Consider an open covering U = {U j } j∈J as in the proof of Lemma 3.1: hence for every j ∈ J there is i ∈ I such that U j ⊆ U i , and on U j there is a local frame s j of E i .
We let Γ j and Γ χ j be the connection forms of D i and D χ i , and we represent h i and h χ i by the matrices H j and H χ j respectively over U j with respect to the local frame s j . Then Proposition 4.9 in Chapter I of [17]). As H χ j = χ · H j it follows that concluding the proof.
Consequence of this is the following: Proposition 3. 19. Let X be a compact complex manifold and g a Kähler metric on X. Let E be an α−twisted holomorphic vector bundle on X and h a Hermitian metric on E. If (E, h) verifies the weak g−Hermite-Einstein condition, then there is a conformal change h χ of h, which is unique up to homothety, such that (E, h χ ) verifies the g−Hermite-Einstein condition.
Proof. Define c := X ϕσ n g X σ n g , which is well-defined real number since X is compact. Let ϕ be the Einstein function of (E, h): as g is a Kähler metric, we know that there is a C ∞ function u : X −→ R such that Let now χ := exp(u), which is then a positive C ∞ real function on X such that iΛ g ∂∂ log(χ) = iΛ g ∂∂(u) = c − ϕ.
As (E, h) verifies the weak g−Hermite-Einstein condition with Einstein function ϕ, by Lemma 3.18 we have that (E, h χ ) verifies the Hermite-Einstein condition with Einstein function which is constant, and we are done. 3.6. Approximate Hermite-Einstein. Let now X be a compact complex manifold of dimension n and g a Kähler metric on X, whose associated Kähler form is σ g . Let E be an α−twisted holomorphic vector bundle of rank r on X, and h a Hermitian metric on E.
Recall that K g (E, h) is a smooth endomorphism of E, so its trace is a smooth function on X, called g−scalar curvature of (E, h). We let which is the mean value of 1 Proof. The proof is almost identical to that of Lemma 3.16. We have Taking the trace we then get Taking the integral over X the statement follows.
As g is a Kähler metric (or more generally if σ n−1 g is ∂∂−closed), the value X γ 1 (E, h) ∧ σ n−1 g only depends on c 1 (E) and the cohomology class [σ g ]. Similarily, the value X σ n g only depends on [σ g ]. As a consequence, c g (E, h) only depends on c 1 (E) and on [σ g ], but not on h: we will use the notation c g (E) for it.
Remark 3.22. If E is g−Hermite-Einstein, then by Lemmas 3.16 and 3.21 the Einstein factor of (E, h) is c g (E). In particular, the Einstein factor does not depend on the Hermitian metric.
Definition 3.23. The g−degree of E is the real number which only depends on c 1 (E).
Lemma 3.21 then reads as We now define which is a smooth function on X. This allows us to give the following: Definition 3.24. We say that an α−twisted holomorphic vector bundle E is approximate g−Hermite-Einstein if for every > 0 there is a Hermitian metric h on E such that First, we have the following: Proposition 3.25. Let E be an α−twisted holomorphic vector bundle on X. If E is g−Hermite-Einstein, then it is approximate g−Hermite-Einstein.
Proof. Let c be the Einstein factor of (E, h). By Remark 3.22 we have c = c g (E). Choose then h = h for every > 0, so that K g (E, h ) = K g (E, h) = c · id E , and we are done.
As for Hermite-Einstein vector bundles, we have some properties about the behavior of approximate g−Hermite-Einstein vector bundles with respect to the usual operations, the proof of which is exactly as the one for untwisted vector bundle (we just provide the proof of one of the following result to show the analogies).
Proposition 3.26. Let E be an α−twisted holomorphic vector bundle on X. If E is approximate g−Hermite-Einstein, then E * is approximate g−Hermite-Einstein.
Proof. Let h be a Hermitian metric on E. We proved in Proposition 3.9 that K g (E * , h * ) = −K g (E, h) * , so and we are done.
Proposition 3.27. For i = 1, 2 let E i be an α i −twisted holomorphic vector bundle on X. If E i is approximate g−Hermite-Einstein, then E 1 ⊗ E 2 is approximate g−Hermite-Einstein.
Proposition 3.28. Let E 1 and E 2 be two α−twisted holomorphic vector bundles on X of respective ranks r 1 and r 2 . If they are approximate g−Hermite-Einstein and we have deg g (E 1 ) then E 1 ⊕ E 2 is approximate g−Hermite-Einstein.
Proposition 3.29. Let E be an α−twisted holomorphic vector bundle on X. If E is approximate g−Hermite-Einstein, then (1) for every p, q ∈ N, E p,q is approximate g−Hermite-Einstein; (2) for every p ∈ N, ∧ p E is approximate g−Hermite-Einstein.
Proposition 3.30. Let f : X −→ Y be anétale covering, and choose a Kähler matric g on Y and the metric f * g on X.
(1) Let E be an α−twisted holomorphic vector bundle on Y . If it is approximate g−Hermite-Einstein, then f * E is approximate f g −Hermite-Einstein.
(2) Let F be an f * α−twisted holomorphic vector bundle on X. If it is approximate f * g−Hermite-Einstein, then f * F is approximate g−Hermite-Einstein.
The last property we will need is the following: Proposition 3.31. Let E 1 and E 2 be two α−twisted holomorphic vector bundles on X of respective ranks r 1 and r 2 , and suppose that such that If E 1 and E 2 are approximate g−Hermite-Einstein, then every morphism f ∈ Hom(E 1 , E 2 ) is zero.
Proof. We know that f is a global section of the untwisted vector bundle E * 1 ⊗ E 2 . As E 1 and E 2 are approximate g−Hermite-Einstein, by Propositions 3.26 and 3.27 we have that E * 1 ⊗E 2 is approximate g−Hermite-Einstein. Now, notice that By Proposition 5.6 in Chapter IV of [17] we then know that E * 1 ⊗ E 2 has no non-zero global sections, and we are done.

Semistability for twisted vector bundles
If A is a domain and M is an A−module of finite type, the homological dimension of M is the length d of a minimal free resolution If A = O C n ,0 , then for every A−module M we have dh(M ) ≤ n. As a consequence, the same holds for A = O X,x where X is a complex manifold of dimension n and x ∈ X.

Singularities of twisted sheaves. Let
The following definition therefore makes sense: As the homological dimension of a module is invariant under isomorphism, the homological dimension dh(E x ) makes sense.

Singularity sets of twisted sheaves. For m ∈ N let
which is called m−th singularity set of E, and we clearly have The subset S n−1 (E) is called singular locus of E, and we have As the function is upper semicontinuous, the same holds for so that S m (E) is a closed subset of X for every m. Scheja's Theorem tells us that the subset is a closed analytic subset of U i of dimension at most m for every i ∈ I and every m.
it follows that S m (E) is a closed analytic subset of X of dimension at most m for every m. As a corollary we get Lemma 4.2. Let E be an α−twisted coherent sheaf on X such that for every x ∈ X there is an open neighborhood U ⊆ X of x and an exact sequence of α−twisted coherent sheaves where F j is a locally free α |U −twisted sheaf on U . Then Proof. If x ∈ U i , up to restricting U we may suppose U ⊆ U i , so the exact sequence in the statement becomes where F j is a locally free sheaf of O U i −modules. But then we know that and as this holds for every i ∈ I, we conclude.
By definition of S n−1 (E) we see that E |X\S n−1 (E) is a locally free α−twisted coherent sheaf, so the rank of E x as a O X,x −module is invariant over X \ S n−1 (E). We will call this number the rank of E. We recall that • every free A−module is reflexive, and every reflexive A−module is torsion free; • every torsion-free A−module is a submodule of a free A−module of the same rank; • for every A−module M we have that M * is torsion-free and that M * * is reflexive;  This is equivalent to T (E) = 0, or even to E i torsion free for every i ∈ I. As a consequence of the previous properties of A−modules, it follows that:

Torsion of twisted sheaves. We recall that if M is an
• a locally free α−twisted coherent sheaf is torsion-free, • every α−twisted coherent subsheaf of a torsion-free α−twisted coherent sheaf if torsion-free, • every torsion-free α−twisted coherent sheaf is an α−twisted subsheaf of a locally free α−twisted coherent sheaf of the same rank, • if E is a torsion-free α−twisted coherent sheaf, then for every x ∈ X we have dh(E x ) ≤ n−1, so by Lemma 4.2 we get that dim(S m (E)) ≤ m − 1. In particular, we see that the singular locus of a torsion-free α−twisted coherent sheaf has codimension at least 2, • The natural morphism σ E : E −→ E * * has kernel equal to T (E), hence E is torsion-free if and only if σ E is injective.
This is equivalent to E i reflexive for every i ∈ I. Again, as before we have that: • all locally-free α−twisted coherent sheaves are reflexive, • all reflexive α−twisted coherent sheaves are torsion-free, • for every α−twisted coherent sheaf E, we have that E * is reflexive and E * * is torsion-free, • if F is reflexive, then dim(S m (F)) ≤ m − 2 for every m. In particular, the singular locus of a reflexive α−twisted coherent sheaf has codimension at least 3 in X, • if E and F are two α−twisted coherent sheaves and F is reflexive, then the coherent sheaf Hom(E, F) is reflexive. We end this section with the following: Normal α−twisted coherent sheaves are useful for the following property, which is an immediate consequence of the untwisted analogue: is an exact sequence of α−twisted coherent sheaves, if F is reflexive and G is torsion-free, then E is normal (and hence reflexive).

4.2.
Semistability for twisted sheaves. Let E be an α−twisted coherent sheaf on a complex manifold X. We know that E admits a finite resolution where E j is a locally free α−twisted sheaf of rank r j (see [3]). If E j is α−twisted holomorphic vector bundle associated to E j , then det(E j ) is an α r j −twisted holomorphic line bundle, whose associated locally free α r j −twisted coherent sheaf is denoted det(E j ) and called determinant of E j .
We then let det(E) := ⊗ m j=0 det(E j ) (−1) j , which is a locally free sheaf of rank 1 twisted by α r , where r is the rank of E. It is called determinant of E, and it does not depend on the chosen resolution.
If E is a torsion-free α−twisted coherent sheaf, then det(E) is canonically isomorphic to (∧ r E) * * , and we have det(E) * det(E * ). Moreover, if f : E −→ F is a morphism of α−twisted coherent sheaves, where E and F are both torsion free and have the same rank, then f induces a morphism det(f ) : det(E) −→ det(F).
Let now E be an α−twisted coherent sheaf of rank r and suppose that X is a compact Kähler manifold of dimension n. As det(E) is a locally free α r −twisted sheaf of rank 1, we may associate to it an α r −twisted holomorphic line bundle L. Choose a Hermitian metric h on L: we will let and call it the first Chern form of (E, h).
As g is Kähler metric (or more generally if σ n−1 g is ∂∂−closed), the cohomology class of γ 1 (E, h) does not depend on h, so we write it as and call it first Chern class of E. The g−degree of E is which again does not depend on h.
If r > 0 is the rank of E, the slope of E with respect to g is Definition 4.7. A torsion-free α−twisted coherent sheaf E is said to be g−semistable if for every α−twisted coherent subsheaf F of E whose rank r is such that 0 < r < r, we have that µ g (F) ≤ µ g (E). If for every α−twisted subsheaf the inaquality is strict, we say that E is g−stable. An α−twisted holomorphic vector bundle E is g−semistable (resp. g−stable) if the associated locally free α−twisted coherent sheaf is g−semistable (resp. g−stable).
Let us now collect some properties of semistable twisted sheaves that will be useful in what follows.
where r is the rank of E and r is the rank of G.
Proof. If r is the rank of F, we have that det(F) is α r −twisted, det(E) is α r −twisted and det(G) is α r −twisted, so that det(E) ⊗ det(G) is α r −twisted (since r + r = r). As the sequence is exact, we have an isomorphism det(E) det(E) ⊗ det(G), and hence c 1 (F) = c 1 (E) + c 1 (G). But then deg g (F) = deg g (E) + deg g (G), hence (r + r )µ g (F) = rµ g (F) = r µ g (E) + r µ g (G), and the statement follows.
As an immediate consequence we have the following: Proposition 4.9. Let E be an torsion-free α−twisted coherent sheaf of rank r on a complex manifold X with a Kähler metric g.
(1) The sheaf E is g−semistable if and only if for every α−twisted quotient G of F of rank 0 < r we have µ g (F) ≤ µ g (G). (2) The sheaf E is g−stable if and only if for every α−twisted quotient G of F of rank 0 < r < r we have µ g (F) < µ g (G).
We now have the following property of torsion twisted sheaves.
Lemma 4.10. If E is a torsion α−twisted coherent sheaf on a compact complex manifold X with a Kähler metric g, then deg g (E) ≥ 0.
Proof. We first show that if E and E are two α−twisted torsion-free coherent sheaves of the same rank r and f : E −→ E is injective, then the induced morphism det(f ) : det(E) −→ det(E ) is injective. Indeed, let A = S n−1 (E) and A = S n−1 (E ): over X \ (A ∪ A ) both E and E are locally free, and f is an injection between two locally free α−twisted sheaves of the same rank. The induced morphism det(f ) is then an isomorphism over X \(A∪A ), so ker(det(f )) is supported on A∪A , and hence it is a torsion α r −twisted sheaf. But moreover ker(det(f )) is an α−twisted coherent subsheaf of det(E), which is torsion-free, so ker(f ) = 0 and f is injective. Now, as by hypothesis E is torsion, over X \A it is locally free and torsion, so it is trivial. It follows that E is supported on A, and it has rank 0. Let be a locally free resolution of E. We let S 1 := E 1 / ker(f 1 ), which is a torsionfree α−twisted coherent sheaf, and we have an exact sequence As the rank of E is 0, det(E) is an untwisted locally free coherent sheaf of rank 1. Moreover S 1 and E 0 are both torsion-free α−twisted coherent sheaves of the same rank r. As f 1 is injective, the previous proof gives that det(f 1 ) : det(S 1 ) −→ det(E 0 ) is injective, i. e. it is a non-trivial morphism of α r −twisted coherent sheaves. Since det(E) det(E 0 ) ⊗ det(S 1 ) * Hom(det(S 1 ), det(E 0 )), we then have that Γ(X, det(E)) Hom(det(S 1 ), det(E 0 )).
But then det(f 1 ) corresponds to a non-trivial holomorphic section of det(E).
Let V be the zero locus of this section, which is a divisor on X whose cohomology class is c 1 (det(E)) = c 1 (E). It then follows that and we are done.
Remark 4.11. The previous proof works even without the Kähler assumption on g, see Proposition 1.3.5 of [21].
As a consequence of this we get the following: Proposition 4.12. Let E be an torsion-free α−twisted coherent sheaf of rank r on a complex manifold X with a Kähler metric g.
(1) The sheaf E is g−semistable if and only if one of the two following conditions is verified: The sheaf E is g−stable if and only if one of the two following conditions is verified: (a) for every α−twisted subsheaf F of E with E/F torsion-free and such that F has rank 0 < r < r, we have µ g (F) ≤ µ g (E). (b) for every torsion-free α−twisted quotient G of E of rank 0 < r < r we have µ g (E) ≤ µ g (G).
Proof. The proof is identical to that of Proposition 7.6 in Chapter V of [17].
The following will be useful in what follows: Proposition 4.13. Let E be an torsion-free α−twisted coherent sheaf of rank r on a complex manifold X with a Kähler metric g.
(2) If L is a locally free α −twisted sheaf of rank 1, then the α−twisted sheaf E is g−(semi)stable if and only if F ⊗ L is g−(semi)stable. Proof. The proof is identical to that of Proposition 7.7 in Chapter V of [17].
The last results we need, whose proofs are identical to those of the corresponding results in the untwisted cases (see Propositions 5.7.9 and 5.7.11, and Corollaries 5.7.12 and 5.7.14), are the following: Proposition 4.14. Let E and F be two torsion-free α−twisted coherent sheaves on a complex manifold X with a Kähler metric g. Then E ⊕ F is g−semistable if and only if E and F are g−semistable and µ g (E) = µ g (F). Proposition 4.15. Let E and F be two torsion-free α−twisted coherent sheaves on a complex manifold X with a Kähler metric g, and let f : E −→ F be a morphism of α−twisted coherent sheaves.
(1) If µ g (E) > µ g (F), then f = 0, (2) If µ g (E) = µ g (F) and E is g−stable, then either f = 0 or E and F have the same rank and f is injective.
(3) If µ g (E) = µ g (F) and F is g−stable, then either f = 0 or E and F have the same rank and f is generically surjective.
This Proposition has two important corollaries: Corollary 4. 16. Let E and F be two torsion-free α−twisted coherent sheaves on a complex manifold X with a Kähler metric g. If they have the same rank and same degree with respect to g, and one of them is g−stable, then every morphism of α−twisted coherent sheaves between them is either 0 or an isomorphism.
Corollary 4.17. Let E be an α−twisted holomorphic vector bundle on a compact complex manifold X with a Kähler metric g. If E is g−stable, then Γ(X, End(E)) C · id E .

4.3.
Hermite-Einstein implies polystable. Let X be a complex manifold of dimension n and g a Kähler metric on X. Let E be an α−twisted coherent sheaf on X of rank r. Let L be the α r −twisted holomorphic line bundle associated to det(E), and choose a Hermitian metric h on L.
We call g−degree form of (E, h) the d−closed real 2n−form on X defined as If X is compact, we then have The first result we need to prove is the following, relating the properties of the g−degree form and the g−Hermite-Einstein condition. (1) If r is the rank of E , r is the rank of E and h := h E is the Hermitian metric induced by h on E , the real d−closed 2n−form (2) If the previous 2n−form is 0, then the exact sequence above splits, and if we let h be the natural Hermitian metric induced by h on E , then (E , h ) and (E , h ) verify the g−Hermite-Einstein condition with Einstein factor c.
Proof. The proof is identical to that of Proposition 8.2 in Chapter V of [17], using the results in section 2.4. The same proof works even only if σ n−1 g is ∂∂−closed (see Proposition 2.3.1 of [21]).
We are now in the position to prove one of the main results of this section, namely: Theorem 4.19. Let E be an α−twisted holomorphic vector bundle on a compact complex manifold X with Kähler metric g. Let h be a Hermitian metric on E and suppose that (E, h) verifies the g−Hermite-Einstein condition with Einstein factor c. Then E is g−semistable, and we have verifies the g−Hermite-Einstein condition with Einstein factor c.
Proof. Let us first prove that E is g−semistable. To do so, let us consider the locally free α−twisted coherent sheaf E associated to E, and let r be its rank: we need to show that E is g−semistable, so let us consider an α−twisted coherent subsheaf F of E and let p be its rank. By Proposition 4.12, in order to prove that E is g−semistable we may suppose that 0 < p < r and that E/F is torsion-free. Let ι : F −→ E be the inclusion, so taking the determinant we get As E is locally free, we have that ∧ p E is locally free, and hence that (∧ p E) * * ∧ p E. The first part of the proof of Lemma 4.10 shows that det(ι) is injective. Moreover, notice that (∧ p E) ⊗ det(F) * is an untwisted sheaf, so tensoring with det(F) * we get that det(ι) defines a global section which is not trivial since ι is injective. This corresponds to finding a holomorphic section of the holomorphic vector bundle ∧ p E ⊗ L * , where L is the α p −twisted holomorphic line bundle corresponding to det(F).
As (E, h) verifies the g−Hermite-Einstein condition with Einstein factor c, by Remark 3.22 we get that c = c g (E), i. e.
By Proposition 3.12 we know that (∧ p E, ∧ p h) verifies the g−Hermite-Einstein condition with Einstein factor pc.
Let h be a Hermitian metric on L. By Propositions 3.8 and 3.19 up to apply a conformal change to h we know that (L, h ) verifies the g−Hermite-Einstein condition with Einstein factor c . Again by Remark 3.22 we have Notice that By Proposition 3.9 we then see that (L * , (h ) * ) verifies the g−Hermite-Einstein condition with Einstein factor −c , and hence by Proposition 3.10 we see that (∧ p E ⊗L * , ∧ p h⊗(h ) * ) verifies the g−Hermite-Einstein condition with Einstein factor pc − c .
But ∧ p E ⊗ L * is a holomorphic vector bundle on a compact Kähler manifold which has a non-trivial holomorphic section. Proposition 3.15 gives then pc − c ≥ 0, i. e.
Let us now prove the second part of the statement. First, if E is g−stable, then we let k = 1 and E = E 1 , and we are done. We will then suppose that E is g−semistable but not g−stable.
This implies that there is an α−twisted coherent subsheaf F of E of rank 0 < p < r with G := E/F torsion-free and µ g (F) = µ g (E). Letting L be the α p −twisted holomorphic vector bundle associated to det(F) and h a Hermitian metric on L, by the previous part of the proof (∧ p E ⊗ L * , ∧ p h ⊗ (h ) * ) verifies the g−Hermite-Einstein condition with Einstein factor 0.
By Proposition 3.15 the mean curvature of (∧ p E ⊗ L * , ∧ p h ⊗ (h ) * ) is negative semidefinite and every holomorphic section of ∧ p E ⊗ L * is parallel with respect to the Chern connection D. In particular the morphism f defined before has to be parallel with respect to D, so L is an α p −twisted holomorphic line subbundle of ∧ p E which is parallel with respect to the Chern connection of (∧ p E, ∧ p h). Now, let X := X \ S n−1 (F) be the open subset of X over which F is locally free, and let F be the α−twisted holomorphic vector bundle associated to F |X . The inclusion ι : F −→ E induces an inclusion ι : F |X −→ E |X , and hence an inclusion j : F −→ E |X of α |X −twisted holomorphic vector bundles. The previous discussion shows that F is a parallel twisted holomorphic subbundle of E with respect to the Chern connection of (E, h).
By Lemma 2.42 we then get an α |X −holomorphic subbundle G of E |X such that E |X = F ⊕ G . Notice that G |X is a locally free α |X −twisted sheaf on X whose associated vector bundle is G . As the exact sequence is splitting, the same holds for the exact sequence As E is locally free, we have that F is reflexive (as a subsheaf of a reflexive sheaf), and we know that G is torsion-free by assumption. By Lemma 4.6 we then get that Hom(E, F) and Hom(F, F) are reflexive, and hence normal. This implies that Γ(X, Hom(E, F)) = Γ(X , Hom(E, F)), and Γ(X, Hom(F, F)) = Γ(X , Hom(F, F)), since S n−1 (F) is a closed analytic subset of codimension at least 2 (since F is reflexive).
As the exact sequence The previous property then implies that there is a unique morphism p : splits too, and we have E = F ⊕ G. But as E is locally free, it follows that F and G are locally free, and hence we have E = F ⊕ G where F and G are the α−twisted holomorphic vector bundles corresponding to F and G. By Lemma 2.42 we then may proceed by induction on the rank, getting the statement.
Let us now present the following definition: Definition 4.20. An α−twisted holomorphic vector bundle E is called g−polystable if it is g−semistable and we have E = E 1 ⊕ · · · ⊕ E k where E 1 , · · · , E k are g−stable α−twisted holomorphic subbundles of E such that µ g (E j ) = µ g (E) for every j = 1, · · · k.
This definition allows us to rephrase Theorem 4.19 as follows, proving one direction of the first part of Theorem 1.1: Theorem 4.21. Let X be a compact Kähler manifold with a Kähler metric g and E an α−twisted holomorphic vector bundle on X. If E is g−Hermite-Einstein, then E is g−polystable.

4.4.
Approximate Hermite-Einstein implies semistable. An adaptation of the proof of Theorem 4.19 allows us to prove the following, which is one direction of the second item of Theorem 1.1: Theorem 4.23. Let E be an α−twisted holomorphic vector bundle on a compact complex manifold X with Kähler metric g. If E is approximate g−Hermite-Einstein, then E is g−semistable.
Proof. Consider an α−twisted coherent subsheaf F of the locally free α−twisted coherent sheaf E associated to E, such that E/F is torsionfree and such that µ g (F) = µ g (E). We let r be the rank of E and p be the rank of F, and we suppose that 0 < p < r. Moreover we let L be the α p −twisted holomorphic vector bundle associated to det(F).
Recall that as E is approximate g−Hermite-Einstein, by Proposition 3.29 then ∧ p E is approximate g−Hermite-Einstein. The α p −twisted holomorphic line bundle L is approximate g−Hermite-Einstein by Propositions 3.8 and 3.25, so by Proposition 3.26 we see that L * is approximate g−Hermite-Einstein. By Proposition 3.27 we then conclude that ∧ p E ⊗ L * is approximate g−Hermite-Einstein.
As in the proof of Theorem 4.19, we have a nontrivial holomorphic section of ∧ p E ⊗ L * , hence deg g (∧ p E ⊗ L * ) ≥ 0 by Proposition 3.31. Notice that But then we get so that µ g (E) ≥ µ g (F), proving that E is g−semistable.

The Kobayashi-Hitchin correspondence
The aim of this section is to complete the proof of point 1 of Theorem 1.1, namely: Theorem 5.1. Let X be a compact Kähler manifold with a Kähler metric g, and E an α−twisted holomorpic vector bundle on X. Then E is g−polystable if and only if it is g−Hermite-Einstein.
By Theorem 4.21 we know that if E is g−Hermite-Einstein, then E is g−polystable. We are left with the proof of the opposite direction, and to do so we follow closely section 3 of [21].
Consider two Hermitian metrics h 0 and h on E, and fix a Kähler metric g on X. We let D 0 be the Chern connection of (E, h 0 ) and D the Chern connection of (E, h). For every i ∈ I we have where D 1,0 0,i denotes the (1, 0)−part of the connection induced by D 0,i on End(E i ) (see as instance section (1.9) of [26]).
If R i (resp. R 0,i ) is the Chern curvature of (E i , h i ) (resp. of (E i , h 0,i )), we then have If we now let R (resp. R 0 ) be the Chern curvature of (E, h) (resp. of (E, h 0 )) we then get

It then follows that
). In conclusion, we see that h is g−Hermite-Einstein if and only if ,0 0 (f ))) = 0. This will be called Hermite-Einstein equation.
For the second point, take any Hermitian metric h on E, and notice that if we let K 0 It follows that there is a smooth function ψ such that iΛ g ∂∂ψ = K 0 g (E, h), and hence there is a function φ such that We let h 1 := h exp(φ) , which is then a Hermitian metric on E such that (see the proof of Lemma 3.18). We then get It is easy to see that T r(K 0 g (E, h 0 )) = 0. Moreover, let so that an easy calculation shows that L h 0 1 (f ) = 0. The final point of the statement is proved exactly as in point (iii) of Lemma (3.2.1) of [21]. Now, the same of Lemma (3.2.3) of [21] shows that if h 0 is as in point 2 of Lemma 5.2, and if for every > 0 and f ∈ End By the identification of Herm(E) with End h 0 (E), the norm L p k on Herm(E) induced a L p k −norm on End h 0 (E). We will write L p k End h 0 (E) for the Banach space completion of End h 0 (E), and L p k End + h 0 (E) for the interior of the closure of End + h 0 (E) in L p k End h 0 (E). Notice that if > 0, then L h 0 ( , ·) is a second order differential operator. It follows that , and by the Sobolev Multiplication Theorem, the Left Composition Lemma and the fact that a multilinear continuous map between Banach spaces is smooth, it follows that the map L h 0 is differentiable. Moreover, as in section 3.2 of [21] for every ∈ (0, 1] and every f ∈ End + h 0 (E) we have that d L h 0 ( , f ) is a linear, second order differential operator extending to . The same proof of Lemma (3.2.4) shows that this differential operator is elliptic, and it is an isomorphism if and only if it is injective or surjective.
Now, let f ∈ End + h 0 (E), so that f 1 2 ∈ End + h 0 (E) (see Lemma 2.10). For every ψ ∈ End(E), write ψ = {ψ i } i∈I , and if f = {f i } i∈I we have Notice that Ad(f

In a similar way one defines
Moreover, we define a new connection D f on End(E) as follows: recall that if D 0 is the Chern connection of (E, h 0 ), it induces the Chern connection of (End(E), End(h 0 )), that we still write D 0 (which is a connection on an untwisted bundle). We let which is again a connection on End(E) compatible with End(h 0 ).
Finally, for f ∈ End + h 0 (E) and ϕ ∈ End h 0 (E) we let The following requires the same proof of Propositions (3.2.5) and (3.2.6) in [21]: where h 0 denotes the Hermitian metric induced by h 0 on End(E).
This allows us to prove the following, as done in Corollary (3.2.7) of [21]:
The aim of this section is to show that if 0 > 0, then f converges to f 0 ∈ End + h 0 (E) as converges to 0 , and that L h 0 0 (f 0 ) = 0. Proposition 5.4 allows us then to extend f to an interval of the form ( 1 , 1] for 1 < 0 , contradicting the maximality of 0 . It will then follow that J = (0, 1].
We will let Moreover, and this will be essential in the whole section, we will suppose that E is simple.
Lemma 5.5. Let E be a simple α−twisted holomorphic vector bundle on X. For every > 0 we have that T r(η ) = 0, and that there is a positive real number C(m ) depending only on m such that But as det(f ) = exp(T r(log(f ))) (see the proof of Lemma 5.2) we get It follows that By definition we have As T r(η ) = 0 it follows that T r(ψ ) = 0, so ψ is L 2 −orthogonal to the identity, and hence since E is simple to ker(∆ ∂ ) (see Remark 7.2.2 of [21]).
As ∆ ∂ is self-adjoint and elliptic, all its eigenvalues are non-negative. If λ 1 is the smallest positive eigenvalue of ∆ ∂ , it follows that We moreover need the following two Lemmas: Lemma 5.7. Let E be a simple α−twisted holomorphic vector bundle on X, h 0 a Hermitian metric on E and f ∈ End + h 0 (E) a solution of L h 0 = 0 for some > 0.
(1) We have (2) If m := max x∈X | log(f )|(x), then we have (3) There is real number C (depending only on g and h 0 ) such that Proof. The proof is identical to that of Lemma (3.3.4) of [21].
Lemma 5.8. Let E be a simple α−twisted holomorphic vector bundle on X. Suppose that there is m ∈ R such that m ≤ m for every ∈ ( 0 , 1]. Then for every p > 1 and every ∈ ( 0 , 1] we have that Proof. The proof is identical to that of Proposition (3.3.5) of [21], where one has to replace Proposition (3.3.3) by Lemma 5.6.
All these results together allow us to prove the following: This implies the existence of a sequence k converging to 0 such that f k converges weakly to some f 0 ∈ L p 2 End(E). The fact that m ≤ m implies that the eigenvalues of f 0 take values in [e −m , e m ], which implies that f 0 ∈ L p 2 End + (E) (see the proof of Lemma 7.3.10 of [21]). Now we know that L p 2 has a compact embedding in L p 1 , hence we may suppose that f k converges strongly to f 0 in L p 1 End + (E). Suppose that that L h 0 0 (f 0 ) = 0. This implies that iΛ g ∂D 1,0 0 (f 0 ) is a multilinear algebraic expression in f , log(f ), D 1,0 0 (f ) and ∂(f ). But since f 0 ∈ L p 2 End(E), it follows that iΛ g ∂D 1,0 0 (f ) ∈ L p 1 End(E). The Elliptic Regularity Theorem then implies that f 0 ∈ L p 3 End(E). Repeating this process, by Rellich's Theorem we then see that f 0 ∈ A 0 (End(E)), i. e. f 0 is a solution of L h 0 0 = 0. We are then left to prove that L h 0 0 (f 0 ) = 0. To do so, we just need to prove that if ζ is any smooth endomorphism of E, we have (L h 0 0 (f 0 ), ζ) L 2 = 0. Recall that L h 0 k (f k ) = 0 for every k, hence we get is continuous, hence we see that ( k log(f k ), ζ) L 2 converges to 0. Moreover we have where β ∈ A 1,0 (End(E)) does not depend on k.
Remark that the map As f k converges strongly to f 0 in L p 1 , we have that Ψ k converges to 0, so (Ψ k , β) L 2 converges to 0. But this implies that As a consequence, if ||f || L 2 is uniformly bounded, then the Hermitian metric (f 0 ) h 0 is a g−Hermite-Einstein metric, and hence E is g−Hermite-Einstein. In the remaining part of this section we prove that if ||f || L 2 are not uniformly bounded, then E is not g−stable. These two results together will prove that if E is a g−stable α−twisted holomorphic vector bundle, then E is g−Hermite-Einstein, proving Theorem 5.1.
The main definition we will need is the following: Definition 5.11. If E is an α−twisted holomorphic vector bundle on X, an element π ∈ L 2 1 End(E) is a weakly holomorphic α−twisted subbundle of E if π * = π = π 2 and (id E − π) • ∂(π) = 0 almost everywhere on X.
The reason for the name is the following: Lemma 5.12. Let E be an α−twisted holomorphic vector bundle on X whose associated locally free α−twisted sheaf is E. If π ∈ L 2 1 End(E) is a weakly holomorphic α−twisted subbundle of E, there is a coherent α−twisted subsheaf F of E and an analytic subset S of X such that: (1) S has codimension at least 2 in X, (2) π |X\S ∈ A 0 (E |X\S ) and we have π * |X\S = π |X\S = π 2 |X\S and (id E|X\S − π |X\S ) • ∂(π |X\S ) = 0, (3) F |X\S is the image of π |X\S and an α−twisted holomorphic subbundle of E.
Proof. We let π = {π i } i∈I , and notice that π i ∈ L 2 1 End(E i ) is such that π * i = π i = π 2 i and (id E i − π i ) • ∂(π i ) = 0 almost everywhere on U i . As E i as a holomorphic vector bundle on U i , by [27] (see Theorem 3.4.3 of [21]) there is a coherent subsheaf F i of E i and an analytic subset S i of U i such that (1) S i has codimension at least 2 in U i , and a holomorphic subbundle of E i . Now, the fact that π is an endomorphism of E imply that F is an α−twisted coherent sheaf and that S i ∩ U ij = S j ∩ U ij (since S i is the locus of U i where F i is not free). Hence the S i 's glue together to give an analytic subset S of X of codimension at least 2, and we are done.
Before going on we need two Lemmas, and we recall that if f ∈ End + h 0 (E), then for every σ ∈ (0, 1] we may define f σ ∈ End + h 0 (E) (see section 2.2.1 and Lemma 2.53).
Proof. The proof is as that of Lemma (3.4.5) of [21].
The two previous Lemmas 5.13 and 5.14 allows us to produce a weakly holomorphic α−twisted subbundle of E. (1) for k → +∞ the endomorphisms ρ( k )f k converge weakly in L 2 1 to an element f ∞ = 0, (2) there is a sequence {σ l } l∈N converging to 0 such that f σ l ∞ converge weakly in L 2 1 to an element f 0 ∞ , (3) letting π := id E − f 0 ∞ , then π is a weakly holomorphic α−twisted subbundle of E.
Proof. The proof is exactly as the one of Proposition (3.4.6) of [21], where one has to replace Lemma (3.4.4) by Lemma 5.13, Lemma (3.3.4) with Lemma 5.7, and Lemma (3.4.5) with Lemma 5.14. Now, by Lemma 5.12 we see that the weakly holomorphic α−twisted subbundle of E defines an α−twisted coherent subsheaf F of E, and with the same proof of Corollary (3.4.7) we see that 0 < rk(F) < rk(E). We conclude this section with the following: Proof. The proof is identical to that of Proposition (3.4.8) of [21].
As a consequence we finally conclude with the proof of Theorem 5.1 Proof. By Theorem 4.19 we have that if E is g−Hermite-Einstein, then E is g−polystable.
Conversely, if E be g−stable, then Corollary 4.17 implies that E is simple, hence by Lemma 5.2 and Propositions 5.4 and 5.9 there is a Hermitian metric h 0 on E for which there is f 1 ∈ End + h 0 (E) which is solution of the perturbed equation L h 0 1 = 0, and we let f : (0, 1] −→ End + h 0 (E) be the unique differentiable solution of the perturbed equation.
If there is no real number C such that ||f || L 2 ≤ C for every ∈ (0, 1], then lim sup
By Proposition 5.16 then E is not g−stable, which is impossible. As a consequence there must be a constant C such that ||f || L 2 ≤ C for every ∈ (0, 1]. By Proposition 5.10 then E is g−Hermite-Einstein.
If now E is g−polystable, then E = E 1 ⊕· · ·⊕E k where E 1 , · · · , E k are all g−stable of the same g−slope. Hence E 1 , · · · , E k are all g−Hermite-Einstein with the same Einstein factor, so by Proposition 3.11 E is g−Hermite-Einstein.
The definition of mean curvature, of Chern classes, of degree, of g−stability and of g−Hermite-Einstein metrics do not depend on the fact that g is a Kähler metric on X, but only on the fact that σ n−1 g is ∂∂−closed, i. e. on the fact that g is a Gauduchon metric on X. By [9] we know that on every compact complex manifold there is a Gauduchon metric.
By Remark 4.22, and by the fact that all the results of this section go through if we suppose that g is a Gauduchon metric on X (see [21]), we conclude the following, providing a generalization of [29].
Theorem 5.17. Let X be a compact complex manifold and g a Gauduchon metric on X. An α−twisted holomorphic vector bundle is g−Hermite-Einstein if and only if E is g−polystable.

Approximate Kobayashi-Hitchin
The aim of this section is to complete the proof of the approximate Kobayashi-Hitchin correspondence for twisted vector bundles, i. e. the following Theorem 6.1. Let E be an α−twisted holomorphic vector bundle over a compact Kähler manifold with Kähler metric g. Then E is g−semistable if and only if it is approximate g−Hermite-Einstein.
Theorem 4.23 tells us that if E is approximate g−Hermite-Einstein, then it is g−semistable. This section is devoted to prove the converse, and for this we follow closely [17] and [13].
First we introduce the Donaldson Lagrangian, and show that the associated evolution equation has a unique smooth solution on R + once the starting Hermitian metric is fixed. As a consequence of this, we show that if the Donaldson Lagrangian of E is bounded from below, then E is approximate g−Hermite-Einstein.
In order to prove Theorem 6.1 we will then just need to prove that the Donaldson Lagrangian for E is bounded from below. To do so we will adapt to twisted sheaves the regularization process described in [13]. 6.1. Donaldson's Lagrangian. Let h, k ∈ Herm + (E), and consider the k−Hermitian endomorphism f k,h ∈ End(E). The determinant det(f k,h ) of f k,h is a smooth function on X, and since by Remark 2.47 we know that f k,h is invertible we see that det(f k,h ) is never zero. If we let it follows that this is a smooth function on X.
Lemma 6.2. For every h, k, l ∈ Herm + (E) we have Proof. By definition, for every h ∈ Herm + (E) we have f h,h = id E , so that Q 1 (h, h) = 0. Moreover, by Remark 2.47 we have f h,k = (f k,h ) −1 , hence Q 1 (h, k) = −Q 1 (k, h). By Remark 2.48 we moreover have f l,k • f k,h = f l,h for every h, k, l ∈ Herm + (E), hence we have and we are done.
For every t ∈ A we let h t := h(t), which is a Hermitian metric on E, and we let D t be the Chern connection of (E, h t ) and R t its curvature. We then have a function If h is differentiable, we have where h t = h (t). We know that h is a geodetic in Herm + (E) from h to k if and only if f h is constant, i. e. if and only if ∂ t f h = 0.
Since f ht,h t ∈ End(E) and R t ∈ A 1,1 (End(E)) for every t ∈ A, we see that f ht,h t · R t ∈ A 1,1 (End(E)). Its trace is then a smooth (1, 1)−form on X depending on t, and we define We now introduce the following notation: if h ∈ Ω a,b h,k (E), we let which is then an element of Ω a,c h,l (E). We have the following: Lemma 6.3. For every h, k, l ∈ Herm + (E) and for every h ∈ Ω a,b h,k (E) and k ∈ Ω b,c k,l we have Proof. Notice that h −1 ∈ Ω a,b k,h (E) and we have By Remark 2.49 it follows that Moreover, as R(h −1 (t)) = R b+a−t , we get The second equality in the statement is trivial.
Finally, for every h, k ∈ Herm + (E) and every h ∈ Ω A h,k (E) we let where n is the dimension of X. The function L h g (h, k) will be called Donaldson Lagrangian of E between h and k along h.
The following is an immediate consequence of Lemmas 6.2 and 6.3: Lemma 6.4. For every h, k, l ∈ Herm + (E) and for every h ∈ Ω a,b h,k (E) and k ∈ Ω b,c k,l we have In this section we prove that L h g (h, k) does not depend on h, but only on h and k. To do so, we first prove the following: Lemma 6.5. Let h be a Hermitian metric on E and h ∈ Ω a,b h,h (E). Then Q h 2 (h, h) ∈ ∂A 0,1 (X) + ∂A 1,0 (X). Proof. Let a 0 , · · · , a m ∈ [a, b] such that a 0 = a, a m = b and a j < a j+1 for every 0 ≤ j ≤ m − 1 be the points where h is not differentiable. Fix now k ∈ Herm + (E), and for every 0 ≤ j ≤ m consider the segment , which is a piecewise differentiable closed curve based at k. By Lemma 6.3 we have for every j = 0, · · · , m − 1, so that As a consequence, if we know that Q h j 2 (k, k) ∈ ∂A 0,1 (X) + ∂A 1,0 (X), it follows that the same holds for Q h 2 (h, h). We then just need to prove the statement for the curve h j . We let ∆ j := [a j , a j+1 ] × [0, 1] and which is a smooth function such that which are Hermitian forms on E, and let t)), i. e. R j (t, s) is the Chern curvature of (E, H j (t, s)). We then have three functions We now define a (1, 0)−form dH j on ∆ j with coefficients in E * ⊗ E * as follows: dH j := ∂ s H j (t) · ds + ∂ t H j (s) · dt (notice that both ∂ s H j (t) and ∂ t H j (s) are Hermitian forms on E, i. e. smooth global sections of the untwisted vector bundle E * ⊗ E * ). We moreover let H −1 j dH j := u j ds + v j dt, which is a (1, 0)−form on ∆ j with coefficients in End(E). We then let H −1 j dH j · R j := u j · R j ds + v j · R j dt, which is then a (1, 0) form on ∆ j whose coefficients are (1, 1)−forms on X with coefficients in End(E). Finally, let dt, which is a (1, 0)−form on ∆ j whose coefficients are (1, 1)−forms on X. Now, we have As v j (t, 0) = 0, we get φ j (t, 0)dt = iT r(v j (t, 0) · R j (t, 0))dt = 0.
Moreover we have φ j (a j , s)ds = iT r(u j (a j , s) · R j (a j , s))ds = iT r(f h j (s),h j (s) · R j (h j (s)))ds,

We then have
and we just need to prove that ∆ j dφ j ∈ ∂A 0,1 (X) + ∂A 1,0 (X).
To do so, we start by letting j (s, t)).
We notice that for every s, t ∈ ∆ j we have v(s, t), u(s, t) ∈ End(E), so that v(s, t) · ∂u(s, t) ∈ A 0,1 (End(E)), and its trace is then a (0, 1)−form on X.
For every (s, t) ∈ ∆ j let D j (s, t) be the Chern connection of (E, H j (s, t)), whose curvature is R j (s, t). We then have D 0,1 j (s, t) = ∂, so α j (s, t) = iT r(v j (s, t) · D 0,1 j (s, t)u j (s, t)). It follows that . As a consequence we see that ). Or aim is to prove that dφ j = −(∂α j + ∂α j )ds ∧ dt + i∂∂T r(v j · u j )ds ∧ dt, which will then conclude the proof. In order to do this we need the following formulas: ∂s∂tH j (the last two formulas are immediate from the definition of u j and v j , while the first two formulas may be proved locally as in the proof of Lemma 3.6 in Chapter VI of [17], and then by gluing the local formulas since the B−field B does not depend neither on t nor on s). Now recall that hence we have The formula we are looking for then follows immediately.
The previous result has an important corollary, which tells us that the Donaldson Lagrangian between two Hermitian metrics does not depend on the chosen path. Corollary 6.6. Let h, k ∈ Herm + (E). For i = 1, 2 and a i < b i two real numbers, let Lemma 6.2 we get Q 1 (h, h) = 0, and hence Now, by Lemma 6.5 there are a (0, 1)−form φ and a (1, 0)−form ψ such that But now notice that by Lemma 6.4 we have , which concludes the proof.
As a consequence on the previous Corollary we will be allowed to use the notation L g (h, k) instead of L h g (h, k), that will simply be called Donaldson Lagrangian between h and k. In particular, for every k ∈ Herm + (E) we get a function L g,k : Herm + (E) −→ R, L g,k (h) := L g (h, k), called Donaldson Lagrangian at k.

6.1.2.
Critical points of the Donaldson Lagrangian. We now want to relate the Donaldson Lagrangian and Hermite-Einstein metrics. We first need the following: Lemma 6.7. Let h be a differentiable curve in Herm + (E) and k ∈ Herm + (E). For every t consider the segment k t in Herm + (E) connecting h t and k. Then we have Along the proof of Lemma 6.5 we proved that if h : [a, b] −→ Herm + (E) is a piecewise differentiable curve and k ∈ Herm + (E), we have This holds for every t ∈ [a, b], and if we derive this with respect to t we get the statement.
Using this, we are able to prove the following: Lemma 6.8. Let h be a differentiable curve in Herm + (E), and fix k ∈ Herm + (E). Then we have Proof. By definition of L g,k (h t ) and Lemma 6.7 we have Since, by definition, we have we then get which proves the statement.
Let h be a differentiable curve in Herm + (E) and let K g (E, h t ) be the mean curvature Hermitian form of (E, h t ): we notice that K g (E, h t ) − c g (E)h t is a Hermitian form on E as well. Recall that for every t we have that h t is a Hermitian form on E, hence h t and K g (E, h t )−c g (E)h t are both tangent vectors of Herm + (E) at h t .
Recall that K g (E, h t ) is a h t −Hermitian endomorphism by Lemma 3.5: by Example 2.54 we then get But this implies that where the last is the Riemannian metric at h t . Lemma 6.8 then gives This allows us to conclude the following: Proposition 6.9. Let k ∈ Herm + (E). An element h ∈ Herm + (E) is a critical point for L g,k if and only if (E, h) is g−Hermite-Einstein.
Proof. The point h is critical for L g,k if and only if for every differentiable curve h from h we have that The previous discussion shows that h is critical for L g,k if and only if As this has to be verified for every and let us look at its differential dL g,k : for a given h ∈ Herm + (E), we have a linear map dL g,k,h : In particular, for every t ∈ [a, b] we have h t ∈ T ht Herm + (E), and we may calculate the value of dL g,k,ht on h t . By definition of the differential, and by the discussion in the previous section, we have Let now gradL g,k be the gradient of L g,k , i. e. the vector field on Herm + (E) which is dual to the differential form dL g,k : the previous formula gives then We now consider the evolution equation in terms of Hermitian forms, or equivalently in terms of endomorphisms. Our present aim is to study the solutions of the evolution equation for a given k ∈ Herm + (E): we will show that there is always a unique smooth solution h : [0, +∞) −→ Herm + (E) with h(0) = k.  i. e. u(t, s) and v(t, s) are endomorphisms of E for every (t, s) ∈ ∆. We moreover let R : ∆ −→ A 1,1 (End(E)) be the function mapping (t, s) ∈ ∆ to the Chern curvature R(t, s) of (E, H(t, s)).
As already remarked in the proof of Lemma 6.5, we have Suppose now that for every t 0 ∈ [0, b] and every x ∈ X the path H(t 0 , s) x is a geodetic in Herm + (E x ). As shown in the proof of Lemma 6.5, we have that u(t, s) = u(t) does not depend on s, i. e. ∂ s u = 0. Moreover, we have As a consequence we have Since Integrating by parts and using ∂ s u = 0 we then get As we are supposing that h 1 and h 2 are two solutions of the evolution equation, we have that We then get ∂ t e = T r(u · (K g (E, h 2,t ) − K g (E, h 1,t ))). Similar calculations give that where R(h) denotes the Chern curvature of (E, h). Notice that ∂∂e is then a (1, 1)−form on X, so we may consider iΛ g (∂∂e), which is a smooth function on X: we will write g e := iΛ g (∂∂e), and we have g e = − 1 0 |D 1,0 u| 2 ds + T r(u · (K g (E, h 2,t ) − K g (E, h 2,t ))).
As a consequence we see that By the Maximum Principle for Parabolic Equations (see Lemma 4.1 in Chapter VI of [17]) we know that the function m is monotone decreasing in t. Now, consider m(0) = max x∈X e 0 (x): recall that e 0 (x) is the energy of the path H(0, s) x in Herm + (E x ) connecting h 1,0,x and h 2,0,x . Since h 1,0 = h 2,0 by hypothesis, we then get e 0 = 0, so that m(0) = 0. Since m is monotone decreasing, it follows that m(t) ≤ 0 for every t ∈ [0, b], so that e t (x) ≤ 0 for every t ∈ [0, b] and every x ∈ X. But since e t (x) ≥ 0, we then get e t (x) = 0 for every x ∈ X and every t ∈ [0, b], i. e. h 1,t = h 2,t for every t ∈ [0, b]. Since this holds for every b < a, we then see that h 1 = h 2 , completing the proof of the first point of the statement.
For the second point, let x ∈ X and consider t, t ∈ [0, a). We let ρ x (t, t ) be the distance between h t,x and h t ,x . Moreover, we let e x (t, t ) be the energy of the (unique) geodesic path in Herm Since the Riemannian metric is complete, in order to prove the statement we just need to prove that the family {h t } t∈[0,a) is uniformly Cauchy, i. e. that for every > 0 there is δ > 0 such that for every t, t ∈ [0, a) such that Fix then > 0. Recall that the family {h t } t∈[0,a) is continuous at 0, hence there is δ > 0 such that for every τ ∈ [0, δ) we have max x∈X e x (0, τ ) < .

Short-time solution.
We now want to show that given two Hermitian metrics h, k on E, then there are δ > 0 and h : [0, δ] −→ Herm + (E) which solves the evolution equation (with respect to k) and which is such that h 0 = h.
If k : [0, a] −→ Herm + (E) is a smooth curve, we let were R kt is the Chern curvature of (E, k t ), and Moreover, we let We view h, v, h + sv as smooth global sections of π * H E . We will let P h (v) := P h+v for every v such that h + v is a family of Hermitian metrics.
For every global section v of π * H E we let It is easy to see that Notice that f h,v is a smooth family of endomorphisms of E. Then iΛ g ∂∂f h,v is a family of endomorphisms of E, and the family of Hermitian forms associated to it and to h will be denoted h v.
By Lemma 3.6 we have that If v ∈ L p q (X/0, a, H E ), then dP h (v) ∈ L p q−2 (X/0, a, H E ), i. e. we may view dP h as a linear operator dP h : L p q (X/0, a, H E ) −→ L p q−2 (X/0, a, H E ). The proof of the following is as that of Lemma 6.5 in Chapter VI of [17]. Lemma 6.11. Let p ≥ 2n + 2 and q ≥ 2. Then dP h : L p q (X/0, a, H E ) −→ L p q−2 (X/0, a, H E ) is an isomorphism.
As a consequence we get the following: Consider an integer p > 2n + 2, where 2n is the real dimension of X. We know by Lemma 6.11 that The Implicit Function Theorem implies that P k maps a neighborhood U of 0 ∈ L p 2 (X/0, a, H E ) onto a neighborhood U of P k (0) in L p 0 (X/0, a, H E ). Let now for some δ > 0 such that w ∈ U (which is possible if δ 1). But then there is v ∈ U such that P k (v) = w. Define then Notice that as w |[0,δ] = 0, we get h is a solution of the evolution equation. The fact that h is smooth can be proved exactly as in Theorem 7.1 in Chapter VI of [17].
We then see that the evolution equation has always a unique smooth solution for short time intervals [0, δ] (for δ 1).
For every t ∈ [a, b] let h t = h(t), and write h t = {h t,i } where h t,i is a Hermitian metric on E i and we have Let D t be the Chern connection of (E, h t ), so that D t = {D t,i } where D t,i is the Chern connection of (E i , h t,i ). For every i ∈ I consider ∂ t D t,i : if Γ t,i is the connection form of D t,i with respect to a given local frame, we have Now, for every i ∈ I we have where we let K t := K g (E, h t ) and c g = c g (E). Restricting to E i we get We then get t,i here denotes the (1, 0)−part of the Chern connection induced by D t,i on End(E i ).
As the K t|E i 's glue together to give K t of E, and as D 1,0 t,i is the restriction to E i of the (1, 0)−part of the Chern connection of (End(E), End(h)), we then see that ∂ t D t = −D 1,0 t K t . Now, if V is any holomorphic vector bundle, h is a Hermitian metric on it and D is the Chern connection of (V, h), we let ). Recall that (see section 2 in Chapter III of [17]) are both monotone decreasing, and in particular bounded.
then for each k ∈ N there is β k ∈ R such that |D k t R t |(x) ≤ β k for every x ∈ X and t ∈ [0, a).
Proof. By Corollary 6.14 we know that (∂ t + ht )|T r(R t )| = 0, and by Lemma 6.15 we know that that (∂ t + ht )|K t | 2 ≤ 0. It then follows from the Maximum Principle for Parabolic Equations (see Lemma 4.1 in Chapter VI of [17]) that both s R and s K are monotone decreasing functions.
For the remaining part, we proceed by induction on k. If k = 1, this is the hypotesis. Consider now k ∈ N, and suppose that for every j < k there is a constant β j such that |D j t R t |(x) ≤ β j for every x ∈ X and t ∈ [0, a), i. e. f 1 2 j (x, t) ≤ β j for every (x, t) ∈ X × [0, a). By Lemma 6.15 there is then a constant A k such that Now, consider the following Cauchy problem and notice that (∂ t + ht )(u) = (∂ t + ht )(1 + u). The differential equation is linear in 1 + u, hence the Cauchy problem has a unique smooth solution u defined for every t ≥ 0.
An easy calculation then shows But then the Maximum Principle for Parabolic Equation (see Lemma 4.1 in Chapter VI of [17]) gives us that the function is monotone decreasing. As f k (0) = u(0) we get that g(0) = 0, so that g(t) ≤ 0 for every t ∈ [0, a). Since e −A k t > 0 for every t, it follows that f k (t) ≤ u(t) for every t, and hence the statement holds.
Another useful consequence of Lemma 6.15 is the following, whose proof is identical to that of Lemma 8.16 in Chapter VI of [17]: Lemma 6.17. Let E be an α−twisted holomorphic vector bundle and consider a smooth solution h : [0, a) −→ Herm + (E) of the evolution equation. If there is q > 3n for which |R t | is uniformly bounded in L q (X) (i. e. independently of t), then |R t | is uniformly bounded in L ∞ (X).
This allows us to prove the following: (1) h t converges in the C 0 −topology to a Hermitian metric h a for t converging to a; (2) the function sup x∈X |K t |(x) is uniformly bounded on [0, a). Then for p < +∞, we have that h is bounded in C 1 (X, a, H E ) and in L p 2 (X, a, H E ), and that R is bounded in L p (X, a). Proof. The proof is almost identical to that of Lemma 8.22 in Chapter VI of [17]. Suppose that h is not bounded in C 1 (X, a, H E ). This implies that there is a sequence {t k } k∈N of points in [0, a) with the two following properties: (1) lim k→+∞ t k = a and (2) if we let M k := sup x∈X |∂h t k |, then lim k→+∞ M k = +∞. We let x k ∈ X be a point where |∂h t k | attains its maximum M k . Taking a subsequence we may suppose that lim k→+∞ x k = x 0 ∈ X. Let i 0 ∈ I be such that x 0 ∈ U i 0 , and choose an open neighborhood U of x 0 contained in U i 0 . Fix local holomorphic coordinates z 1 , · · · , z n on U .
Choose a local frame s of E i 0 over U , and represent K t|U by a matrix of smooth functions whose entries are denoted K t,rs , and h t,i 0 by a matrix H t,i 0 of smooth functions whose entries are denoted h t,i 0 ,pq . We let h pq t,i 0 be the entries of H −1 t,i 0 , and represent the Kähler metric g by a matrix G on U , whose entries are denoted g αβ . We let g αβ be the entries of G −1 .
The proof now follows the same lines as that of Lemma 8.22 in Chapter VI of [17], where one has to replace the formula (8.23) (expressing K t in terms of h t ) with the following formula The proof works since B i 0 does not depend on t, so iΛ g B i 0 is bounded on [0, a).
We are now in the position to prove the main result of this section: Suppose now that [0, a) is the largest interval on which a solution exists. By point 2 of Proposition 6.10 we know that h t converges in the C 0 topology to a Hermitian metric h a as t converges to a. By point 1 of Corollary 6.16 we know that sup x∈X |K t |(x) is bounded on [0, a). We then may apply Lemma 6.18, getting that h is bounded in C 1 (X, a, H E ), and that R is bounded in L p (X, a) for every p < +∞.
The boundedness of R implies by Lemma 6.17 that |R t | is uniformly bounded in L ∞ on [0, a), so Point 2 of Corollary 6.16 implies that for every k the functions |D k t R t | are uniformly bounded for t ∈ [0, a). The boundedness of the family h allows us to show that h is bounded in C k (X, a, H E ) for every k ∈ N. The proof goes by induction on k. Suppose that h is bounded in C k−1 (X, a, H E ). Then the families of the first order partial derivatives of h t are bounded in C k−2 (X, a, H E ). Since K t is uniformly bounded in C l (X, a, H E ) for all l ∈ N, and hence in particular in C k−2 (X, a, H E ), equation (1) and Elliptic Regularity imply that h is bounded in C k (X, a, H E ).
It follows that h is bounded in C ∞ (X, a, H E ). But since h t converges to h a in the C 0 −topology, it follows that the convergence is in the C ∞ −topology, i. e. we can extend h : [0, a) −→ Herm + (E) to a smooth solution over [0, a].
By Proposition 6.12, starting with h a we may extend h to a unique smooth solution of the evolution equation defined over [a, a ) for some a > a, and hence we extend h to a smooth solution on [0, a ). But this contradicts the fact that [0, a) is the largest interval over which h exists, concluding the proof. (1) For every k ∈ Herm + (E) the function is monotone decreasing.
(2) The function Proof. As seen in section 5.1.3 we have Since h is a solution of the evolution equation, we have For the second point, by Corollary 6.14 we know that ∂ t K t = − ht K t . Let us now calculate ||K t − c g (E)id E || 2 . To do so, we first have 6.3. Regularization of twisted sheaves. We start by describing a regularization process that was first used by Buchdahl in [2] for holomorphic vector bundles on surfaces, and then by Jacob in [13] for holomorphic vector bundles on compact Kähler manifolds. We adapt it here for holomorphic twisted vector bundles. As we will see, this construction will allows us to prove the boundedness of the Donaldson Lagrangian of a g−semistable twisted vector bundle.
6.3.1. Blow-ups and regularization of subsheaves. Let X be a compact Kähler manifold with Kähler metric g. The starting point of the construction is an exact sequence of α−twisted coherent sheaves, where E is locally free of rank r (we will consider it as an α−twisted holomorphic vector bundle), S is torsion-free of rank s and Q is torsion-free of rank q = r − s.
We let Z be the singular set of Q: on X \ Z the sheaves S, E and Q are all α−twisted holomorphic vector bundles, and the morphisms f and p are morphisms of α−twisted holomorphic vector bundles. As the locus of X where Q is locally free is where the rank of f is maximal, i. e. equal to s, we get We notice that Z k ⊆ Z k+1 for every k ∈ N, that Z 0 = ∅, that Z s−1 = Z and that Z k = X for k ≥ s.
Let us now choose k 0 ∈ N such that Z k 0 is the smallest non-empty set among the Z k 's, and choose x ∈ Z k 0 , so that rk x (f ) = k 0 . Let us write If x ∈ U i , we may choose local frames s i of S i and e i of E i so that the matrix F i representing f i with respect to these local frames is −matrix whose entries are holomorphic functions vanishing identically on Z k 0 .
Suppose that x ∈ U ij , and represent φ ij (resp. ψ ij ) by a r × r−matrix a ij (resp. a s × s−matrix b ij ) with respect to e i and e j (resp. s i and s j ). As f is a morphism of twisted sheaves, we have a ij F i = F j b ij . From this relation and the form of F i and F j , we get an invertible (r − k 0 ) × (r − k 0 )−matrix a ij and an invertible ( Let us now consider the blow up π : X −→ X of X along Z k 0 (with reduced structure). We let U i := π −1 (U i ), and π i := π | U i : we then have that Let U be an open subset of X over which the exceptional divisor of π is given by equation w = 0.
Over U ∩ U i we then have that π * i f i is represented by the matrix where π * i G i is a (s − k 0 ) × (r − k 0 )−matrix whose entries are holomorphic functions which are all multiples of w. We let m i be the largest power of w one can pull out of π * i G i . Since a ij G i = G j b ij and a ij and b ij are invertible, we get m i = m j . We then let m k 0 (f ) := m i , and call it the vanishing multiplicity of f along Z k 0 . We will use the notation m(f ) for m k 0 (f ) if no confusion is possible.
Notice that there is a (r −k 0 )×(s−k 0 )−matrix G ij which does not vanish identically on the exceptional divisor of π i , such that We let which is a coherent subsheaf of O U i −modules of S i . Lemma 6.22. For every i, j ∈ I there is an isomorphism ψ ij : S i −→ S j such that S = { S i , φ ij } i,j∈I is an α−twisted coherent sheaf, and such that is an injective morphism of α−twisted coherent sheaves.
Proof. First, we define ψ ii := id S i . Now, let us consider i = j ∈ I. We represent ψ ij on a local frame by a s × s−matrix b ij = [β ij,pq ]. We let b ij = [ β ij,pq ] be defined as To this matrix we associate a morphism ψ ij : S i −→ S j , and we prove that it is an isomorphism whose inverse is ψ ji . To do so, let us calculate b ij · b ji , i. e. we calculate the entry γ p,q in position (p, q) of this product.
• If p, q ≤ k 0 we have where the last equality comes from the fact that the last sum is the entry in position (p, q) of b ij · b ji , which is a matrix representing φ ij • φ ji = id.
• If p > k 0 and q ≤ k 0 we have where the last equality is again as before (since p = q).
• If p ≤ k 0 and q > k 0 we have In conclusion b ij · b ji = I s , so ψ ij is an isomorphism whose inverse is ψ ji .
We are left to show that ψ ij • ψ jk • ψ ki = α ijk · id E i . A calculation similar to the previous one shows that b ij b jk b ki = α ijk I, which completes the proof of the fact that S is an α−twisted coherent sheaf.
The fact that f is a morphism of α−twisted coherent sheaves is as follows: represent f i by a matrix F i , f i by a matrix F i and t i by amatrix T i . Moreover, represent φ ij by a ij : we then have A calculation similar to the previous one for the product of b ij 's shows that T j · π * b ij = b ij . This shows that π * φ ij • f i = f j • ψ ij , so that f is a morphism of α−twisted coherent sheaves. As f i and t i are both injective, it follows that f is injective.
As a consequence, we now have a new exact sequence of π * α−twisted coherent sheaves on X, where S is again torsion-free of rank s and Q is torsion-free of rank q. The matrix F i representing f i is We have two possible cases: , m x is the maximal ideal of x, p is the smallest integer such that m p x ⊆ V ( G i ) (the ideal generated by the entries of G i ), and q is the smallest integer such that m q π(x) ⊆ V (G i ), then p < q.
We are now in the position to prove the following: Proposition 6.23. Let X be a compact manifold, E and α−twisted holomorphic vector bundle and S a torsion-free coherent subsheaf of E with torsion-free quotient. Then there exists a finite number of blow-ups and for each 1 ≤ k ≤ N there is a π * k • · · · • π * 1 (α)−twisted torsion-free coherent sheaf S k with an injective morphism f k : S k −→ π * k • · · · • π * 1 (E) verifying the two following properties: (1) for every i ∈ I there is a morphism t k,i : π * k S k−1 −→ π * k S k−1 such that (a) π * k f k−1,i = f k,i t k,i and (b) for every x ∈ X k , if the exceptional divisor of π k around x has equation w = 0, then t k,i is represented (with respect to a local frame) by a diagonal matrix whose entries are monomials in w. (2) The rank of f N is constant, so S N is a π * N • · · · • π * 1 (α)−twisted holomorphic subbundle of π * N • · · · • π * 1 E, and the corresponding quotient is a π * N • · · · • π * 1 (α)−twisted holomorphic bundle. Proof. The construction provided above shows us that after the blow-up of Z k 0 in X we get f : S −→ π * E. Now, for every k ∈ N let and let k 0 be the smallest k such that Z k = ∅. Then i) and ii) give If this last case happens, with a finite number of blow-ups we reduce to the case where k 0 > k 0 . We now repeat the construction, and the statement is proved.
The sequence of blow-ups described in the statement of Proposition 6.23 is called regularization of the exact sequence and each blow-up in the process is a regularization step.
6.3.2. Metrics, curvatures and regularization. As seen in section 2.7, if is an exact sequence of α−twisted holomorphic vector bundles on X and h is a Hermitian metric on E, then h induces Hermitian metrics h S on S and h Q on Q. Moreover, we have a splitting morphism ϕ : Q −→ E of α−twisted C ∞ vector bundles.
If we choose a local frame for E i and represent h i by a matrix H i , f i by a matrix F i and ϕ i by a matrix Φ i , let H S i be the matrix representing h S i and H Q i be the matrix representing h Q i : we then have Using the same notation of the previous section, let π : X −→ X be the blow-up of X along Z k 0 with reduced structure, and let , p, q > k 0 As a consequence of the previous Lemma we get: Corollary 6.25. Let R be the Chern curvature of (Q, h Q ), and R the Chern curvature of ( Q, h Q ). Let w be a local equation of the exceptional divisor of π, and let Proof. Consider the curvature R i of (Q i , h Q i ). By Lemma 1 of [13] we have where R i is the curvature of ( Q i , h Q i ). As the Chern curvature of (Q, h) is obtained locally as R i − B i id E i and the Chern curvature of ( Q, h Q ) is obtained locally as R i − π * i B i id π * i E i , the statement follows. Let now g be a Kähler metric on X and σ g its Kähler form. The metric π * g, whose associated (1, 1)−form is π * σ g , is not a Kähler metric on X since it is degenerate on the exceptional divisor of π. Anyway, if E is an α−twisted holomorphic vector bundle, and h is a Hermitian metric on it, then we may define the degree of E with respect to π * g as follows: Definition 6.26. The π * g−degree of π * E is deg π * g (π * E) := X π * γ 1 (E, h) ∧ π * σ n−1 g . Again, as σ g is closed and the exceptional divisor of π is contracted by π, the previous definition does not depend on h. Using the fact that by definition we have where R is the Chern curvature of (E, h), we get The same definition makes sense for every torsion-free α−twisted coherent sheaf whose Chern curvature is L 1 on the locally-free locus. We show that this is the case for S and Q as before. To do so, recall that given as before, on X \Z there is an element C ∈ A 0,1 (Hom(Q, S)), the second fundamental form of the induced metric. We then have C ∧C * ∈ A 1,1 (End(Q)), so that we let γ g (Q) := iΛ g (C ∧ C * ) ∈ End(Q).
Notice that T r(γ g (Q)) is a smooth function on X \ Z, and we have: Proposition 6.27. The second fundamental form C is in L 2 , and X\Z T r(γ g (Q))σ n g is finite. Moreover, the Chern curvatures of (S, h S ) and (Q, h Q ) are in L 1 .
Proof. The proof of the first part is identical to that of Proposition 1 of [13] (where one replaces Lemma 1 by Corollary 6.25). The second part is a consequence of the first part, of the Gauss-Codazzi equations for the curvatures on the locally free part, and of the fact that the Chern curvature of an α−twisted locally free sheaf is smooth.
As a consequence, the definition of degree of E with respect to g (resp. of degree of π * E with respect to π * g) extends to S and Q (resp. to S and Q). The following is then an immediate consequence of this definition and of Corollary 6.25 (see Lemma 2 of [13]).
be an exact sequence of α−twisted coherent sheaves, where E is locally free of rank r and S and Q are torsion-free of respective rank s and q. Let π : X −→ X be a regularization, and let be the regularized sequence. If g be a Kähler metric on X, then deg π * g ( S) = deg g (S), deg π * g ( Q) = deg g (Q). 6.4. Regularization and Donaldson's Lagrangian. We now define the Donaldson Lagrangian for the subsheaves S and Q in the exact sequence (2) 0 where E is an α−twisted locally free coherent sheaf and S and Q are torsionfree.
Recall that if h, k ∈ Herm + (E) and h ∈ Ω 0,1 h,k (E), then we defined where h t = h(t) and R t is the Chern curvature of (E, h t ). For a Kähler metric g on X whose associated Kähler form is σ g , we defined Notice that the previous definition may however depend on the choice of the regularization, which is not unique. The following tells us that it is not the case: Proposition 6.31. If π : X −→ X and π : X −→ X are two blowups producing regularizations of S and Q, then L S π * g = L S (π ) * g and L Q π * g = L Q (π ) * g . Proof. The proof is identical to that of Proposition 5 of [13].
This allows us to simplify the notation, and let L S g = L S π * g and L Q g = L π * g . 6.4.2. Relations between the Lagrangians. We now describe the relation between L g , L S g and L Q g . We will use the following notation: for every h ∈ Herm + (E) we let C h ∈ A 0,1 (Hom(Q, S)) be the second fundamental form of the metric h Q induced by h on Q.
If π : X −→ X is a regularization of the exact sequence (2), and if C π * h ∈ A 0,1 (Hom( Q, S)) is the second fundamental form of the metric h Q induced by π * h on Q, then the same proof of Proposition 1 of [13] shows that X T r(C π * h ∧ C * π * h ) ∧ π * σ n−1 g is a real number which does not depend on the chosen regularization. We let ||C h || 2 L 2 := X T r(C π * h ∧ C * π * h ) ∧ π * σ n−1 g , and call it the L 2 −norm of C h (we know that C h is in L 2 by Proposition 6.27). Using this notation we state now the following, which is the relation between the various Donaldson Lagrangians.
Proposition 6.32. In the exact sequence (2) suppose that µ g (S) = µ g (E). Then for every h, k ∈ Herm + (E) we have L g (h, k) = L S g (h, k) + L Q g (h, k) + ||C h || 2 L 2 − ||C k || 2 L 2 . Proof. Let us first suppose that S and Q are both locally free. We let Q S 1 (h, k) := log(det(f k S ,h S )), Q Q 1 (h, k) := log(det(f k Q ,h Q )). For every i ∈ I, choose a local frame of S i and a local frame of Q i , whose images in E i under f i and ϕ i give a local frame of E i , where ϕ = {ϕ i } is the splitting morphism ϕ : Q −→ E.
With respect to these local frames we represent h i by a matrix H i , k i by a matrix K i , f i by a matrix F i , ϕ i by a matrix P i , h S i by a matrix H S i , k S = −T r(f h S and as in the proof of Proposition 1 of [13] we have ||C π * k || 2 L 2 = ||C k || 2 L 2 , ||C π * h || 2 L 2 = ||C h || 2 L 2 , and we are done. 6.5. A lower bound. In this section we prove the following result: Proposition 6.33. Let X be a compact Kähler manifold with Kähler metric g, and consider the exact sequence (2), where we suppose that µ g (S) = µ g (E) and that S has minimal rank among the torsion-free α−twisted coherent subsheaves of E with this property. If π : X −→ X is regularization of the exact sequence (2) and S the twisted sheaf induced by S, then there is M ∈ R such that L S π * g (h, k) ≥ M for every h, k ∈ Herm + ( S). The proof will be as follows: we first assume that g is a Kähler metric on X with volume 1, and suppose that π consists of a single blow-up. If we take h, k ∈ Herm + (E) and let h S and k S be the induced Hermitian metrics on S, by definition we have L S g (h, k) = L S π * g (h S , k S ).
By Remark 6.30 we may assume that L S π * g ( h S , k S ) is calculated using a path h S ∈ Ω 0,1 h S ,k S ( S). We will let h t := h S (t) for every t ∈ [0, 1]: notice that this Hermitian metric is not necessarily induced by a Hermitian metric on π * E.
Let R t be the Chern curvature of ( S, h t ), and let K S t = iΛ π * g R S t : we remark that the metric that we are using to define the mean curvature is not a Kähler metric, since it blows-up along the exceptional divisor of π. Now, the evolution equation has the form f ht, h t = −( K t − c π * g ( S)id S ).
By Proposition 6.28 we know that c g (S) = c π * g ( S), hence the evolution equation is (4) f ht, h t = −( K t − c g (S)id S ).
Suppose now that h S is a solution of the evolution equation, and let L(t) : [0, 1] −→ R, L(t) := L S π * g ( h t , k S ). Then But then L is decreasing along any solution of the evolution equation, so that L S π * g is bounded from below along a solution of the evolution equation for every initial Hermitian metric on S coming from a Hermitian metric on E, and then it is bounded from below in general.
The previous strategy works if we are able to show that the evolution equation has a solution starting from any Hermitian metric on S coming from a Hermitian metric on E. Even if S is locally free, this cannot be concluded from 6.19 since K t is defined using π * g, which is not a Kähler metric. Proof. Let us first suppose that the regularization π : X −→ X of the exact sequence (2) is a single blow-up. Fix a Fubini-Study metric σ on the exceptional divisor of π, and take > 0 a small real number such that g := π * g + σ is a Kähler metric on X.
For every Hermitian metrics h, k ∈ Herm + ( S) we let L ( h, k) := L S g ( h, k), which is a Donaldson Lagrangian of a π * α−twisted holomorphic vector bundle with respect to a Kähler metric. By Proposition 6.19, the evolution equation for a given initial Hermitian metric. We show that there is a sequence m converging to 0 such that h m converges to a solution of the evolution equation (4). To do so, let K the mean curvature of ( S, h) with respect to g , i. e. K = iΛ g R where R is the Chern curvature of ( S, h). By definition we have X T r(− K + c g ( S)id S )π * σ n g = 0, hence it follows that the equation has a smooth solution, denoted ϕ . We now let h := e ϕ h, which is a Hermitian metric on S (since it is a conformal change of h. The evolution equation (5)  such that h (0) = h . We let R ,t be the Chern curvature of ( S, h ,t ), where h ,t = h (t), and K ,t will be its mean curvature. We now make the following: Claim: for every 0 < t 1 ≤ t 2 < +∞ there is a constant N ∈ R such that | R ,t | ≤ N for every and every t ∈ [t 1 , t 2 ].
Let us first show that if this claim holds, then we are done. Indeed, as in Corollary 6.16 the claim implies that for every k there is then a constant N k ∈ R such that | D k ,t R ,t | ≤ N k , where D ,t is the Chern connection of ( S, h ,t ).
This gives a a uniform bound for the C k −norm of R ,t . By compactness we then see that there is a subsequence m converging to 0 such that the solution h m,t converges to a Hermitian metric h 0,t for every t ∈ [t 1 , t 2 ], and hence we get a solution The subsequence m depends on t 1 and t 2 , so this does not yet give the desired solution. Anyway, if we take a sequence t n going to +∞, for each n we will find a subsequence n,m of n−1,m converging to 0 and such that h n,m,t converges to h 0,t for every t ∈ [t 1 , t n ]. Letting t n going to +∞ we get a solution h : [t 1 , +∞) −→ Herm + ( S) of the evolution equation (4). Up to rescaling, we are then done.
In conclusion, we are left to prove the claim, i. e. that for every 0 < t 1 ≤ t 2 < +∞ there is a constant N ∈ R such that | R ,t | ≤ N for every and every t ∈ [t 1 , t 2 ]. This requires some steps.
Step 2 : the same proof of Proposition 7 of [13] shows that the existence of a uniform bound for || K ,t || L 1 implies the existence of a uniform bound for || K ,t || L ∞ . As in Proposition 8 of [13], this implies a uniform bound on T r(f h ,t, h ,t ) (i. e. independent of > 0 and t ∈ [t 1 , t 2 ] for every 0 < t 1 ≤ t 2 < +∞). As in Lemma 6.18 this proves the uniform bound of the Chern curvatures we are looking for.
This concludes the proof of the statement in the case of a single blow-up. Suppose now that π = π k • π k−1 • · · · • π 1 , where π j : X j −→ X j−1 is a single blow-up (here we let X 0 := X, so that X k = X). For every j we let σ j be a Fubini-Study metric on the exceptional divisor of π j • · · · • π 1 , and for every choice of 1 , · · · , k > 0 we define a Kähler metric on X j in a recursive way as g j := π * j g j−1 + j σ j , where we let g 0 = g. If we let k go to 0, by the previous part of the proof we find a smooth solution of the evolution equation for the metric π * k g k−1 , defined on some interval [t k−1 , +∞). Repeating this process we will then find a smooth solution of the evolution equation with respect to π * g, defined over some interval [t 0 , +∞). Up to rescaling, we are done. 6.5.2. Bounds and subsheaves. Before giving the proof of Proposition 6.33 we need to recall some results of [25] and [26]. The first result is the following: Lemma 6.35. Let E be an α−twisted holomorphic vector bundle on a compact Kähler manifold X, and let h, k ∈ Herm + (E). Then ∆(log(T r(f h,k ))) ≤ 2(|K g (E, h)| + |K g (E, k)|).
Proof. The proof is essentially the same of that of point (d) of Lemma 3.1 in [25]. See even the proof of equation (1.9.2) of [26].
The second result, which is the content of Propositions 2.1 and 2.2 of [25], is the following: Lemma 6.36. Suppose that X is either a compact Kähler manifold with a Kähler metric g, or a Zariski open subset of a compact Kähler manifold with a metric g which is the restriction of a smooth metric. Let n be the dimension of X. Then the following properties hold.
(1) X has finite volume with respect to g.
Consider now the push-forward π * S: it is an α−twisted coherent sheaf on X which is locally free outside the closed subset Z we blow-up to obtain π : X −→ X. As π −1 (Z) has measure zero and X \ Z is isomorphic, via π, to X \ π −1 (Z), the same argument presented in section 6.5.2 shows that L S π * g, h 0 This induces a map F −→ π * E of α−twisted sheaves, and composing with the natural map π * π * E −→ E we then get a morphism j : π * F −→ E of π * α−twisted sheaves, which is injective over π −1 (Z). Hence ker(j) is a π * α−twisted sheaf whose support is contained in π −1 (Z).
But now notice that π * F/ ker(j) is a π * α−twisted coherent subsheaf of E, and as E is π * g−semistable we get finally that µ π * g (π * g F/ ker(j)) = µ π * g (E). Since the rank of π * F/ ker(j) equals the rank of π * F, and this is smaller than the rank of S, we get a contradiction with respect to the minimality of S. We may now apply the same strategy as in the proof of Proposition 6.33 to get the lower boundedness of L S π * g . 6.6. Conclusion of the proof. We are now in the position to prove Theorem 6.1, namely that an α−twisted holomorphic vector bundle E on a compact Kähler manifold with Kähler metric g is g−semistable if and only if it is approximate g−Hermite-Einstein.
By Theorem 4.23 we know that if E is approximate g−Hermite-Einstein, then it is g−semistable. We need to prove the converse, and by Proposition 6.21 it is sufficient to prove that the Donaldson Lagrangian L g,k is bounded below for every k ∈ Herm + (E). We will need the following: Lemma 6.39. Let X be a compact Kähler manifold with Kähler metric g, π : X −→ X a blow-up.
(1) If E is a g−semistable α−twisted holomorphic vector bundle on X, then π * E is a π * g−semistable π * α−twisted holomorphic vector bundle on X.
Proof. As degree and slope are defined even with respect to π * g, the notion of π * g−semistability is available as well, and has the same definition as the one with respect to a Kähler metric. Suppose first that π * E is not π * g−semistable. There is then a proper π * α−twisted coherent subsheaf F of π * E such that 0 < rk(F) < rk(π * E) and µ π * g (F) > µ π * g (π * E). As π is an isomorphism between X \ Z and X \ π −1 (Z), it follows that µ g (π * F) > µ g (E).
On X \ Z we have that π * F is a proper α−twisted coherent subsheaf of E. Since the codimension of Z is at least two, as already explained (see the proof of Proposition 6.38) we then have that π * F is a proper α−twisted coherent subsheaf of E on X, getting a contradiction, and hence showing the first point of the statement.
We are now in the position to prove Theorem 6.1: Proof. By Theorem 4.23 we just need to show that if E is g−semistable, then it is approximate g−Hermite-Einstein. By Proposition 6.21 we just need to prove that the Donaldson Lagrangian of E with respect to g is bounded below, i. e. we need to prove that for every k ∈ Herm + (E) there is a constant M k ∈ R such that L g (h, k) ≥ M k for every h ∈ Herm + (E). Fix then k ∈ Herm + (E).
If E is g−stable, by Theorem 5.1 we know that E is g−Hermite-Einstein, and hence approximate g−Hermite-Einstein by Proposition 3.25, and we are done. Suppose then that E is not g−stable, and let S be a torsion-free α−twisted coherent subsheaf of E such that µ g (S) = µ g (E) and such that Q = E/S is torsion-free. Suppose moreover that S has minimal rank among all the subsheaves with these properties.
Consider the exact sequence By minimality of S, we see that S is g−stable, hence by Proposition 6.21 the Donaldson Lagrangian L S g is bounded below, i. e. for every Hermitian metric k on E there is a constant B k ∈ R such that L S g,k (h) ≥ B k for every h ∈ Herm + (E). By Proposition 6.32 we have L g (h, k) = L S g (h, k) + L Q g (h, k) + ||C h || 2 L 2 − ||C k || 2 L 2 . Notice that ||C h || 2 L 2 is positive and ||C k || 2 L 2 is fixed once k is fixed. We then have L g (h, k) ≥ B k + ||C k || 2 L 2 + L Q g (h, k), so we just need to prove that there is a constant N k ∈ R such that L Q g (h, k) ≥ N k for every h ∈ Herm + (E).
By Lemma 6.39 we conclude Q is π * g−semistable. If it is π * g−stable, we are done.
Suppose then that Q is not π * g−semistable. Let S 1 be a torsion-free π * α−twisted coherent subsheaf of Q such that 0 < rk(S 1 ) < rk( Q), µ π * g (S 1 ) = µ π * g ( Q), and the quotient Q/S 1 is torsion-free. Choose S 1 to have minimal rank among all the subsheaves with these properties. We then see that S 1 is π * g−stable, and hence from Proposition 6.38 we know that there is a constant P k such that L S 1 π * g (h, k) ≥ P k for every h ∈ Herm + ( Q). Again we have L Q π * g (h, k) = L S 1 π * g (h, k) + L Q 1 π * g (h, k) + ||C h || 2 where Q 1 = Q/S 1 . As before, the problem is reduced to prove that there is a constant W k such that L Q 1 π * g (h, k) ≥ W k for every h ∈ Herm + ( Q). To do so we blow up again π 1 : X 1 −→ X in order to provide a regularization of Q 1 . By Lemma 6.39 we know that Q 1 is π * 1 π * g−semistable, and we have rk( Q 1 ) < rk( Q). After a finite number of steps of this type we then have to stop, concluding the proof. 6.7. Corollaries. We conclude the present paper with some easy corollaries of the (approximate) Kobayashi-Hitchin correspondence which may prove to be useful for some applications. The first two corollaries are immediate applications of Theorem 1.1 and Propositions 3.12, 3.29, 3.10 and 3.27 Corollary 6.40. Let E be an α−twisted holomorphic vector bundle.
(1) If E is g−stable, then ∧ p E and Sym p E are g−polystable.
(2) If E is g−semistable, then ∧ p E and Sym p E are g−semistable.
Corollary 6.41. Let E be an α−twisted holomorphic vector bundle and F a β−twisted holomorphic vector bundle.
(1) If E and F are g−stable, then E ⊗ F are g−polystable.
(2) If E and F are g−semistable, then E ⊗ F are g−semistable.
The following is known as Bogomolov's inequality.
Proof. The proof is identical to that of Theorems 4.7 and 5.7 in Chapter IV of [17]. Se even Theorem 2.2.3 and Corollary 2.2.4 of [21].