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Approximate Hermitian-Yang-Mills structures on semistable Higgs bundles

We review the notions of (weak) Hermitian-Yang-Mills structure and approximate Hermitian-Yang-Mills structure for Higgs bundles. Then, we construct the Donaldson functional for Higgs bundles over compact K\"ahler manifolds and we present some basic properties of it. In particular, we show that its gradient flow can be written in terms of the mean curvature of the Hitchin-Simpson connection. We also study some properties of the evolution equation associated to that functional. Next, we study the problem of the existence of approximate Hermitian-Yang-Mills structure and its relation with the algebro-geometric notion of Mumford-Takemoto semistability and we show that for a Higgs bundle over a compact Riemann surface, the notion of approximate Hermitian-Yang- Mills structure is in fact the differential-geometric counterpart of the notion of semistability. Finally, using Donaldson heat flow, we show that the semistability of a Higgs bundle over a compact K\"ahler manifold (of every dimension) implies the existence of an approximate Hermitian-Yang-Mills structure. As a consequence of this we deduce that many results about Higgs bundles written in terms of approximate Hermitian-Yang-Mills structures can be translated in terms of semistability.

Finally, using Donaldson heat flow, we show that the semistability of a Higgs bundle over a compact Kähler manifold (of every dimension) implies the existence of an approximate Hermitian-Yang-Mills structure. As a consequence of this we deduce that many results about Higgs bundles written in terms of approximate Hermitian-Yang-Mills structures can be translated in terms of semistability.

Some historical background
The notion of holomorphic vector bundle is common to some branches of mathematics and theoretical physics. In particular, this notion plays a fundamental role in complex differential geomtry, algebraic geometry, conformal string and Yang-Mills theories. Moreover, the study of holomorphic vector bundles involves techniques from geometric analysis, partial differential equations and topology. In this thesis we study Higgs bundles and some of their main properties. We restrict our study to the case when the complex manifold is compact Kähler. On the one hand, complex manifolds provide a rich class of geometric objects, which behave rather differently than real smooth manifolds.
In complex geometry, the Hitchin-Kobayashi correspondence asserts that the notion of (Mumford-Takemoto) stability, originally introduced in algebraic geometry, has a differential-geometric equivalent in terms of special metrics. In its classical version, this correspondence is established for holomorphic vector bundles over compact Kähler manifolds and says that such bundles are polystable if and only if they admit a Hermitian-Einstein 1 structure. This correspondence also holds for Higgs bundles.
The history of this correspondence probably starts in 1965, when Narasimhan and Seshadri [30] proved that a holomorphic bundle over a Riemann surface is stable if and only if it corresponds to a projective irreducible representation of the fundamental group of the surface. Then, in the 80's Kobayashi [23] introduced for the first time the notion of Hermitian-Einstein structure in a holomorphic vector bundle, as a generalization of Kähler-Einstein metric in a tangent bundle. Shortly after, Kobayashi [24] and Lübke [29] proved that a bundle with a irreducible Hermitian-Eintein structure must be necessarly stable. Donaldson [12] showed that the result of Narasimhan and Seshadri [30] can be formulated in term of metrics and proved that the concepts of stability and existence of Hermitian-Einstein metrics are equivalent for holomorphic vector bundles over Riemann surfaces. Then, Kobayashi and Hitchin conjectured that the equivalence sholud be true in general for holomorphic vector bundles over Kähler manifolds.
The existence of a Hermitian-Einstein structure in a stable holomorphic vec-tor bundle was proved by Donaldson for projective algebraic surfaces in [13] and for projective algebraic manifolds in [14]. Finally, Uhlenbeck and Yau showed in [34] proved for general compact Kähler manifolds using some techniques from analysis and Yang-Mills theory. Hitchin [21], while studying the self-duality equations over a compact Riemann surface, introduced the notion of Higgs field and showed that the result of Donaldson for Riemann surfaces could be modified to include the presence of a Higgs field. Following the result of Hitchin, Simpson [31] defined a Higgs bundle to be a holomorphic vector bundle together with a Higgs field and proved the Hitchin-Kobayashi correspondence for such objects. Actually, using some sophisticated techniques in partial differential equations and Yang-Mills theory, he proved the correspondence even for non-compact Kähler manifolds, if they satisfy some analytic conditions. As an application of this, Simpson [32] later studied in detail a one-to-one correspondence between stable Higgs bundles over a compact Kähler manifold with vanishing Chern classes and irreducible representations of the foundamental group of that Kähler manifold.
The Hitchin-Kobayashi correspondence has been extended in several directions. Lübke and Teleman [28] studied the correspondence for compact complex manifolds. Bando and Siu [1] extended the correspondence to torsion-free sheaves over compact Kähler manifolds and introduced the notion of admissible Hermitian metric for such objects. Following the ideas of Bando and Siu, Biswas and Schumacher [3] introduced the notion of admissible Hermitian-Yang-Mills metric in the Higgs case and generalized this extension to torsion-free Higgs sheaves.
In [13] and [14] Donaldson introduced a functional, which is indeed known as the Donaldson functional, and later Simpson [31] extended this functional in his study of the Hitchin-Kobayashi correspondence for Higgs bundles. Kobayashi in [25] constructed the same functional in a different form and showed that it plays a foundamental role in a possible extension of the Hitchin-Simpson correspondence. In fact, he proved in [25] that for holomorphic vector bundles over projective algebraic manifolds, the counterpart of semistability is the notion of approximate Hermitian-Yang-Mills structure.
The correspondence between semistability and the existence of approximate Hermitian-Yang-Mills structure in the ordinary case has been originally proposed by Kobayashi. In [25] he proved that for a holomorphic vector bundle over a compact Kähler manifold a boundedness property of the Donaldson functional implies the existence of a Hermitian-Einstein structure and that implies the semistability of the bundle. Then, using some properties of the Donaldson functional and the Mehta-Ramanathan Theorem, he established the boundedness propery of the Donaldson functional for semistable holomorphic bundles over compact algebraic manifolds. As a consequence of this, he obtained the correspondence between semistability and the existence of approximate Hermitian-Einstein structures when the base manifold was projective. Then he conjectured that all three conditions (the boundedness property, the existence of an approximate Hermitian-Einsten structure and the semistability) should be equivalent in general, that is, independently from whether the manifold was algebraic or not.
In the Higgs case and when the manifold is one-dimensional (a compact Riemann surface), the boundedness property of the Donaldson functional follows from the semistability in a similar way to the classical case, since we need to consider only Higgs subbundles and their quotients and we have a decomposition of the Donaldson functional in terms of these objects. The existence of approximate Hermitian-Einsten metrics for semistable holomorphic vector bundles has been recently studied in [22] using some techniques developed by Buchdahl [7], [8] for the desingularization of sheaves in the case of compact complex surfaces. One of the main difficulties in the study of this correspondence in higher dimensions arises from the notion of stability, since for compact Kähler manifolds with dimension greater or equal than two, it is necessary to consider subsheaves and not only subbundles. On the other hand, properties of the Donaldson functional commonly involve holomorphic bundles. Although all these difficulties appears also in the Higgs case, using Donaldson heat flow, Li and Zhanh [17] recently showed that the semistability of a Higgs bundle over a compact Kähler manifold implies the existence of approximate Hermitian-Einsten structure. It remains to be proved that, in dimension greater or equal than two, the semistability of the Higgs bundle implies the boundedness propery of the Donaldson functional. In order to prove this, it seems natural to introduce first the notion of admissible Hermitian metrics on Higgs sheaves. Then, to define the Donaldson functional for such objects using these metrics and finally, following [22] to study how this functional defined for a semistable Higgs bunde can be decomposed in terms of Higgs subsheaves and their quotients.

About this thesis
This thesis is organized as follows. In Chapter 1 we start with some basic definitions and results, in particular we introduce complex manifolds and almost complex structures on differentiable manifolds. Then we summarize the main properties of complex differential forms over complex manifolds. Finally, we introduce Kähler manifolds. After studying some properties of these objects, we give an exemple of complex manifolds which cannot be given a Kählerian structure.
In Chapter 2 we present some definitions and results on principal fibre bundles over a differentiable manifold M. In particular we study connections and curvatures in principal fibre bundles and present some important results such as the structure equation and the Bianchi identity.
Although our primary interest lies in holomorphic vector bundles, we begin Chapter 3 with the study of connections in differentiable complex bundles. Then, we introduce connections in complex vector bundles over complex manifolds and we characterize those complex vector bundles which admit holomorphic structures. Moreover, we define Hermitian structures in complex vector bundles over (real or complex) manifolds and we study those connections which are compatible with the holomorphic structure and the Hermitian metric. Finally, we summarize the main properties of subbundles and quotient bundles. We study how Hermitian metrics on holomorphic vector bundles (over complex manifolds) induce metrics on subbundles and quotient bundles. Moreover, we give the definition and study the main properties of the second fundamental form of subbundles and quotient bundles.
In Chapter 4 we introduce Chern classes. First of all we define the first Chern class of a line bundle. Then, using the splitting principle, we define higher Chern classes for complex vector bundles of any rank. Moreover, we present the axiomatic approach to Chern classes: this enables us to separate differential geometry aspects of Chern classes from their topological aspects. Finally, via de Rahm theory, we give a representation of Chern classes in terms of the curvature form of a connection in a complex vector bundle.
In Chapter 5 we summarize some basic properties of coherent sheaves over compact Kähler manifolds. Then, we review the definition of singularity sets for coherent sheaves and briefly comment some of their main properties concerning to the codimension of these singularity sets. We review the construction of the determinant bundle of a coherent sheaf and we write some facts on determinant bundles that are used through this work. Moreover, we present some useful analytic results such as the Fredholm alternative Theorem and the Maximum principle for parabolic equations.
In Chapter 6 we study the basics results of Higgs sheaves. These results are important mainly because the notion of stability in higher dimension (greater than one) makes reference to Higgs subsheaves and not only Higgs subbundles. Moreover, we summarized some properties of metrics and connections on Higgs bundles and itroduce the space of Hermitian structures, which is the space where the Donaldson functional is defined. Finally, we construct the Donaldson functional for Higgs bundles over compact Kähler manifolds following a construction similar to that of Kobayashi and we present some basic properties of it. In particular, we prove that the critical points of this functional are precisely the Hermitian-Yang-Mills structures, and we show also that its gradient flow can be written in terms of the mean curvature of the Hitchin-Simpson connection. We also establish some properties of the solution of the evolution equation associated with that functional. Next, we study the problem of the existence of approximate Hermitian-Yang-Mills structure and its relation with the algebro-geometric notion of Mumford-Takemoto semistability. We prove that if the Donaldson functional of a Higgs bundle over a compact Kähler manifold is bounded from below, then there exists an approximate Hermitian-Yang-Mills structure on it. This fact, together with a result of Bruzzo and Graña Otero [6], implies the semistability of the Higgs bundle. Then we show that a semistable but non-stable Higgs bundle can be included into a short exact sequence with a stable Higgs subsheaf and a semistable Higgs quotient. We use this result in the final part of this chapter when we show that for a Higgs bundle over a compact Riemann surface, the notion of approximate Hermitian-Yang-Mills structure is in fact the differential-geometric counterpart of the notion of semistability.
In Chapter 7, using Donaldson heat flow, we show that the semistability of a Higgs bundle over a compact Kähler manifold (of every dimension) implies the existence of an approximate Hermitian-Yang-Mills structure; this proof is due to Jiayu and Zhang [17] and essentially follows from Simpson [31].

Chapter 1
Basics on Complex Manifolds

Complex manifolds
In this first chapter we start with some basic definitions and results, in particular we introduce complex manifolds and almost complex structures on differentiable manifolds.
is a biholomorphim. A complex structure on M is an equivalence class A of complex n-atlases. Definition 1.1.3. Let M be a T2, N2 topological space and let A a complex structure on M. We say that the pair (M, A) is a complex manifold. We also define the complex dimension of M as dim C = n.
α is holomorphic. Holomorphic functions have some interesting properties, see [10] and [16] for more details. The gradient of f is the unique vector field ∇f ∈ χ(M ) such that for every vector field X ∈ χ(M ) g(∇f, X) = df (X).
A point P ∈ M is said to be a critical point of f if (∇f ) P = 0. Note 1.1.2. In local coordinates we have Definition 1.1.6. Let (M, g) be a Riemannian manifold of (real) dimension n and let X ∈ χ(M ) be a vector field on M. Let ∇ be the Levi-Civita connection associated with the Riemannian metric g. The divergence of X the function on M obtained by contraction of the (1, 1)-tensor field ∇X, i.e., the function div(X) = C 1 1 (∇X).
where Γ i jl are the Christoffel's symbols of the Levi-Civita connection ∇ associated with g. Definition 1.1.7. Let (M, g) be a Riemannian manifold of (real) dimension n and let f be a C ∞ complex valued function on M. We define the Laplace-Beltrami operator as the divergence of the gradient: ∆f = div(∇f ). Definition 1.1.8. Let M be a complex manifold of complex dimension n and let f : M −→ C be a C ∞ function. We say that f is harmonic if, regarded as a function on the real Riemannian manifold M of real dimension 2n, ∆f = 0. We now define and study almost complex manifolds. Definition 1.1.9. Let M be a differentiable manifold. A pair (M, J) is called an almost complex manifold if J is a tensor field of type (1, 1) such that at each point P ∈ M, we have: J 2 P = −1 P . The tensor field J is called the almost complex structure of M.
Let us look at the tangent space of a complex manifold M with dim C M = n. The tangent space T P M , regarded as a 2n-real vector space, is spanned by 2n-real vectors {∂/∂x 1 , . . . , ∂/∂x n , . . . , ∂/∂y 1 . . . , ∂/∂y n }, where z k = x k + iy k are the complex coordinates in a complex chart (U, φ). With the same coordinates the dual space T * P M as 2n-dimensional (real) vector space is spanned by {dx 1 , . . . , dx n , dy 1 , . . . , dy n }.
Let us define 2n-vectors where 1 k n. Clearly they form a basis of the 2n-dimensional (complex) vector space They are dual to ∂/∂z k and ∂/∂z k . Proposition 1.1.5. Let M be a complex manifold of (complex) dimension n. Regarded as a 2n-differentiable manifold, M admits an almost complex structure J.
Proof. Let M be a complex manifold and define a linear map J P : T P M −→ T P M by: J P is a real tensor of type (1, 1) and J 2 P = −1 P where 1 P is the identity map on T P M. With respect to the basis {∂/∂x 1 , . . . , ∂/∂x n , . . . , ∂/∂y 1 . . . , ∂/∂y n }, J P takes the form: We only have to show that the action of J P is independent of the chart. In fact, let (U, φ) and (V, ψ) be overlapping charts with local coordinates z k = x k + iy k and w k = u k + iv k . On U ∩ V, the functions z k = z k (w) satisfy the Cauchy-Riemann relations. Then we find: We also find J(∂/∂v k ) = −∂/∂u k and this completes the proof.
Let (M, J) be an almost complex manifold. J P may be defined on T P M C by setting Then we have the identities Thus we have an expression for J P in (anti-) holomorphic bases, Let Z ∈ T P M C be a vector of the form Z = Z k ∂/∂z k . Then Z is an eigenvector of J P , in fact we have J P Z = iZ. In the same way if W = W k ∂/∂z k , it satisfies J P W = −iW. In this way T P M C is decomposed into a direct sum Definition 1.1.10. Z ∈ T P M C+ is called a vector of type (1, 0), while W ∈ T P M C− is called a vector of type (0, 1).
We can easy verify that Note 1.1.6. We have the following identity for the dimensions of these complex vector spaces: Definition 1.1.11. Let M be a complex manifold of (complex) dimension n.
We define X (M ) C = X (M ) ⊗ C. Given a complex vector field Z ∈ X (M ) C , Z is naturally decomposed as Z = Z + + Z − . Z + is called the component of type (1, 0), while Z − of type (0, 1). Accordingly once J is given, X (M ) C is decomposed uniquely as Definition 1.1.12. Let (M, J) be an almost complex manifold. If the Lie bracket of any vector fields of type (1, 0) X, Y ∈ X (M ) C+ is again of type (1, 0), i.e. [X, Y ] ∈ X (M ) C+ , the almost complex structure J is said to be integrable.
Theorem 1.1.7. Let (M, J) be an almost complex manifold. If the almost complex structure J is integrable then M admits a complex structure and such complex structure is unique up to biholomorphism.

Complex differential forms
In this section we define the main properties of complex differential forms over a complex manifold.
In local coordinates ω ∈ A p has the form: here ω j1...jp are C ∞ (U, C) functions on the local chart (U ; x 1 , . . . , x n ).
Now we define complex differential forms on a complex manifold M and then we introduce the Dolbeault complex and the Dolbeault cohomology groups. Definition 1.2.2. Let M be a complex manifold of (complex) dimension n and let p, q 0. We define the space of (p, q)-forms over M. As in the previous definition, in local coordinates ω ∈ A p,q has the form: where ω i1...ipj1...jp are C ∞ (U, C) functions on (U ; z 1 , . . . , z n , z 1 , . . . , z n ).
So we have A r = p+q=r A p,q and the exterior differential d has the form d = d + d , where d : A p,q −→ A p+1,q and d : A p,q −→ A p,q+1 . Now we can define the Dolbeault cohomology groups. Definition 1.2.3. Let M be a complex manifold of (complex) dimension n and let r 0. The sequence of C-linear maps is the Dolbeault complex. The set of d -closed (p, q)-forms is called the is denoted by Z p,q (M ) and its element are called (p, q)-cocycles, while the set of d -exact (p, q)-forms is denoted by B p,q (M ) and its element are called (p, q)-coboundaries. Since d • d = 0, we have B p,q (M ) ⊆ Z p,q (M ). The complex vector space is called the (p, q)th Dolbeault cohomology group of the complex manifold M.
In the real case every closed differential form is locally exact (Poincaré's Lemma). As seen in [16] in the complex case we have a similar result due to Grothendieck. Every d -closed complex (p,q)-form, with q 1, is locally d -exact. Definition 1.2.4. Let r 1 , . . . , r n ∈ (0, +∞]. A polydisc in C n is a set ∆ = {(z 1 , . . . , z n ) ∈ C n ||z j | < r j }.

Kähler manifolds
Let M be a complex manifold with dim C M = n and let g be a Riemann metric on M as a differentiable manifold. Take Z = X + iY and W = U + iV and extend g to T P M C so that The components of g with respect to the bases (1.1) of T P M C+ and (1.2) of T P M C− are: g ij (P ) = g P (∂/∂z i , ∂/∂z j ) g i (P ) = g P (∂/∂z i , ∂/∂z j ), gī j (P ) = g P (∂/∂z i , ∂/∂z j ), gī(P ) = g P (∂/∂z i , ∂/∂z j ).
We writeī and to stress that g is now meant to be extended to T P M C . One checks that: If a Riemannian metric g of a complex manifold M satisfies at each point P ∈ M and for any X, Y ∈ T P M, it is said to be a Hermitian metric. The pair (M, g) is called a Hermitian manifold. Proof. Let g be any Riemannian metric of a complex manifold M . Define a new metricg byg Clearlyg P (JX, JY ) =g P (X, Y ). Moreover,g is positive definite provided that g is. Henceg is a Hermitian metric on M.
Let g be a Hermitian metric on a complex maifold M . From (1.6), we find that g ij = gī. Thus the Hermitian metric g takes the form It is easy to see that h P is a positive-definite Hermitian form in T P M C+ . In fact: and h(X, X) = g(X, X) = g(X 1 , X 1 ) + g(X 2 , X 2 ) 0 for X = X 1 + iX 2 . This is why a metric g satisfying (1.6) is called Hermitian.
Let (M, J, g) a Hermitian manifold. Define a tensor field ω whose action on Note that ω is antisymmetric because we have identity hence ω is a 2-form. Proof. Let J be the almost complex structure and let X, Y ∈ T M C+ . From the definition of T M C+ we have JX = iX and JY = iY. Then we have Hence, ω(X, Y ) = 0. In the same way, if X, Y ∈ T M C− , ω(X, Y ) = 0, proving that the Kähler form ω is of type (1, 1). Proof. In local coordinates we have Proof. In local coordinates the Kähler form is ω = ig i dz i ∧ dz j . Then we have A Hermitian manifold (M, g) of (complex) dimension n regarded as a 2n-dimensional differentiable manifold is always orientable.
Proof. Let ω the Kähler form. Then ω n is a 2n non-vanishing real form, in fact ω n = (i) n n! det(g i )dz 1 ∧ dz 1 ∧ · · · ∧ dz n ∧ dz n and det(g i ) is nowhere vanishing. 1. g is a Kähler metric, i.e., dω = 0, 4. There exists a locally defined real function f such that ω = id d f, i.e., Proof. See [25] for a detailed proof.

The Hopf manifold
Let ∆ be the discrete group generated by z k → 2z k , 1 k n. The quotient manifold (C n \ {0})/∆ is a complex manifold called the Hopf manifold. It is homeomorphic to S 1 × S 2n−1 . Now, we want to prove that for n 2 the Hopf manifold cannot be given a Kählerian structure. In fact compact Kähler manifolds have strong topological restrictions. Perhaps the simplest among them is the following: Proof. Since n 2, Then, for n 2 the second Betti number of the Hopf manifold S 1 × S 2n−1 is 0, and this shows that for n 2 the Hopf manifold cannot be given a Kählerian structure.

Principal Fibre Bundles
In this chapter we present some definitions and results on principal fibre bundles over a differentiable manifold M. See [27] for more details.

Basics on principal G-bundles
Definition 2.1.1. Let M be a differentiable manifold and G a Lie group. A (differentiable) principal fibre bundle over M with group G consists of a manifold P and an action G on P satisfying the following conditions: 1. G acts freely on P on the right, 2. M is the quotient space of P by the equivalence relation induced by G, i.e., M = P G and the canonical projection π : P −→ M is differentiable, 3. P is locally trivial, that is, every point x of M has a neighborhood U such that π −1 (U ) is isomorphic with U × G in the sense that there is a diffeomorphism ψ : π −1 (U ) −→ U × G such that ψ(u) = (π(u), ϕ(u)) where ϕ is a mapping of π −1 (U ) into G satifsfying ϕ(ua) = (ϕ(u))a for all u ∈ π −1 (U ) and a ∈ G.
A principal fibre bundle will be denoted by P (M, G), or simply P. We call P the total space, M the base space, G the structure group and π the projection.
For each x ∈ M, π −1 (x) is a closed submanifold of P called the fibre over x. If u ∈ π −1 (x) is a point in the fibre over x, then π −1 (x) is the set of point ua with a ∈ G. Every fibre is diffeomorphic to G.
Definition 2.1.2. Let P (M, G) and Q(N, H) be principal fibre bundles over the (differentiable) manifolds M and N respectively. A morphism of principal fibre bundles consist of a C ∞ mapping f : P −→ Q and a homomorphism ψ : G −→ H such that f (ua) = f (u)ψ(a) for all u ∈ P and a ∈ G. For the sake of simplicity, we shall denote f and ψ by the same letter f. Every morphism f : P −→ Q maps each fibre of P into a fibre of Q and hence it induces a mapping of M into N, which will be also denoted by f.  ), the action of G on P induces a homomorphism σ of the Lie algebra g of G into the Lie algebra X (P ) of vector fields on P (see Proposition 4.1 in [27]). For each A ∈ g, A * = σ(A) is called the foundamental vector field corresponding to A.
In order to relate our intrinsic definition of a principal fibre bundle to the definition and the construction by means of an open cover, we need the concept of transition functions. By Conversely, we have the following: we can construct a (differentiable) principal fibre bundle P (M, G) with transition functions ψ αβ . Definition 2.1.5. Let P (M, G) a principal fibre bundle over the differentiable manifold M and let f : N −→ M be a C ∞ mapping. The pull-back of P (via f ) is the principal fibre bundle over N defined as the fibre product f * P = N × M P = {(n, u) ∈ N × P |f (n) = π(u)}.

Connections in a principal fibre bundle
Definition 2.2.1. Let P (M, G) be a principal fibre bundle over a manifold M with structure group G. For each u ∈ P, let T u P be the tangent space of P at the point u and G u ⊆ T u P the subspace of T u P consisting of vectors tangent to the fibre through u. A connection Γ in P is a C ∞ assignment of a subspace Q u of T u P for each u ∈ P such that 2. Q ua = (R a ) * Q u for every u ∈ P and a ∈ G.
The second condition means that the distribution u −→ Q u is invariant under the action of G over the total space P. We call G u the vertical subspace and Q u the horizontal subspace of T u P. In this way a vector X u ∈ T u P can be uniquely written as direct sum of his vertical and horizontal component: Definition 2.2.2. Given a connection Γ in a principal fibre bundle P, we define a 1-form ω on P with values in the Lie algebra g of G as follows. For each X u ∈ T u P, we define ω(X) to be the unique A ∈ g such that (A * ) u is equal to the vertical component of X u . The form ω is called the connection form of the given connection Γ.
Proposition 2.2.1. The connection form ω of a connection Γ in P satisfies the following conditions: for every a ∈ G and every vector field X on P, where ad denotes the adjoint representation of G on g.
The projection π : P −→ M induces a linear mapping π : T u P −→ T x M for each u ∈ P, where x = π(u). When a connection is given, π maps the horizontal subspace Q u isomorphically onto T x M. Definition 2.2.3. Given a connecion Γ in P, the horizontal lift of a vector field X on M is the unique vector field X * on P which is horizontal and which projects onto X, i.e. π(X * u ) = X π(u) for every u ∈ P. Proposition 2.2.2. Let Γ be a connection in P and let X be a vector field on M, there is a unique horizontal lift X * of X. The lift X * is invariant by R a for every a ∈ G. Conversely, every horizontal vector field X * on P invariant by G is the lift of a vector field X on M.
Proposition 2.2.3. Let X * and Y * be the horizontal lifts of X and Y respectively. Then 1. X * + Y * is the horizontal lift of X + Y.

Curvature form and structure equation
Such a form is called a tensorial form if it is horizontal in the sense that ϕ(X 1 , . . . , X r ) = 0 whenever at least one of the tangent vectors X i is vertical, i.e., tangent to a fibre.
By the above definition we see that if Γ is a connection in a principal fibre bundle P with structure group G, then its connection form ω is a pseudotensorial 1-form of type (ad, g). Now we present the main results of this section (see [27], p.76-78 for details). Let Γ be a connection in P and let G u and Q u be the vertical and the horizontal subspaces of T u P, respectively. Let h : T u P −→ Q u be the projection.  Then

Connections in Vector Bundles
Although our primary interest lies in holomorphic vector bundles, we begin this chapter with the study of connections in differentiable complex vector bundles.
Definition 3.0.4. Let M be a differentiable manifold of (real) dimension n. A C ∞ complex vector bundle over M of rank r is differentiable manifold E together with a surjective C ∞ map π : E −→ M such that exists a countable open covering {U α } satisfying: 1. There are diffeomorphisms φ α : π −1 (U α ) −→ U α × C r such that the diagrams commute Here pr 1 is the projection on the first factor.

When
In fact it is a bijective C-linear endomorphism of C r .
Note 3.0.5. In the same manner, if M is complex manifold, we can define a holomorphic vector bundle E over M in the obvious way.
As in the case of principal fibre bundles, we have the following result: we can construct a complex vector bundle of rank r over the manifold M with transition functions ψ αβ .

For every
Finally, if f is a diffeomorphism, we say that the complex vector bundles (E, M, π E ) and (F, N, π F ) are isomorphic.
Definition 3.0.6. Let (E, M, π E ) be a complex vector bundle of rank r over the differentiable manifold M. A subbundle of E is a complex vector bundle is commutative. In particular (f * E, M, pr 1 ) is a complex vector bundle over M of rank r and pr 2 : f * E −→ E is a morphism of complex vector bundles.
Note 3.0.8. In the same way we can define morphisms, subbundles and pull-back bundles of holomorphic vector bundles over complex manifolds.

Connections in complex vector bundles (over real manifolds)
Let M be a differentiable manifold of (real) dimension n and let E a C ∞ complex vector bundle of rank r over M. We make use of the following notation: By abuse of notation we omit ⊗ and write D(f σ) = σdf + f Dσ.
We call the matrix 1-form ω = (ω j i ) the connection form of D with respect to the frame fiels s.
We call Dζ the covariant derivative of ζ.
Evaluating D on a tangent vector X of M at the point x, we obtain an element of the fibre E x denoted by

4)
In terms of the local frame field s, (3.4) can be written as a system of ordinary differential equations We shall now study how the connection form ω changes when we change the local frame field s. Let s = (s 1 , . . . , s r ) be another local frame field over U. It is related to s by s = s a, Let ω be the connection form of D with respect to s . Then In fact we have, We extend a connection D : by setting Definition 3.1.5. Using this extended D, we define the curvature R of the connection D to be Then R is A 0 -linear. In fact, if f ∈ A 0 and σ ∈ A 0 (E), then Hence R is a 2-form on M with values in End(E), i.e., R ∈ A 2 (End(E)).
Definition 3.1.6. Using the matrix notations of (3.3) the curvature form Ω of D with respect to the frame field s is defined by Then Exterior differentiation of (3.6) gives Bianchi identity: If ω is the connection form of D relative to another frame field s = sa −1 , the corresponing curvature form Ω is related to Ω by In fact, is the C ∞ transition function. Then we have: Conversely, given a system of gl(r, C)-valued 1-forms ω α on U α satisfying (3.9), we obtain a connection D in E having {ω α } as connection forms. If Ω U is the curvature form of D relative to s α , then (3.8) means Given a connection D in E, we define a connection, also denoted by D, in E * by the following formula: Proposition 3.1.1. Let E and F two complex vector bundles over the same manifold M. Let D E and D F be connections in E and F, respectively. Then we can define connections If we denote the curvatures of D E and D F by R E and R F , then we have If s = (s 1 , . . . , s r ) is a local frame field of E and t = (t 1 , . . . , t p ) is a local frame field of F and ω E , ω F , Ω E , Ω F , are the connecionts and the curvature forms with respect to these frame fields, then in a natural manner the connection and curvature forms of D E ⊕ D F are given by and while those of D E⊗F are given by Here I r and I p denote the identity matrices of rank r and p. All these formulas extend in an obvious way to the direct sum and the tensor product of any number of vector bundles and so they give formulas for the connection and curvature in Let E be a complex vector bundle over M and let N be another manifold. Given a C ∞ mapping f : N −→ M, we obtain an induced vector bundle f * E over N.
Since in the cathegory of C ∞ bundles there is an isomorphism If ω and Ω are respectively the connection form and the curvature form of D over a local frame field (U α , s α ), then f * ω and f * Ω are the connection form and the curvature form with respect to the pull-back local frame field (f −1 U, f * s).

Connections in complex vector bundles (over complex manifolds)
Let M be a complex manifold with (complex) dimension n and E a C ∞ complex vector bundle of rank r over M. In addition to the notations A p and A p (E) introduced in the previous section, we use the following: Decomposing D according to the bidegree, we have, for σ ∈ A 0 (E) and Let R be the curvature of D, i.e., R = D • D ∈ A 2 (End(E)). Then Let s be a local frame field of E and let ω and Ω be the connection and the curvature forms of D with respect to s. We can write Let M be a complex manifold of (complex) dimension n. and let E be a holomorphic vector bundle of rank r over M. Let {U α } be an oper cover which trivializes E and let s α = (s α 1 , . . . , s α r ) be a holomorphic frame field on U α . Let Proof. First of all we prove that d E is well defined. Let s β = (s β 1 , . . . , s β r ) be a holomorphic frame field on U β . On the overlapping open set U α ∩ U β , s β is related to s α by and this proves that d E is well defined.
After a straightforward computation we find Finally, from d • d = 0 we have d E • d E = 0, and this completes the proof.
We shall now describe those complex vector bundles which admit holomorphic structures. Let A p,q be the sheaf of complex (p, q)-forms over M and let A p,q (E) = E ⊗ A p,q be the sheaf of complex (p, q)-forms over M with values in E. Then we have the following Theorem 3.2.2. Let M be a complex manifold of (complex) dimension n and let E be a complex vector bundle of rank r over M. The following conditions are equivalent: 1. There exists a holomorphic vector bundle structure on E,

There exists an operator
Moreover, we have the following results: Proof. See [25] for a detailed proof. Proposition 3.2.5. For a connection D in a holomorphic vector bundle E, the following conditions are equivalent: 2. For every local holomorphic section s, Ds is of degree (1, 0), 3. With respect to a local holomorphic frame field, the connection form ω is of degree (1, 0).

Connections in Hermitian vector bundles
Definition 3.3.1. Let E be a C ∞ complex vector bundle over a (real or complex) manifold M. A Hermitian structure or Hermitian metric h in E is a C ∞ field of Hermitian inner products in the fibre of E. Thus, We call (E, h) a Hermitian vector bundle.
Given a local frame field s α = (s 1 , . . . , s r ) of E over U α , we set Then H α is a positive definite Hermitian matrix at every point of U α . When we are working with a single frame field, we often drop the subscript α. We say that s α is a unitary frame field or orthonormal frame field if H α is the identity matrix. Under a change of frame field given by s α = s β g βα , we have , if it makes h parallel in the following sense: Let ω = (ω i j ) be the connection form relative to the local frame field s U . Then setting ζ = s i and η = s j in (3.11), we obtain (3.12) So in matrix notation we have Applying d to (3.3) we obtain If E is a holomorphic vector bundle over a complex manifold M, then a Hermitian structure h determines a natural h-connection satisfying D = d E . Namely, we have Proof.
1. First af all we prove the uniqueness. Let D be such a connection and let s α = (s 1 , . . . , s r ) be a local frame field on U α . Since This determines the connection form ω α , i.e., Hence, we have proved uniqueness.
2. Now, we want to prove existence. Let ω t α = d H α H −1 α . By a straightforward calculation we can see that the collection {ω α } satisfies (3.9) and this completes the proof.
The second part of this Proposition follows 3.2.4. (See [25], p. 9 and p. 12 for more details).
With respect to a local holomorphic frame field the connection form ω = ω i j is of degree (1, 0). Since the curvature form Ω s equal to the (1, 1)-component of dω + ω ∧ ω, we obtain In local coordinates we write

Subbundles and quotient bundles
Let E be a holomorphic vector bundle of rank r over a complex manifold M of (complex) dimension n and let S be a holomorphic subbundle of rank p of E. Then the quotient bundle Q = E S is a holomorphic vector bundle of rank r − p. We can express this situation as an exact sequence Let h be a Hermitian structure in E. Restricting h to S, we obtain a Hermitian structure h S in S. Taking the orthogonal complement of S in E with respect to h, we obtain a complex subbundle S ⊥ of E.
As a C ∞ complex vector bundle, Q is naturally isomorphic to S ⊥ . Hence, we obtain also a Hermitian structure h Q in a natural way.

Proposition 3.4.2.
Under the hypothesis of the previous definition, we have the following results: 1. D S is the Hermitian connection of (S, h S ),
Proof. Let f be a function on M.
Replacing ζ by f ζ in (3.14), we obtain On the other hand, Comparing the components of the two decompositions of D(f ζ), we conclude The first equality says that D S is a connection and the second says that A is a 1-form with values in Hom(S, and this proves that D S preserves h S . With the identification Q = S ⊥ , we can consider A as an element of A 1,0 (Hom(S, Q)).
Similarly, we define D S ⊥ and B by setting In the hypothesis of the previous definitions and constructions we have the following results: Proof. The proof is very similar to the previous one. See [25] for more details.

Chapter 4
Chern Classes

Chern classes of a line bundle
In this section we define the first Chern class of a line bundle. We start with some useful definitions.
and consider the holomorphic line bundle over P n (C) with transition functions g ij = z i z j . We will call it the universal line bundle over P n (C). Definition 4.1.2. Let L be the universal line bundle over P n (C), and let H = {a 0 z 0 + · · · a n z n = 0} be an hyperplane in P n (C). If we take nonhomogeneous coordinates we have Because of this origin the latter bundle, to be denoted by H, is called the hyperplane section bundle of P n (C). It is the dual bundle of the universal line bundle over P n (C).
where j : Z −→ A is simply the natural injection and e : A −→ A * is the exponential map This induces an exact sequence of cohomology groups where δ is the connecting homomorphism. Since A is a fine sheaf over the differentiable manifold M, we have Then δ : We also set c 0 (L) = 1 and c(L) = 1 + c 1 (L).
(ln g αβ + ln g βγ + ln g γα ). (4.1) Note 4.1.3. If M is a complex manifold, we can define Chern classes of holomorphic line bundles in a similar way. (See Chern [11], Chapter VI for more details).

Chern classes of a complex vector bundle
In this section we define higher Chern classes for complex vector bundles of any rank. We proceed in two steps: 1. We first define Chern classes of vector bundles that are direct sums of line bundles, 2. We show that we can always reduce the computation of Chern classes to the previous case.
Definition 4.2.2. Let M be a differentiable manifold and let E a complex vector bundle of rank r over M. Let us assume E = L 1 ⊕ · · · ⊕ L r is the direct sum of complex line bundles over M.
and ∪ denotes the cup product.
Step 2 relies on the following result, sometimes called the splitting principle. (See Bott-Tu [4], p. 273-278, for more details). Our proof uses the de Rham cohomology of M, but with some little changes one can prove the splitting principle also for the cohomology with coefficient in Z. 1. If E has rank 1, there is nothing to prove.
2. If E has rank 2, we can take as a splitting manifold F (E) the projective bundle P(E), which is by definition the fibre bundle over M whose fibre at a point P ∈ M is the projective space P(E P ) and whose transition functions areg αβ : U α ∩ U β −→ P GL(r, C), induced from the transition functions g αβ of E. As on the projectivization of a vector space, on P(E) there are several tautological bundles: the pull-back bundle π * E, the universal subbundle S E and the universal quotient bundle Q E .
Here the universal subbundle S E over P(E) is defined by If E has rank 2 we have π * E = S E ⊕ Q E , which is a direct sum of line bundles.
3. Now suppose E has rank 3. Over P(E) the line bundle S E splits off as before. The quotient bundle Q E over P(E) has rank 2 and so can be split into a direct sum of line bundles when pulled back to P(Q E ).
The pattern is now clear, we split off one subbundle at a time by pulling back to the projectivization of a quotient bundle (see Figure 4.1 for details).
Setting F (E) = P(Q n−2 ), this is the splitting manifold and this completes the proof. Proof. In view of the splitting principle, it is enough to prove the properties (1) and (2) and this completes the proof of (2).
3. By the splitting principle we can assume E = L 1 ⊕ · · · ⊕ L r and F = L r+1 ⊕ · · · ⊕ F r+q are direct sum of line bundles. Then 4. It follows from a general results: for any divisors D ∈ Div(X) in a compact Riemann surface where c 1 (D) is the Chern class of the line bundle associated with the divisor D. Then, if H is the hyperplane line bundle over P 1 (C), we find  Proof. It follows immediately from (4.1).

From te splitting principle we may assume
and then we have Corollary 4.2.5. Let E be a complex line bundle over M and let E * its dual bundle. Then and this completes the proof.
Corollary 4.2.6. Let E be a complex vector bundle of rank r over the differentiable manifold M, and let E * be its dual bundle. Then for i = 1, . . . , r we have Proof. From the splitting principle we may assume E = L 1 ⊕ · · · ⊕ L r is direct sum of line bundles over M. Then E * = L * 1 ⊕ · · · ⊕ L * r . From the previous corollary and elementary properites of symmetric functions we obtain

Axiomatic approach to Chern classes
In order to minimize topological prerequisites, in this section we take the axiomatic approach to Chern classes. This enables us to separate differential geometry aspects of Chern classes from their topological aspects. We consider the cathegory of complex vector bundles over real manifolds. For more references see Kobayashi [25] and Kobayashi-Nomizu vol. 1 [26].
. , E q be complex line bundles over M and let E 1 ⊕ · · · ⊕ E q be their Whitney sum. Then

Chern classes in terms of curvature
In the previous section we introduced the i-th Chern class c i (E) of a complex vector bundle as an element of H 2i (M, R). Via the de-Rham theory we should be able to represent c i (E) by a closed 2i-form γ i . In this section we shall construct such a γ i using the curvature form of a connection in E. For convenience, we imbed H 2i (M, R) into H 2i (M, C) and represent c i (E) by a closed complex 2i-form γ i . i.e., the Lie algebra of all r × r complex matrices. Let G be GL(r, C) acting on gl(r, C) by the adjoint action, i.e., a ∈ GL(r, C).
Now we define homogeneous polynomials f k on gl(r, C) of degree k = 1, . . . , r by . . , f r are GL(r, C)-invariants. It is known that these polynomials generate the algebra of GL(r, C)-invariant polynomials on gl(r, C).
Since GL(r, C) is a connected Lie group, the GL(r, C)-invariance can be expressed infinitesimally. In fact, we have the following result:  Proof. If s = sa −1 is another frame field, then the corresponding curvature form is given by aΩa −1 . Since f is GL(r, C)-invariant, it immediately follows that γ is a globally defined differential form of degree 2k.  Proof. We consider two connections D 0 and D 1 in E and connect them by a segment of connections Let ω t and Ω t be the connection form and the curvature form of D t with respect to a local frame field s. Then we write and Then Finally we set From (3.5) we see that the difference α of two connection forms transforms in the same way as the curvature form under a transformation of the local frame field s. It follows that f (α, Ω t , . . . , Ω t ) is indipendent of s and hence a globally defined (2k − 1)-form on M. Therefore, ϕ is a (2k − 1)-form on M. From Bianchi identity D t Ω t = 0, we obtain Hence, which proves that the cohomology class of γ does not depend on the connection D. In other words, Note 4.4.5. After some linear algebra calculations we find In particular and Proof. We have to show that the cohomology classes represented by the closed 2k-forms γ k satisfy the four axioms given in the previous section.
2. For Axiom 2, in the vector bundle f * E induced from E by the C ∞ mapping f : N −→ M, we use the connection f * D induced from a connection D in E. Then its curvature form is given by f * Ω. Since f k (f * Ω) = f * (f k (Ω)) = f * γ k , Axiom 2 is satisfied.

4.
We take a natural Hermitian structure in the tautological line bundle L over P 1 (C), i.e., the one arising from the natural inner product in C 2 . Since a fibre of L is a complex line through the origin of C 2 , each element ζ ∈ L is represented by a vector (ζ 0 , ζ 1 ) in C 2 . Then the Hermitian structure h is defined by: Considering [ζ 0 , ζ 1 ] as a homogeneous coordinate in P 1 (C), let z = ζ 1 /ζ 0 be the inhomogeneous coordinate in the local chart Let s be the frame field of L over U 0 defined by With respect to s, h is given by the function Hence the connection form and the curvature form of the Hermitian connection D are given by Using polar coordinates (r, t) defined by z = re 2πit , 0 t 1 we write Then integrating γ 1 over P 1 (C), we obtain

Algebraic and Analytic Tools
In this chapter we present some algebraic notions such as coherent sheaves, torsion-free sheaves and locally-free sheaves over a compact Kähler manifold (X, ω). Moreover, we introduce the notion of ω-stable and ω-semistable torsionfree sheaf of O X -modules over a compact Kähler manifold (X, ω). For more detailed definitions and proofs see [19], [25] and [35].
On the other hand, in the last section of this chapter we present some useful analytic tools which are involved in the proofs of the main results of this work: the Fredholm alternative Theorem and the Maximum Principle. See [15] and [25] for more details.

Torsion-free and locally-free analytic coherent sheaves
Definition 5.1.1. Let X be complex manifold of (complex) dimension n and let O = O X be the structure sheaf of X, i.e., the sheaf of germs of holomorphic functions on X. We write An analytic sheaf over X is a sheaf of O X -modules over X.
Definition 5.1.2. Let S be an analytic sheaf over a complex manifold X of (complex) dimension n. S is coherent if, for every point P ∈ X, there exists a neighborhood U of P in X and an exact sequence of sheaves Definition 5.1.3. Let S be an analytic sheaf over a complex manifold X of (complex) dimension n. We have the following definitions: 1. S is free if it is isomorphic to a direct sum of copies of the structure sheaf O X , 2. S is locally-free if X can be covered by open sets {U } for which S| U is a free O| U -module.
If S is locally-free, the rank of S on such an open set is the number of copies of the structure sheaf needed (finite or infinite). If X is connected, the rank of a locally-free sheaf is the same everywhere. A locally-free sheaf of rank 1 is also called an invertible sheaf.
Proposition 5.1.1. Let S be a coherent sheaf over a complex manifold X of (complex) dimension n. Then S has finite rank.
Proof. It follows essentially from Oka Lemma and the Syzygy Theorem. (See Gunning-Rossi [16] for details).
Definition 5.1.4. Let S be an analytic sheaf over a complex manifold X of (complex) dimension n. S is torsion-free if for every point P ∈ X, the corresponding stalk S P is a torsion-free O P -module.
Proposition 5.1.2. Every subsheaf of an analytic torsion-free sheaf is also torsion-free.
Proof. Let F ⊆ S be a subsheaf of the analytic sheaf S. For every point P ∈ X the stalk F P is an O P -submodule of S P . Since S P is a torsion-free O P -module, we conclude that F P is a torsion-free O P -module, and this completes the proof.
Proposition 5.1.3. Let S be a coherent sheaf over a complex manifold X of (complex) dimension n. If S is locally-free then it is torsion-free.
Proof. Let P ∈ X a point of X. We only have to show that the stalk S P is a torsion-free O P -module. Since S is locally-free, we can find a neighborhood U ⊆ X of P in X such that S| U = O q | U . Since S is coherent, according to Proposition 5.1.1, the rank on U is finite. If we consider the stalk at the point P we have S P = O q P and then, since O P is UFD, we have that the stalk S P is a torsion-free O P -module.
The following result demonstrates a deep relationship between vector bundles and locally-free sheaves. For the proof see Wells [35].
Theorem 5.1.4. Let X be a complex connected manifold of (complex) dimensions n. There is a one-to-one correspondence between isomorphism classes of holomorphic vector bundles over X and isomorphism classes of locally-free coherent sheaves over X.
Definition 5.1.5. Let S be a coherent sheaf over a complex manifold X of complex dimension n. The singularity set of the sheaf S is S = S n−1 (S) = {P ∈ X|S P is not free} Theorem 5.1.5. Let S be coherent torsion-free sheaf over a complex manifold X of (complex) dimension n. The singularity set S is a closed analytic subset of X of dimension dim C S n − 2.
Corollary 5.1.6. Let S be a coherent torsion-free sheaf over a complex manifold X of (complex) dimension 1, i.e, a Riemann surface. Then S is locally-free.
Proof. Clearly S is locally-free outside the singularity set S. From the previous Theorem, since X is a Riemann surface, we conclude that S is empty and this completes the proof.

Stable and semistable sheaves
According to Theorem 5.1.4, every exact sequance of holomorphic vector bundles over a complex manifold X of (complex) dimension n, induces an exact sequence of locally-free coherent sheaves over X where E i denotes the sheaf O(E i ) of germs of holomorphic sections of E i . Conversely, every exact sequence (5.2) of locally-free sheaves over a complex manifold X comes from an exact sequence (5.1) of the corresponding holomorphic vector bundles.
Definition 5.2.1. Given a coherent sheaf S over a complex manifold X, we shall define its determinant bundle det S. Let be a locally-free resolution of S| U , where U is a small open set in the base manifold X. Thanks to Syzygy Theorem and Oka Lemma such a resolution always exists. Let E i denote the holomorphic vector bundle corresponding to the sheaf E i . We set One should check that det S| U is indipendent of the choice of the resolution (5.3), for details see Kobayashi [25], p. 163-165.
Proposition 5.2.1. Let S be a coherent sheaf over a complex manifold X of complex dimension n. Then detS is a line bundle over X.
Proof. From (5.4) we know that detS is locally a holomorphic line bundle over X and then, thanks to Theorem 5.1.4, it is a locally-free sheaf over X of rank 1. Hence, also from Theorem 5.1.4 we deduce that detS is in one-to-one correspondence with an isomorphism class of holomorphic line bundle over X. holds for all coherent subsheaf S with 0 < rk(S ) < rk(S), we say that S is ω-stable.

Analytic tools
Proof. First, from x m −→ x in X we deduce that the sequence {x m } is bounded in X, then there exists M > 0 such that and M does not depend on m. Let ϕ ∈ Y , then ϕL m −→ ϕL in X , in fact So that, since x m x in X and since ϕL, ϕL m ∈ X ,

Approximate Hermitian-Yang-Mills Metrics and Semistability
In this chapter we review the notions of (weak) Hermitian-Yang-Mills structure and approximate Hermitian-Yang-Mills structure for Higgs bundles. Then, we construct the Donaldson functional for Higgs bundles over compact Kähler manifolds and present some basic properties of it. In particular, we study the properites of the Donaldson heat flow and we establish a relation between this gradient flow and the mean curvature of the Hitchin-Simpson connection. We also study some properties of the solutions of the evolution equation associated with that functional. Finally, we study the problem of the existence of approximate Hermitian-Yang-Mills strucutres and its relation with the notion of semistability.
In particular in this chapter we show that for a Higgs bundle E = (E, φ) over a compact Riemann surface X with Kähler form ω, thefollowing conditions are equivalent: 1. There exists an approximate Hermitian-Yang-Mills metric structure, 2. The Higgs bundle E = (E, φ) is ω-semistable.

Higgs sheaves and Higgs bundles
We start this section with some basic definitions. Definition 6.1.1. Let (X, ω) be a compact Kähler manifold of (complex) dimension n. We define Ω 1 X the holomorphic cotangent bundle to X. It is the dual of the holomorphic tangent bundle to X. Note 6.1.1. From the decomposition T X C = T X C+ ⊕ T X C− we identify the holomorphic tangent bundle to X with the sheaf of holomorphic sections of T X C+ . Definition 6.1.2. Let (X, ω) be a compact Kähler manifold of (complex) dimension n. A Higgs sheaf E over X is a coherent sheaf E over X, together with a morphism of O X -modules φ : E −→ E ⊗ Ω 1 X , such that the morphism φ ∧ φ : E −→ E ⊗ Ω 2 X vanishes, i.e., φ ∧ φ = 0. The morphism φ is called the Higgs field of E. Definition 6.1.3. A Higgs sheaf E is said to be torsion-free if the sheaf E is torsion free. A Higgs bundle E is just a Higgs sheaf in which the sheaf E is locally-free.
Definition 6.1.5. Let (X, ω) be a compact Kähler manifold and let E = (E, φ) be a Higgs sheaf over X. A section s ∈ Γ(X, E) is φ-invariant if there exists a section λ of Ω 1 X such that φ(s) = s ⊗ λ.
Definition 6.1.6. Let (X, ω) be a Kähler manifold of (complex) dimension n and let E 1 and E 2 be two Higgs sheaves over X. A morphism between E 1 and E 2 is a map f : is commutative. We will denote such a morphism by f : E 1 −→ E 2 . A sequence of Higgs sheaves is a sequence of their corresponding coherent sheaves where each map is a morphism of Higgs sheaves. A short exact sequence of Higgs sheaves is defined in the obvious way.
Note 6.1.2. Using local coordinates on X we can write φ = φ α dz α , where the index takes values α = 1, . . . , n and each φ α is an endomorphism of E. The condition φ ∧ φ = 0 is then equivalent to the commutativity of the endomorphisms φ α .
Note 6.1.3. Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E = (E, φ) be a Higgs bundle of rnak r over X. If (s 1 , . . . , s r ) is a local frame field on X and η 1 , . . . , η r is its dual, using local coordinates on X one can write φ = φ γ αβ s γ ⊗ η β ⊗ dz α , where the indexes take values α = 1, . . . , n and β, γ = 1, . . . , r while φ γ αβ are functions locally defined on X. We give the definitions of dual Higgs bundle and Higgs pull-back bundle. Definition 6.1.7. Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E = (E, φ) be a Higgs bundle of rank r over X. Let us consider the Higgs field φ as a section of End(E) ⊗ Ω 1 X . Since End(E * ) ∼ = End(E) there is a natural dual morphism φ * : E * −→ E * ⊗ Ω 1 X . From this it follows that E * = (E * , φ * ) is a Higgs bundle, called the dual Higgs bundle of E = (E, φ). Definition 6.1.8. Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E = (E, φ) be a Higgs bundle of rank r over X. Let Y be another compact Kähler manifold and let f : Y −→ X be a holomorphic map. Since in the cathegory of holomorphic bundles there is an isomorphism if φ is a Higgs field on E, there is a pull-back Higgs field f * φ on f * E defined by is a Higgs bundle on Y, called th.e pull-back bundle of E = (E, φ). Definition 6.1.9. Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E be a Higgs sheaf over X of rank r. The degree of E is defined by and the slope of E is defined by As in the ordinary case (see Kobayashi [25] for more details) there is a notion of stability for Higgs sheaves, which depends on the Kähler form ω and makes reference only to Higgs subsheaves. Namely we have: Definition 6.1.10. Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E be a Higgs sheaf over X of rank r. E is ω-stable (resp. ω-semistable) if it is torsion-free and for any Higgs subsheaf F with 0 < rk(F) < rk(E) one has the inequality µ(F) < µ(E) (resp. µ(F) µ(E)).
Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let let E = (E, φ) be a Higgs bundle of rank r over X. Let h be an Hermitian metric on this bundle. Let D h = D h + D be the Hermitian connection on E. From section 3.3 we already know that this connection exists and is the unique connection compatible with the metric h and the holomorphic structure of the bundle E. (For more details see [25]). Here D h and D = d E are the components of type (1, 0) and (0, 1). Using this decomposition of D h and the Higgs field φ, Simpson [31] introduced a connection on E in the following way: where φ h is the usual adjoint of the Higgs field with respect to the hermitian structure h, and it is defined by the formula  From the previous definition we immediately have then using the decomposition D h = D h + D and defining we obtain the following formula of the Hitchin-Simpson curvature in terms of the curvature of the Hermitian connection D h In the previous formula, D h (φ) and D (φ h ) are the components of type (2, 0) and (0, 2), respectively, while the (1, 1) component is given by We denote by Herm(E) the space of Hermitian forms in E and by Herm + (E) the space of Hermitian structures (i.e., the positive definite Hermitian forms) in E. For any Hermitian structure h it is possible to identify Herm(E) with the tangent space of Herm + (E) at the point h (see Kobayashi [25] for more details). Hence Herm(E) ∼ = T h Herm + (E). We can also define a Riemann structure in Herm + (E). From (6.1), for any v, v ∈ Herm(E) we define the inner product We have some natural properties associated with tensor products and direct sums. In particular we have Proposition 6.1.5. Let E 1 and E 2 two Higgs bundles with Higgs fields φ 1 and φ 2 respectively. Then 1. The pair E 1 ⊗ E 2 = (E 1 ⊗ E 2 , φ) is a Higgs bundle with Higgs field φ = φ 1 ⊗ I 2 + I 1 ⊗ φ 2 .

If pr
is a Higgs bundle with φ = pr * 1 φ 1 + pr * 2 φ 2 . Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E = (E, φ) be a Higgs bundle of rank r over X. In a similar way as in the ordinary case (see Kobayashi,[25]) we have a notion of Hermitian-Yang-Mills metric structure for the Higgs bundle E. Let us consider the usual operator * : A p,q −→ A p+1,q+1 and the operator L : A p,q −→ A p+1,q+1 defined by Lϕ = ω ∧ ϕ, where ϕ ∈ A p,q is a form on X of type (p, q). Then we define as usual the operator Λ = * • L * −1 : A p,q −→ A p−1,q−1 .
Definition 6.1.13. Consider now a Hermitian metric h ∈ Herm + (E) and let R h be its Hitchin-Simpson curvature. We can define the mean curvature of the Hitchin-Simpson connection D h just by contraction of this curvature with the operator iΛ. In other words, or equivalently inR h ∧ ω n−1 = Kω n . Definition 6.1.14. We define the operators Note that operator˜ h depends on the Kähler form ω via the the action of the operator Λ and also on the metric h, while 0 depends on the Kähler form ω via the application of Λ.
Proposition 6.1.7. K h ∈ A 0 (End(E)). In particular the (2, 0) and (0, 2) components of R h do not contribute to K h .
Proof. We already know that R h ∈ A 2 (End(E)) and that the operator iΛ kills the components of (2, 0) and (0, 2) type in R h . Then, from (6.2) we have Definition 6.1.15. We say that a Hermitian metric h ∈ Herm + (E) is a weak Hermitian-Yang-Mills structure with factor γ for E if where γ is a real valued function on X and I E is the identity endomorphism on E. If γ = c is a real positive constant, we say that h is a Hermitian-Yang-Mills structure. where s, s are section of the Higgs bundle E = (E, φ).

Weak Hermitian-Yang-Mills structures: elementary results
Let (X, ω) be a compact Kähler manifold of (complex) dimension n. From the expression of the curvature of tensor product and direct sum of Higgs bundles over X, we immediately have the following Proposition 6.2.1.
1. If h 1 and h 2 are two weak Hermitian-Yang-Mills structures with factors γ 1 and γ 2 for Higgs bundles E 1 and E 2 over X, then h 1 ⊗ h 2 is a weak Hermitian-Yang-Mills structure with factor γ 1 + γ 2 for the tensor product bundle E 1 ⊗ E 2 .
2. The metric h 1 ⊕ h 2 is a weak Hermitian-Yang-Mills structure with factor γ for the Whitney sum E 1 ⊕ E 2 if and only if both metrics h 1 and h 2 are weak Hermitian-Yang-Mills structures with the same factor γ for E 1 and E 2 .

Proof.
1. As in the classical case condider the formula for the curvature of the Hitchin-Simpson connection in a tensor product Hence, taking the trace with respect to ω (that is, applying the operator iΛ) we have the following expression involving the mean curvature of a tensor product 2. To prove (2) we use the following identity From the previous result and the definition of the dual Higgs bundle we have the following Corollary 6.2.2. Let h ∈ Herm + (E) be a (weak) Hermitian-Yang-Mills structure with factor γ for the Higgs bundle E over X. Then 1. The induced metric on the tensor product E ⊗p ⊗E * ⊗q is a (weak) Hermitian-Yang-Mills structure with factor (p − q)γ,

The induced Hermitian metric on
p E is a (weak) Hermitian-Yang-Mills structure with factor pγ for every 0 p r = rk(E).
In general, if h is a weak Hermitian-Yang-Mills structure with factor γ, the slope µ(E) can be written in terms of γ. In fact, we have Proposition 6.2.3. Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E = (E, φ) be a Higgs bundle of rank r over X. If h ∈ Herm + (E) is a weak Hermitian-Yang-Mills structure with factor γ, then

5)
Proof. Let R h be the Hitchin-Simpson curvature of the Hitchin-Simpson connection D h associated with the Hermitian connection D h . From (6.3) we have the identity inR h ∧ ω n−1 = K h ω n .
Lemma 6.2.5. If h ∈ Herm + (E) is a weak Hermitian-Yang-Mills structure with factor γ, then there exists a conformal change h = ah such that h is a Hermitian-Yang-Mills structure with constant factor c, given by Such a conformal change is unique up to homotety.
Proof. Since X is compact the integrals X γω n and X ω n are finite real numbers. Then there exists a finite real number c such that Hence, It is sufficient to prove that there exists a function u satisying the equation But this equality always holds from the choice of the constant c, and this completes the proof.
From the previous Lemma we immediately see that if a Higgs bundle admits a weak Hermitian-Yang-Mills structure, then it also admits, by an appropriate conformal change of the metric, a Hermitian-Yang-Mills structure. In particular, if the Higgs bundle has rank 1, we have the following Corollary 6.2.6. Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E = (E, φ) be a Higgs bundle of rank 1 over X. Then E admits a Hermitian-Yang-Mills structure.
Proof. Let h ∈ Herm + (E) be a Hermitian metric on E, which always exists from a partition of unity argument. Since E has rank 1 and since K h is selfadjoint, where γ is a positive real function on the manifold X. Then the thesis comes from Lemma 6.2.5.

Approximate Hermitian-Yang-Mills metrics
In this section we define approximate Hermitian-Yang-Mills metric structures on Higgs bundles, and study some of their properties. Definition 6.3.1. Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E = (E, φ) be a Higgs bundle of rank r over X. We define a positive real constant c as . (6.8) Definition 6.3.2. Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E = (E, φ) be a Higgs bundle of rank r over X. Let h be a Hermitian metric on E and let K be its Hitchin-Simpson mean curvature. We define the lenght of the endomorphism K − cI E by the formula Since K is selfadjoint with respect to the metric h and c is real, |K − cI E | 2 is a real function on X and |K − cI E | 2 0.
In the hypotheses of the previous definition, we introduce the following norms: (6.10) Definition 6.3.4. Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E = (E, φ) be a Higgs bundle of rank r over X. We say that E admits an approximate Hermitian-Yang-Mills structure if for any > 0 there exists a metric h such that Here K h is the mean curvature of the Hitchin-Simpson connection associated with h .
This notion satisfies some simple properties with respect to tensor product an direct sums. Proposition 6.3.1. Let (X, ω) be a compact Kähler manifold of (complex) dimension n. If the Higgs bundles E 1 and E 2 over X admit approximate Hermitian-Yang-Mills structures, so does their tensor product E 1 ⊗ E 2 . Furthermore, if µ(E 1 ) = µ(E 2 ), so does their Whitney sum.

Proof.
1. Assume that E 1 and E 2 admit approximate Hermitian-Yang-Mills structures with factors c 1 and c 2 , respectively. Let > 0. Then there exist h 1 and h 2 such that where K 1 and K 2 are the mean curvature endomorphisms of the Hitchin-Simpson connection associated with h 1 and to h 2 , respectively. Defining h = h 1 ⊗ h 2 and setting c = c 1 + c 2 , I = I E1 ⊗ I E2 , the mean curvature and this completes the proof.
Corollary 6.3.2. If E admits an approximate Hermitian-Yang-Mills stucture, so do the tensor product E ⊗p ⊗E * ⊗q and the exterior product bundle p E whenever 0 p r = rk(E). Corollary 6.3.3. Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E = (E, φ) be a Higgs bundle of rank r over X. If deg(E) < 0, then E has no nonzero φ-invariant sections.
Proof. Suppose E admits an approximate Hermitian-Yang-Mills structure, then for each there exists a Hermitian metric h on E such that where K h is the Hitchin-Simpson mean curvature endomorphism associated with h .
This implies that the mean curvature, seen as a Hermitian form, satisfies the following inequality If we assume that deg(E) < 0, then c < 0 in the above inequality and then for some sufficient small the mean curvature K h is negative definite. Then the result follows from a vanishing Theorem for Higgs bundles. (See Theorem 3.2 in [9] for details).

The Donaldson functional
Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E = (E, φ) a be Higgs bundle of rank r X. Let h, k ∈ Herm + (E). Since Herm + (E) is a connected Riemannian manifold (see Chapter VI in [25] for more details), we connect h and k by a curve h t , 0 t 1, in Herm + (E) so that h 0 = k and h 1 = h. We set where v t = h −1 t ∂ t h t and R t denotes the curvature of the Hitchin-Simpson connection associated with h t . Since h and k are Hermitian structures, det(k −1 h) is a strictly positive real function.
Notice that Q 1 (h, k) does not involve the curve h t . On the other hand to define Q 2 (h, k) we use explicitly the curve h t . Definition 6.4.1. Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E = (E, φ) be a Higgs bundle of rank r over X. Let h, k ∈ Herm + (E). We define the Donaldson functional by where c is, as usual, the constant given by .
We want to prove that the Donaldson functional does not depend on the curve joining the metrics h an k. First of all we prove the following result Lemma 6.4.2. Let (X, ω) be a compact Kähler manifold of (complex) dimension n. Let η ∈ d A 0,1 + d A 1,0 , one has X η ∧ ω n−1 = 0.
Proof. We write η = d(α + β) = d α + d β where α ∈ A 1,0 and β ∈ A 0,1 . Since X is Kähler, i.e., dω = 0, we have The following Lemma and the subsequent Propositions are straightforward generalizations of a result of Kobayashi (see [25], Chapter VI) to the Higgs case. Part of the proof is similar to the proof presented in [25]. However, some differences arise because of the term involving the commutator of the Higgs field in the Hitchin-Simpson curvature. Let h t (for a t b) be any differentiable curve in Herm + (E) and k any fixed Hermitian structure of E. Then the (1, 1) component of Proof. Following [25], we consider the domain ∆ in R 2 defined by ∆ = {(t, s)|a t b, 0 s 1}, and let h : ∆ −→ Herm + (E) be a smooth mapping such that h(t, 0) = k and h(t, 1) = h t for a t b. Let h(a, s) and h(b, s) line segments from k to h a and from h b to k, respectively. There is a simple expression for h given by The right hand side of the above expression can be computed straightforwardly from definition. In fact, after a short computation we obtain Therefore, we need to show that the left hand side of (6.12) is an element of d A 0,1 + d A 1,0 , and hence, it suffices to show thatdΨ ∈ d A 0,1 + d A 1,0 . Now, from the definition of Ψ we havẽ On the other hand, a simple computation shows that Replacing these expressions in the formula fordΨ and writing R = R + [φ, φ h ] (since the (2, 0) and (0, 2) components do not contribute), we havẽ The first trace in the expression above does not depend on the Higgs field φ (in fact, it is the same expression that it is found in [25] for the classical case). Then we only have to show that the second trace is identically zero. First of all, we need explicit expressions for ∂ t φ h and ∂ s φ h . Omitting the parameter t for simplicity, from [31] we know that where u 0 is a selfadjoint endomorphism such that h s+δs = h s + u 0 . Now and hence, at the first order in δs, we obtain u 0 = 1 + δs and consequently In a similar way we obtain the formula ∂ t φ h = [φ h , v]. Therefore, using these relations, the Jacobi identity and the cyclic property of the trace, we see that the second trace is identically zero. On the other hand, the term involving the curvature R can be written in terms of u, v and their covariant derivatives. So, finally, we get As it is shown in [25] in the classical case, defining the (0, 1)-form α = itr[vd u] we finally obtaind and hencedΨ is an element of d A 0,1 + d A 1,0 .
As a consequence of the above Lemma we have an important result for piecewise differentiable closed curves. Namely, we have Proposition 6.4.4. Let h t , α t β, be a piecewise differentiable closed curve in Herm + (E). Then Proof. Let α = a 0 < a 1 < · · · < a p = β be the values of t where the curve h t is not differentiable. Now take a fixed metric k in Herm + (E). We have From the previous Lemma, for each j = 1, . . . , p, we have and this completes the proof. Proof. Let γ 1 and γ 2 be two differentiable curves from h to k, and let L γ1 (h, k) and L γ2 (h, k) be the Donaldson functionals computed along the curves γ 1 and γ 2 , respectively. We have to show that From the definition of Donaldson functional we have But γ 1 − γ 2 is a piecewise differentiable closed curve in Herm + (E), then by the previous Proposition so from Lemma 6.4.2 we finally have and this completes the proof.
where φ t = φ ht is just an abbreviation. Therefore the (1, 1) component of Q 2 (h, ah) becomes and hence, from the above formula we obtain r ln aVol(X) = 0.
Lemma 6.4.7. For any differentiable curve h t in in the Riemannian manifold Herm + (E) and any fixed point k ∈ Herm + (E) we have Proof. 1. Since k does not depend on t, we get 2. It suffices to consider b as a variable in the equation and differentiate this expression with respect to b.
Let h be a metric in Herm + (E) and let K be the mean curvature of the Hitchin-Simpson connection associated with h. Notice that the endomorphism K ∈ A 0 (End(E)) can be written as K = h −1 K(·, ·), where K(·, ·) denotes the mean curvature as a Hermitian form. Theorem 6.4.8. Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E = (E, φ) be a Higgs bundle of rank r over X. Let Herm + (E) be the Riemannian manifold of the Hermitian metric on E, and let k be a fixed element in Herm + (E). Then, h is a critical point of L, i.e., a critical point of the function L( * , k) −→ R, if and only if K − ch = 0, if and only if h is an Hermitian-Yang-Mills structure for E.
Proof. By using the above Lemma, from inR 1,1 t ∧ ω n−1 = K t ω n we have the following formula for the derivative with respect to t of the Donaldson functional We consider the endomorphism K t as a Hermitian form by defining K t (s, s ) = h t (s, K t s ), where s and s are section of the Higgs bundle E = (E, φ).
for any fixed Hermitian metric k ∈ Herm + (E) and any differentiable curve h t in Herm + (E) we obtain where K t is considered here as a form and (·, ·) is the inner product in the Riemannian manifold Herm + (E). For each t, we can consider ∂ t h t ∈ Herm(E) as a tangent vector of Herm + (E) at h t . (See [25], Chapter VI for more details). Therefore, the differential dL of the functional evaluated at ∂ t h t is given by Then, the gradient of L( * , k) is given by ∇L = K − ch, and this completes the proof. Notice that here K still denotes the mean curvature as an Hermitian form.
Now, using the decomposition D = D h + D , we show that all critical points of L correspond to an absolute minimum. Proof. The second derivative of L is Here K t is an element of A 0 (End(E)). Since h 0 is a critical point of the Donaldson functional, K t − cI E = 0 at t = 0, hence On the oher hand, ∂ t K t can be written in terms of v t . In fact, we have .
Therefore, since the operator iΛ kills the (2, 0) and (0, 2) components, we have From the previous equations and since the linear operators iΛ and ∂ t commute, we obtain Hence, replacing this in the expression for the second derivative of L we find That is, h 0 must be a local minimum of L. Now, suppose in addition that h 1 is an arbitrary element of Herm + (E) and assume h t is a geodesic which joins the point h 0 and h 1 . Hence, ∂ t v t = 0 (see [25] for more details). Therefore, for such a geodesic we have Following the same procedure we have done before, but this time for t arbitrary, we get for 0 t 1 Note that the right hand side implicity depends on t via D ht . It follows that L(h 0 , k) L(h 1 , k). If we assume h 1 is also a critical point of L, we necessarily obtain the equality L(h 0 , k) = L(h 1 , k), so the local minimum defined for any critical point of L is an absolute minimum.

The evolution equation
For the construction of approximate Hermitian-Yang-Mills structures, the standard procedure is to start with a fixed Hermitian metric h 0 and try to find from it an approximate Hermitian-Yang-Mills metric structure using a curve h t , 0 t < +∞. In other words, we try to find the approximate structure by deforming h 0 through the 1-parameter family of Hermitian metrics h t . Theorem 6.5.1. Given a Hermitian metric h 0 on the Higgs bundle E, the non-linear evolution problem has a unique smooth solution defined for every positive time 0 t < +∞. Notice that here K t is the Hermitian form associated with the mean curvature of the Hitchin-Simpson connection of h t .
Proof. See Kobayashi [25], p. 205-223 for details. Note 6.5.2. Let k be a fixed Hermitian metric on the Higgs bundle E. We have ∇L = K − ch, where K is the Hermitian form associated with the mean Hitchin-Simpson curvature of h. Hence, we can rewrite the evolution problem in the gradient form Here ∇L is a vector field on the Riemannian manifold Herm + (E). This nonlinear evolution problem is called the Donaldson heat flow problem.
In this section we study some properties of the solutions of the Donaldson heat flow problem. In particular, we are interested in the study of the mean curvature when the parameter t goes to infinity. Proposition 6.5.3. Let h t , 0 t < +∞ be the solution of the Donaldson heat flow with initial condition h 0 . Then: 1. For any fixed Hermitian metric k ∈ Herm + (E), the functional L(h t , k) is a monotone decreasing function of t; that is, Here K t ∈ A 0 (End(E)) is the Hitchin-Simpson mean curvature endomorphism.
Proof. The proofs of (2) and (3) are similar to the proofs in the classical case [25], but this time we need to work with the operator˜ h = iΛD D h instead of the operator h = iΛD D h .
1. From a previous calculation we already know that where K t is now thought as a Hermitian form. Since h t is the solution of the Donaldson heat flow, from the definition of the Riemannian structure in Herm + (E) we get 2. Let K t ∈ A 0 (End(E)) be the mean curvature of the Hitchin-Simpson connection associated with the metric h t and let v t = h −1 t ∂ t h t . Consider the operator˜ h = iΛD D h which depends on the Kähler form ω via the adjoint multiplication Λ and also on the metric h. Using˜ h , we can rewrite where˜ t =˜ ht and the subscript t remember us the dependence on the metric h t . From the evolution equation we have v t = −(K t − cI E ), and hence we get˜ t v t = −˜ t K t . Therefore we obtain On the other hand, since tr(AB) = tr(BA), Applying the iΛ operator (this will kill the (2, 0) and (0, 2) components), from tr(AB) = tr(BA) and since the operator iΛ and the trace commute we get is a positive real valued function on X. (See p. 225 in [25] for details in the classical case). So, finally we obtain and (2) follows from the Maximum Principle 5.3.2. (1) and (2) in a similar way to the classical case [25]. Integrating the equality (6.14) from 0 to s, we obtain

Finally (3) follows from
Since L(h s , k) is bounded below by a constant A indipendent of s and since is a monotone decreasing of s, there exists the limit where L is a finite real number. Hence In particular we deduce Let χ = χ(x, y, t) be the heat kernel for the differential operator ∂ t +˜ t .
Now fix t 0 ∈ [0, +∞) and set where dy is the volume form dy = ω n n! . Then u(x, y) is of class C ∞ on X × (t 0 , +∞) and extends to a continuous function on X × [t 0 , +∞). From the definition of the heat kernel we immediately have Combined with the inequality (6.15) this yields By the Maximun Principle 5.3.2 and the properties of u(x, t) we find Fix a, say a = 1, and let t 0 −→ +∞. Since L(h t , k) is bounded below, using (6.16) we conclude and this completes the proof. Proof. Let h t , 0 t 1, be a curve in Herm + (E) such that h 0 = k and h 1 = h. From a previous calculation we already know that where K t is now thought as a Hermitian form and (·, ·) is the inner product of Herm + (E). So that (K t − ch t , ∂ t h t ) ∈ R for any t ∈ [0, 1] (see p.196-197 in [25] for details), and then At this point we introduce the main result of this section. This establishes a relation among the boundedness property of Donaldson functional, semistability and the existence of approximate Hermitian-Yang-Mills metric structures on the Higgs bundle E. Theorem 6.5.5. Let (X, ω) be a compact Kähelr manifold of (complex) dimension n and let E = (E, φ) be a Higgs bundle of rank r over X. We have implications (1) −→ (2) −→ (3) for the following statements: 1. for any fixed Hermitian metric structure k ∈ Herm + (E), there exists a constant B such that L(h, k) B for all Hermitian metrics h ∈ Herm + (E), 2. E admits an approximate Hermitian-Yang-Mills structure, i.e., for each > 0 there exists an Hermitian metric h in E such that where h depends on and K is the mean curvature endomorphism of the Hitchin-Simpson connection associated with the metric h, Proof.
1. Assume (1). The Donaldson functional is bounded below by a constant B. Let h 0 be a fixed Hermitian metric in E and let h t be the solution of the Donaldson heat flow with initial condition h 0 . Then from the previous Proposition there exists a real consant B such that L(h t , h 0 ) B > −∞ for every positive time 0 t < +∞. From Theorem 6.5.5, where K t is the mean curvature endomorphism K t ∈ A 0 (End(E)). Hence, there exists an approximate Hermitian-Yang-Mills structure.
2. On the other hand, that (2) impies (3) has been proved in [6] by Bruzzo and Graña Otero. Here we reproduce their proof. Assume (2) and let F be a proper nontrivial Higgs subsheaf of E. Then rk(F ) = p for some 0 < p < r = rk(E) and the inclusion F −→ E induces a morphism det F −→ p E. Tensoring by (det F) −1 we have a nonzero section s of the Higgs bundle If ψ represents the Higgs field naturally defined on G by the Higgs field of E and F, then s is ψ-invariant, i.e., ψ(s) = s. Now, since by hypothesis E admits an approximate Hermitian-Yang-Mills structure, from Proposition 6.3.1 we know that so does G and in particular

The one-dimensional case
In this section we establish a boundedness property for the Donaldson functional for semistable Higgs bundles over compact Riemann surfaces. As a consequence we get that in the one-dimensional case all three conditions in Theorem 6.5.5 are equivalent.
We introduce some properites that will be useful in proving some statements using induction on the rank of Higgs bundles. This section is essentially an extension to Higgs bundles of the classical case. (See p. 226-233 in [25] for more details). Now, let (X, ω) be a compact Kähler manifold of (complex) dimension n and let be an exact sequence of Higgs bundles over X. As in the ordinary case (see Chapter I, §6 in [25]), a Hermitian metric h in E induces Hermitian structures h and h in E and E , respectively. We have also a second foundamental form A h ∈ A 1,0 (Hom(E , E )) and its adjoint B h ∈ A 0,1 (Hom(E , E )), where B * h = −A h . As usual, some properties which hold in the ordinary case, also hold in the Higgs case. Proposition 6.6.1. Given an exact sequence (6.17) and a pair of Hermitian structures h and k in E, the function Q 1 (h, k) and the form Q 2 (h, k) satisfy the following relations: Proof.
1. (6.18) is straightforward from the definition of Q 1 . Since h = h ⊕h and k = k ⊕ k , we immediately have 2. On the other hand, (6.19) follows from an analysis similar to the ordinary case. Since the sequence (6.17) in particular is an exact sequence of holomorphic vector bundles over the complex manifold X, for any Hermitian metric h we have a splitting of the exact sequence by C ∞ homomorphisms µ h : E −→ E and λ h : E −→ E. In particular We consider now a curve of Hermitian structures h t for 0 t 1 such that h 0 = k and h 1 = h. Corresponding to h t we have a family of homomorphisms µ t and λ t . We define the homomorphism S t : E −→ E given by A short computation shows that ∂ t B t = d (∂ t S t ). Choosing convenient orthonormal local frame fields for E and E (see [25] for more details), Here v t and v t are the natural endomorphisms associated with h t and h t , respectively. Now, from the ordinary case we have where R t and R t are the Hermitian curvatures of E and E associated with the metrics h t and h t , respectively.
. Since E and E are Higgs subbundles of E, we obtain an expression for the (1,1)-component of the Hitchin-Simpson curvature Hence we can compute the trace The last four terms are the same as in the ordinary case [25]. After a short computation we finally get that Then, multiplying the last expression by i and integrating from t = 0 to t = 1 we finally obtain (6.19).
Definition 6.6.1. In the hypothesis of the previous Proposition, we define |B| as the nonnegative real function on the compact Kähler manifold (X, ω) that satisfies |B| 2 ω n = −intr(B ∧ B * ) ∧ ω n−1 .
Lemma 6.6.2. Let (X, ω) be a compact Kähler manifold of (complex) dimension n. Given an exact sequence of Higgs bundles over X, with µ(E) = µ(E ) and a pair of Hermitian structures h and k in E, we have the identity Proof. First of all, from µ(E) = µ(E ) we have c = c = c . From Lemma 6.4.2 and from (6.18) and (6.19) we obtain In dimension greater or equal than two, the notion stability (resp. semistability) depends on the Kähler form, as the degree depends on it. Now, in dimension one, the degree does not depend on the Kähler form and hence the notion of stability (resp. semistability) does not depend on it and we can establish all our results without any explicit reference to ω. Since the degree and the rank of any Higgs sheaf is the same degree and rank of the corresponding coherent sheaf, we have the following (see [25], Ch. V, Lemma 7.3). From Lemma 6.6.3 it follows that the condition of stability (resp. semistability) can be written in terms of quotient Higgs sheaves instead of Higgs subsheaves. To be precise we have Corollary 6.6.4. Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E be a torsion-free Higgs sheaf over X. Then E is ω-stable (resp. semistable) if for every quotient Higgs sheaf G with 0 < rk(G) < rk(E) it follows µ(G) < µ(E) (resp. µ(G) µ(E)).
From the definition of degree, one has that any torsion Higgs sheaf T has deg(T) 0. Therefore, in a similar way to the classical case, this implies that in the definition of stability (resp. semistability) we do not have to consider all quotient Higgs sheaves. To be precise we have Proposition 6.6.5. Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E be a torsion-free Higgs sheaf over X. Then 2. E is ω-stable (resp. semistable) if and only if µ(E) < µ(G) (resp. ) for any torsion-free quotient Higgs sheaf G with 0 < rk(G) < µ(E).
Proof. Is true in one direction. For the converse, suppose the inequality between slopes in (1) (resp. in (2)) holds for proper Higgs subsheaves with torsion-free quotient (resp. for torsion-free quotient Higgs sheaves) and let consider an exact sequence of Higgs sheaves Let E = (E, φ) and denote by φ F and φ G the Higgs fields of F and G respectively. That is F = (F, φ F ) and G = (G, φ G ). Now, let T be the torsion subsheaf of G.
Since the Higgs field satisfies φ(T ) ⊆ T ⊗ Ω 1 X , the pair T = (T, φ| T ) is a Higgs subsheaf of E with Higgs quotient, say G 1 . Then if we define F 1 by the kernel of the Higgs morphism E −→ G 1 , we have the following commutative diagram of Higgs sheaves Id in which all rows and columns are exact. From this diagram we have that F is a Higgs subsheaf of F 1 with T = F 1 F . Since T is a torsion Higgs sheaf deg(T) 0 and we also obtain Now, since T is torsion we have rk(G) = rk(G 1 ) and rk(F 1 ) = rk(F) and hence finally we obtain µ(F) µ(F 1 ) and µ(G 1 ) µ(G).
At this point, the converse direction in (1) and (2) follows from the hypothesis and the last two inequalities. Now we can establish a boundedness property for the Donaldson functional for semistable Higgs bundles in the one-dimensional case, i.e., for compact Riemann surfaces. Namely we have Theorem 6.6.6. Let (X, ω) be a compact Riemann surface and let E = (E, φ) be a Higgs bundle of rank r over X. If E is ω-semistable, then for any fixed Hermitian metric k ∈ Herm + (E) the set {L(h, k)|h ∈ Herm + (E)} is bounded below.
Proof. Fix k and assume E is ω-semistable. The proof runs by induction on the rank of E. Then the set {L(h, k)|h ∈ Herm + (E)} is bounded below.
2. Let us assume rk(E) 2 and let us assume the thesis true for every Higgs bundle F over X such that 0 < rk(F) < rk(E).
We have to distinguish between two cases.
(a) If E is ω-stable, there exists a Hermitian-Yang-Mills structure h 0 on it (see [31] for a detailed proof) and the Donaldson functional must attain an absolute minimum at h 0 , i.e., for any other Hermitian metric h L(h, k) L(h 0 , k).
(b) Suppose E is ω-semistable, but not ω-stable, with rk(E) 2. Among all proper nontrivial Higgs subsheaves with 0 < rk(F) < rk(E), torsion-free quotien and the same slope of E we choose one, say E , with minimal rank. Since µ(E ) = µ(E), the sheaf E is necessarly ω-stable. If not, there exists a proper Higgs subsheaf F of E with µ(F ) µ(E ). Since F is clearly a subsheaf of E and E is ω-semistable, we obtain µ(E ) µ(F ) µ(E) = µ(E ), which is a contradiction, because E was chosen of minimal rank. Now let E = E E . Using Lemma 7.3 in [25] it follows that µ(E) = µ(E ) = µ(E ) and E is ω-semistable. Hence we have the following exact sequence of Higgs sheaves where both E and E are torsion-free. Since dim C X = 1, from Corollary 5.1.6 we deduce that they are also locally-free. Hence the above sequence is in fact an exact sequence of Higgs bundles. Assume now h is an arbitrary metric on E, by applying Lemma 6.6.2 to the metrics h an k we obtain where h , k and h , k are the Hermitian structures induced by h and k in E and E , respectively. By induction, L(h , k ) and L(h , k ) are bounded below by consants depending only on k and k . Then, from (6.21) it follows that L(h, k) is bounded below by a constant depending only on k.
As consequence of this Theorem we obtain the following Theorem 6.6.7. Let (X, ω) be a compact Riemann surface and let E = (E, φ) be a Higgs sheaf of rank r over X. The following are equivalent: 2. E admits an approximate Hermitian-Yang-Mills structure.
Proof. Since X is a Riemann surface, from 5.1.6 we deduce that (E, φ) is a Higgs bundle over X. Hence, the thesis comes from the previous result and Theorem 6.5.5.
As a consequence of this Theorem we immediately deduce that in the onedimensional case, many results about Higgs bundles written in terms of approximate Hermitian-Yang-Mills structures can be translated in terms of semistability. In particular we have the following Corollary 6.6.8. If (X, ω) is a compact Riemann surface and E 1 , E 2 are ω-semistable Higgs bundles over X, then so is their tensor product E 1 ⊗ E 2 . Furthermore, if µ(E 1 ) = µ(E 2 ), so is the Whitney sum E 1 ⊕ E 2 .

Proof.
1. Let E 1 and E 2 be ω-semistable Higgs bundles over X. From the previous Theorem E 1 and E 2 admit approximate Hermitian-Yang-Mills metric structures, so does their tensor product E 1 ⊗E 2 (Proposition 6.3.1). Hence, using the above Theorem we conclude that E 1 ⊗E 2 is ω-semistable.
2. It is similar to (1), using the second part of Proposition 6.3.1.
Corollary 6.6.9. If E is ω-semistable, then so is the tensor product bundle E ⊗p ⊗ E * ⊗q and the exterior product bundle p E whenever 0 p r = rk(E).

General Case
In the previous chapter we have proved the equivalence between semistability and the existence of approximate Hermitian-Yang-Mills metric structures for Higgs bundles over compact Riemann surfaces. In this chapter we extend this result to higher dimensions. Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E = (E, φ) be a Higgs bundle of rank r over X. Let h t be the solution of the Donaldson heat flow with initial metric h 0 , i.e., the solution of the non-linear evolution problem From Theorem 6.5.1 we know that this problem has a unique solution h t defined for every positive time 0 t < +∞, and L(h t , h 0 ) is a real monotone decreasing function on t for 0 t < +∞. If we assume that (E, φ) is ω-semistable, we can distinguish between two cases: 1. L(h t , h 0 ) is bounded below. From Theorem 6.5.5 there exists an approximate Hermitian-Yang-Mills structure.
Under the assumption that (E, φ) is ω-semistable, we can still prove the existence of approximate Hermitian-Yang-Mills metrics.

Some useful results
Lemma 7.1.1. Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E = (E, φ) be a Higgs bundle of rank r over X. Let h ∈ Herm + (E) be a Hermitian metric and let R and K be its Hitchin-Simpson curvature and mean curvature, respectively. Then Proof. First we note that iΛtr(R) = tr(iΛR), since iΛ and the trace are linear differential operator. From Stokes' formula and the representation of Chern classes in terms of curvature we have: Given a metric h ∈ Herm + (E), let K be the mean curvature of the Hitchin-Simpson connection associated with h. There always exists a real positive function a = a(x) on X such that, setting h = ah, tr(K − cI E ) = 0. To be more precise, we have the following: Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E = (E, φ) be a Higgs bundle of rank r over X. Let h ∈ Herm + (E) be a Hermitian metric on E, and let K be the mean curvature of the Hitchin-Simpson connection associated with h. By an appropriate conformal change of the metric h, we can assume tr(K − cI E ) = 0.
Proof. Consider a real positive function a = a(x) on X and set h = ah. Then h = ah defines another Hermitian metric h ∈ Herm + (E). From the identity K ω n = Kω n + 0 ln(a)I E ω n (7.2) and since the Kähler form is nowhere vanishing, taking the trace in the previous identity we obtain tr(K − cI E ) = tr(K − cI E ) + r 0 ln(a).
Hence, it suffices to prove that there is a function u satisfying the equation  We have where K t is the mean curvature of the Hitchin-Simpson connection associated with h t and˜ t =˜ ht . Here the subscript t remember us the dependence on the metric h t .
Proof. We already know that Since the trace and the linear operators˜ t and ∂ t commute, we obtain Let Herm + int (E) denote the set of all Hermitian metrics h satifsfying where K h is the mean curvature of the Hitchin-Simpson connection of h. If X is compact this space coincides with Herm + (E). The space Herm + int (E) was studied by Simpson in [31]. It is an analytic manifold, which in general is not connected, and has the following properties. If k ∈ Herm + int (E) is a fixed element, then any other metric in the same connected component is given by h = k exp(a) with a a smooth endomorphism of E which is selfadjoint with respect to k. Moreover Simpson showed in [31] that for any Higgs bundle E = (E, φ) over a compact Kähler manifold X the solution of the Donaldson heat flow remains in the same connected component of the initial metric. To be precise, if k is a fixed metric in Herm + int (E), then the unique solution of the Donaldson heat flow h t with h 0 = k is contained in the same connected component as k.
Now, let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E = (E, φ) be a Higgs bundle of rank r over X. Let h 0 ∈ Herm + (E) be an initial metric with the condition tr(K 0 − cI E ) = 0, and let h t , 0 t < +∞, be the corresponding solution of the Donaldson heat flow. Then h t will be of the form h t = h 0 exp(S(t)) for some section S(t) of End(E) over X, and S(t) will be selfadjoint with respect to h 0 . Definition 7.1.1. Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E = (E, φ) be a Higgs bundle of rank r over X. Let h 0 ∈ Herm + (E) be an initial metric with the condition tr(K 0 − cI E ) = 0, where K 0 is the mean curvature of the Hitchin-Simpson connection associated with h 0 . Let h t , 0 t < +∞, be the solution of the Donaldson heat flow with initial condition h 0 , and let S(t) be a section of End(E) such that h t = h 0 exp(S(t)) and S(t) is selfadjoint with respect to h 0 . Then |S(t)| 2 = tr(S(t) · S(t)) is a nonnegative real valued function on X and we have the following norms: Moreover, since h t is differentiable for 0 < t < +∞, so does S(t).
Lemma 7.1.7. Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E = (E, φ) be a Higgs bundle of rank r over X. Let h 0 ∈ Herm + (E) be a metric with the condition tr(K 0 − cI E ) = 0, where K 0 is the mean curvature of the Histchin-Simpson connection of h 0 . Let h t be the solution of the Donaldson heat flow with initial condition h 0 and let S(t) be a section of End(E) such that h t = h 0 exp(S(t)) and S(t) is selfadjoint with respect to h 0 . The following inequality holds: Proof. Let V = Vol(X) and r = rk(E). First of all, from the Hölder inequality we have Set H t = e S(t) = h −1 0 h t and let λ 1 (t), . . . , λ r (t) be the eigenvalues of S(t). Since S(t) is selfadjoint with respect to h 0 we deduce that, for each t 0, they are real valued functions on X and then, using Lemma 7.1.6, we obtain |S(t)| = λ 2 1 (t) + · · · + λ 2 r (t) r 1/2 ln(e λ1(t) + · · · + e λr(t) + e −λ1(t) + · · · + e −λr(t) ) = = r 1/2 ln(trH t + trH −1 t ), Since S(t) is selfadjoint with respect to h 0 , it follows that H t = e S(t) and H −1 t = e −S(t) are also selfadjoint with respect to h 0 . Moreover, from the definition of H t and the properties of the exponential of matrices we deduce that H t and H −1 t are positive definite and then trH t > 0 and trH −1 t > 0. In fact trH t = e λ1(t) + · · · + e λr(t) > 0 and trH −1 t = e −λ1(t) + · · · + e −λr(t) > 0.
We also notice that H −1 By direct calculation, from Lemma 7.1.5, since ∂ t and the trace commute and since the trace is GL(r, C)-invariant we find (7.10) On one hand, since K t − cI E L 2 is a nonnegative real valued monotone decreasing function on t, 0 t < +∞, we have and then, for t > 0 On the other hand, since X is compact we can use the Fubini-Tonelli Theorem. Integrating over X, from (7.10) and (7.9) and from the Hölder inequality we have Combining (7.13) and (7.12), since K t − cI E L 2 0 finally we obtain and this proves (7.6).
Now we review some constructions involving Hermitian matrices. Let E be a Higgs bundle with a fixed Hermitian metric k, and let S = S(E) be the real vector bundle of selfadjoint endomorphisms of E. Suppose ϕ : R −→ R is a smooth function. Then we define a map of fibre bundles over X ϕ : S −→ S as follows: suppose s ∈ S, then, at each point in X, choose an orthonormal frame {e i } for E with s(e i ) = λ i e i , and set ϕ(s)(e i ) = ϕ(λ i )e i .
Suppose Ψ : R × R −→ R is a smoot function of two variables. Then we define a map of fibre bundles Ψ : S −→ S(End(E)), where S(End(E)) consists of elements of End(End(E)) which are selfadjoint with respect to the usual metric tr(A · B * ). The function Ψ is described as follows. Suppose s ∈ S and A ∈ End(E). Choose an orthonormal frame field {e i } of eigenvectors of s with eigenvalues λ i . Let {ê i } be the dual frame field in E * , and write A = i,j A ijêi ⊗ e j . Then set Again this is well defined, smooth and linear in A.
If the functions ϕ and Ψ are analytic, then we can express the constructions above as a power series. If ϕ(λ) = a n λ n then ϕ(s) = a n s n . If which is taken as (dϕ/dλ)(λ 1 ) if λ 1 = λ 2 . If s ∈ S, then D ϕ(s) = dϕ(s)(D s) where the right side uses the obvious extension to form-coefficient in the second variable. To see this for example when ϕ is analytic, note that if ϕ(λ) = λ n then dϕ(λ 1 , λ 2 ) = i+j=n−1 The construction ϕ(s) and Ψ(S) retain the same positivity properties as ϕ and Ψ. For example if ϕ(λ) > 0 for all λ, then ϕ(s) is positive definite for all s. And if Ψ(λ 1 , λ 2 ) > 0 for all λ 1 , λ 2 then tr(Ψ(A)·A * ) > 0 for all s and all A ∈ End(E).
We will describe how these constructions behave with respect to Sobolev norms. Fix a Hermitian metric k on a Higgs bundle E. Using the metric we can define the space where s is selfadjoint with respect to the metric k and |s| 2 = tr(s·s). In particular L 2 0 (S) is a Hilbert space with the inner product s, u = X tr(s · u) ω n n! (7.14) and then s 2 L 2 = X tr(s · s) ω n n! .
(7.15) Note 7.1.8. In the above formulas s and u are selfadjoint with respect to the metric k since they are element of S. Then X tr(s · u) ω n n! ∈ R, in fact, since the Kähler form is real, from tr(A) = tr(A t ) and tr(AB) = BA, we have From (7.14) and (7.15) we can deduce the following Lemma 7.1.9. Let k be a fixed Hermitian metric and let K k be the mean curvature of the Hitchin-Simpson connection associated with the metric k. If Moreover, if tr(u m ) = 0 for each m, then tr(u ∞ ) = 0.
Proof. From (7.14) and (7.15) we immediately deduce that is a continuous linear functional on L 2 0 (S) and then (7.16) follows from the definition of weak convergence in Hilbert spaces.
In order to prove that tr(u ∞ ) = 0 if tr(u m ) = 0 for each m let us consider the set Since u ∞ is a continuous section of End(E) and it is selfadjoint with respect to the metric k we have that tr(u ∞ ) is a continuous real valued function on X, so that Ω is an open subset of X. Then, since X is compact and the volume form ω n n! defines a finite measure on X, if we apply (7.16) to the open set Ω we obtain Ω tr(u m ) ω n n! = Ω tr(u m · I E ) ω n n! −→ Ω tr(u ∞ · I E ) ω n n! = Ω tr(u ∞ ) ω n n! .
But the continuous function tr(u ∞ ) is strictly positive on the open set Ω and then we conclude that Ω = ∅. In a similar way the open set Θ = {x ∈ X|tr(u ∞ (x)) < 0} is empty, so tr(u ∞ ) = 0 on X.
Let L p 1 (S) denote the space of sections s ∈ S such that s ∈ L p (S) and D s ∈ L p (E). Note that this is a condition on d E s = D s and also a growth condition involving the Higgs field φ if X is noncompact. For a given number b denote the closed subspaces of sections s ∈ S with |s| b by L p 0,b and L p 1,b . Finally let P (S) be the normed space of smooth sections s ∈ S with norm s p = max X |s| + D s L 2 + D k s L 1 .
The construcions ϕ and Ψ behave in a rather delicate fashion on L p (S) and L p 1 (S) as it is shown in the following Proposition. They behave better on P (S) since their · L ∞ -norm is controlled. 2. The map Ψ extends to a map Ψ : L p 0,b (S) −→ Hom(L p (End(E)), L q (End(E))) for q p, and for q < p it is continuous in the norm operator topology. For the proof see Proposition 4.1 in [31]. Now, let k ∈ Herm + (E) be a fixed Hermitian structure. We already know that any Hermitian metric h will be of the form k exp(v) for some section v of End(E) over X. Moreover, v is selfadjoint with respect to k. We can join k to h by the geodesic h τ = k exp(τ v) where 0 τ 1. Note that here v τ = h −1 τ ∂ τ h τ = v is constant, i.e., it does not depend on τ. (See Chapter VI, §1 and §2 in [25] for more details). Now, in the proof of Theorem 6.4.9 we got an expression for the second derivative ∂ 2 τ L(h τ , k) for any curve h τ = k exp(τ v), namely: where K τ is the mean curvature endomorphism of the Hitchin-Simpson connection associated with the metric h τ = k exp(τ v). Notice that in our case, the chosen curve is such that h 0 = k, since it is also a geodesic ∂ τ v τ = 0 we have where β 1 (t), . . . , β r (t) are the eigenvalues of S(t). Following Simpson in [31] we can rewrite (7.19) in the equivalent form L(h t , h 0 ) = X tr(S(t) · K 0 ) ω n n! + X Ψ(S(t))(D S(t)), D S(t) h0 ω n−1 (n − 1)! .
(7.20) From the previous constructions it follows that if the Donaldson functional L(h t , h 0 ) is not bounded below, then S(t) L ∞ −→ +∞ as t −→ +∞. Namely we have Lemma 7.1.11. Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E = (E, φ) be a Higgs bundle of rank r 2 over X. Let h 0 ∈ Herm + (E) be a Hermitian metric such that tr(K 0 − cI E ) = 0, where K 0 is the mean curvature of the Hitchin-Simpson connection associated with h 0 . Let h t be the solution of the Donaldson heat flow with initial metric h 0 and assume h t = h 0 e S(t) , where S(t) is a section of End(E) and it is selfadjoint with respect Proof. We will show that if the limit does not hold, there is a sequence t m −→ +∞ such that L(h tm , h 0 ) is bounded below. Suppose the required limit does not hold. Hence, we can find a positive constant B > 0 and a sequence t m −→ +∞ such that From the theory of the Gerschgorin circles (see [2] for details), there exists a positive radius R > 0, which does not depend on m, such that and C does not depend on m. Therefore, from (7.19) and since X is compact and we can locally consider D = D + φ as a matrix of forms of degree 1 which does not depend on m, we have Lemma 7.1.12. Up to extracting a subsequence, u m u ∞ weakly in L 2 1 (S). The limit u ∞ is not 0. If Φ : R × R −→ R is a nonnegative smooth function with compact support such that Φ(x 1 , x 2 ) (x 1 − x 2 ) −1 whenever x 1 > x 2 , then Moreover, from u m u ∞ weakly in L 2 1 (S), we have D u m D u ∞ weakly in L 2 0 (S) and hence, from Lemma 5.3.4 we find that for any q < 2. So that, from Lemma 5.3.3 for any q < 2.
tr(u ∞ · K k ) ω n n! + V This works for any q < 2 and then, in the limit, using some continuity and boundedness arguments, this implies inequality (7.21). Since (7.21) holds for all nonnegative smooth functions Φ : R × R −→ R with compact support such that Φ(x 1 , x 2 ) < (x 1 − x 2 ) −1 whenever x 1 > x 2 , again using some continuity and boundedness arguments, we can conclude that the inequality in the Lemma also holds if we assume Φ(x 1 , x 2 ) (x 1 − x 2 ) −1 whenever x 1 > x 2 .
Lemma 7.1.13. In the hypothesis of the previous Lemma, letΦ : R × R −→ R be a nonnegative smooth function such that Lemma 7.1.14. (Key Lemma) Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E = (E, φ) be a Higgs bundle of rank r 2 over X. Let h t be the solution of the Donaldson heat flow with initial condition h 0 , and suppose tr(K 0 − cI E ) = 0, where K 0 is the mean curvature endomorphism of the Hitchin-Simpson connection associated with h 0 . Let us assume (E, φ) is ω-semistable and L(h t , h 0 ) is not bounded below, i.e. L(h t , h 0 ) −→ −∞ as t −→ +∞. Then where S(t) is a section of End(E) such that h t = h 0 e S(t) and S(t) is selfdajoint with respect to h 0 .
Proof. Let K t be the mean curvature of the Hitchin-Simpson connection associated with the metric h t . We will show that if the estimate does not hold, there is a Higgs subsheaf which condradicts semistability. Suppose the required estimate does not hold. Hence, we can find a positive constant > 0 and a sequence t m −→ +∞ such that Since max X |K t − cI E | is a monotone decreasing function (see Proposition 6.5.3 for details), from max we deduce that max X |K t | is uniformly bounded with respect to t. So from Lemma (3.1)(d) in [31] and the hypotesis that max X |K t | are uniformly bounded, we have the following Simpson's estimate (p. 885 in [31]) where C 1 and C 2 depend only on the curvature of the initial metric h 0 and the Käher form ω. Since L(h t , h 0 ) is not bounded below, L(h tm , h 0 ) −→ −∞ as t m −→ +∞ and then S(t m ) L 1 −→ +∞. In fact, from Lemma 7.1.11 and (7.28) we have Set u m = l −1 m S(t m ), where l m = S(t m ) L 1 , then u m L 1 = 1. Since S(t) is selfadjoint with respect to h 0 , the u m are also selfadjoint with respect to h 0 . From the hypothesis tr(K 0 − cI E ) = 0 and from (7.5) we deduce that Since S(t m ) L 1 −→ +∞, from (7.28) we find and then for m largely enough, u j L ∞ C 1 + 1.
Hence, up to extracing a subsequence, from Lemma 5.4 in [31] u m u ∞ weakly in L 2 1 , and the limit is not 0. Moreover, from (7.29) and from Lemma 7.1.9 we deduce that tru ∞ = 0. From Lemma 5.5 in [31] we know that the eigenvalues of u ∞ are real and constant almost everywhere, in other words, there are λ 1 < · · · < λ l which are the distinc eigenvalues of u ∞ (x) for almost all x ∈ X.
Since tru ∞ = 0 and since u ∞ is not 0 we must have l 2, otherwise if tru ∞ = 0 and l = 1 it follows that u ∞ = 0 contraddicting the fact that u ∞ = 0.
The weak limit u ∞ gives rise to a flag of L 2 1 (S)-subbundles. For any integer 1 α < l, define C ∞ functions P α : R −→ R sucht that and set π α = P α (u ∞ ).
Then we can write  Therefore, from (7.31) and since D (π α ) = dP α (u ∞ )(D u ∞ ), we find is positive, so we can divide both terms of (7.6) by (7.34) obtaining . Now, applying again Lemma 7.1.14 as t −→ +∞ and this completes the proof.

Proof of the main Theorem
In this section we prove the equivalence between semistability and the existance of approximate Hermitian-Yang-Mills metric structures for Higgs bundles in every dimension. First of all we need to prove a preliminary result under the assumption tr(K 0 − cI E ) = 0.
By the Maximum Principle 5.3.2 and the properties of u(x, t) we find max X (|K t − cI E | 2 − u(x, t)) max X (|K t0 − cI E | 2 − u(x, t 0 )) = 0, t t 0 . Hence, where C a = max X×X χ(x, y, a).
Fix a, say a = 1, and let t 0 −→ +∞. Using Lemma 7.1.15 we conclude and this completes the proof. Now, using the previous Proposition and Lemma 7.1.2 we can give the proof of the main Theorem of this section: Theorem 7.2.2. Let (X, ω) be a compact Kähler manifold of (complex) dimension n and let E = (E, φ) be a Higgs bundle of rank r over X. If (E, φ) is ω-semistable, then it admits an approximate Hermitian-Yang-Mills structure.
Proof. Let h 0 ∈ Herm + (E) be a fixed Hermitian metric structure on E, and let K 0 be the mean curvature of the Hitchin-Simpson connection associated with h 0 . From Lemma 7.1.2 we may assume tr(K 0 − cI E ) = 0. Let h t be a solution of the Donaldson heat flow with initial condition h 0 . We already know that h t is defined for every positive time 0 t < +∞ and L(h t , h 0 ) is a real monotone decreasing function of t for 0 t < +∞. Now we can distinguish between three cases: