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 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 solutions of the evolution equation associated with that functional. Next, we study the problem of the existence of approximate HermitianYang-Mills structures and its relation with the algebro-geometric notion of semistability and we show that for a compact Riemann surface, the notion of approximate Hermitian-Yang-Mills structure is in fact the differentialgeometric counterpart of the notion of semistability. Finally, we review the notion of admissible Hermitian structure on a torsion-free Higgs sheaf and define the Donaldson functional for such an object. IntroductionIn 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 an Hermitian-Einstein 1 structure. This correspondence is also true for Higgs bundles.The history of this correspondence starts in 1965, when Narasimhan and Seshadri [12] proved that a holomorphic bundle on 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 [8] introduced * Electronic address: sholguin@sissa.it 1 In the literature Hermitian-Einstein, Einstein-Hermite and Hermitian-Yang-Mills are all synonymous. Sometimes, also the terminology Hermitian-Yang-Mills-Higgs is used [5].

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 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 solutions of the evolution equation associated with that functional. Next, we study the problem of the existence of approximate HermitianYang-Mills structures and its relation with the algebro-geometric notion of semistability and we show that for a compact Riemann surface, the notion of approximate Hermitian-Yang-Mills structure is in fact the differentialgeometric counterpart of the notion of semistability. Finally, we review the notion of admissible Hermitian structure on a torsion-free Higgs sheaf and define the Donaldson functional for such an object. IntroductionIn 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 an Hermitian-Einstein 1 structure. This correspondence is also true for Higgs bundles.The history of this correspondence starts in 1965, when Narasimhan and Seshadri [12] proved that a holomorphic bundle on 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 [8] introduced * Electronic address: sholguin@sissa.it 1 In the literature Hermitian-Einstein, Einstein-Hermite and Hermitian-Yang-Mills are all synonymous. Sometimes, also the terminology Hermitian-Yang-Mills-Higgs is used [5].

We introduce the notion of T -stability for torsion-free Higgs sheaves as a natural generalization of the notion of T -stability for torsion-free coherent sheaves over compact complex manifolds. We prove similar properties to the classical ones for Higgs sheaves. In particular, we show that only saturated flags of torsion-free Higgs sheaves are important in the definition of T -stability. Using this, we show that this notion is preserved under dualization and tensor product with an arbitrary Higgs line bundle. Then, we prove that for a torsion-free Higgs sheaf over a compact Kähler manifold, ω-stability implies T -stability. As a consequence of this, we obtain the T -semistability of any reflexive Higgs sheaf with an admissible Hermitian-Yang-Mills metric. Finally, we prove that T -stability implies ω-stability if, as in the classical case, some additional requirements on the base manifold are assumed. In that case, we obtain the existence of admissible Hermitian-Yang-Mills metrics on any T -stable reflexive sheaf.

In this article we study an integrable deformation of the Kapustin-Witten equations. Using the Weyl-Wigner-Moyal-Groenewold description an integrable ⋆-deformation of a Kapustin-Witten system is obtained. Starting from known solutions of the original equations, some solutions to these deformed equations are obtained.

We introduce the notion of Hermitian Higgs bundle as a natural generalization of the notion of Hermitian vector bundle and we study some vanishing theorems concerning Hermitian Higgs bundles when the base manifold is a compact complex manifold. We show that a first vanishing result, proved for these objects when the base manifold was K\"ahler, also holds when the manifold is compact complex. From this fact and some basic properties of Hermitian Higgs bundles, we conclude several results. In particular we show that, in analogy to the classical case, there are vanishing theorems for invariant sections of tensor products of Higgs bundles. Then, we prove that a Higgs bundle admits no nonzero invariant sections if there is a condition of negativity on the greatest eigenvalue of the Hitchin-Simpson mean curvature. Finally, we prove that invariant sections of certain tensor products of a weak Hermitian-Yang-Mills Higgs bundle are all parallel in the classical sense.Comment: 10 Pages, some typos corrected and minor change