Approximate Hermitian–Yang–Mills structures and semistability for Higgs bundles. I: generalities and the one-dimensional case

Abstract

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].

Approximate Hermitian–Yang–Mills structures on semistable principal Higgs bundles

“…The converse implication was proved in [9] when dim(X) = 1, and in [16] for arbitrary dimension. More exactly, [23] and was further studied in [9,10,16].…”

“…More exactly, [23] and was further studied in [9,10,16]. Let us denote by H + (E) the space of Hermitian metrics on the vector bundle E. This is a Hilbert manifold modelled on the space H(E) of Hermitian endomorphism of E suitably completed to an L 2 Hilbert space.…”

On Gieseker stability for Higgs sheaves

“…This shows that in the one-dimensional case these additional terms in (8) are all zero and the notions of stability and Gieseker stability coincide. 4 Now, since in the one-dimensional case there exist examples of stable Higgs bundles that are not stable in the classical sense (see [11] for details), we know that the notion of Gieseker stability in the Higgs case is not the same classical Gieseker stability.…”

The Hitchin–Kobayashi Correspondence for Quiver Bundles over Generalized Kähler Manifolds

“…There are several generalizations for this correspondence along different directions. For example, one replaces base manifolds with Hermitian manifolds with Gauduchon metric [22] or non-compact Kähler manifolds satisfying some analytic conditions [26]; one generalizes Yang-Mills system to other gauge theoretic systems, such as introducing Higgs fields or vortex fields via dimensional reduction [14,27,4], introducing singularities for Hermitian-Einstein connection and parabolic structure on vector bundle [24,25]; introducing frame structure via vacuum expectation value of the scalar fields in N = 2 vector multiplet [8]; one changes the stability condition, typically relaxes to semistability and approximate Hermitian-Einstein metric [6,7,21]; one considers an analog of such correspondence in positive characteristic or mixed characteristic [9,20].…”