Metrics on semistable and numerically effective Higgs bundles

Abstract

Abstract. We provide notions of numerical effectiveness and numerical flatness for Higgs vector bundles on compact Kähler manifolds in terms of fibre metrics. We prove several properties of bundles satisfying such conditions and in particular we show that numerically flat Higgs bundles have vanishing Chern classes, and that they admit filtrations whose quotients are stable flat Higgs bundles. We compare these definitions with those previously given in the case of projective varieties. Finally we study the relations between numerically effectiveness and semistability, establishing semistability criteria for Higgs bundles on projective manifolds of any dimension.

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

“…By Theorem 2.6 on [8] we conclude that Ad(E) is semistable. But this holds true if and only if E is semistable.…”

“…A sketch of the proof of this result is as follows. If the principal Higgs bundle E = (E, φ) admits an approximate Hermitian-Yang-Mills structure, one shows that the same holds for the adjoint Higgs bundle Ad(E); the latter then is semistable [8], so that E is semistable as well. The proof of the converse result relies on an analysis of the flow of the Donaldson functional defined on the space of Hermitian metrics on the Higgs bundle Ad(E), showing that the flow preserves the condition that an Hermitian metric on Ad(E) comes from a reduction of the structure group of E to a maximal compact subgroup K (this implements in the case of principal Higgs bundles the ideas used in [6] for principal bundles, but we provide here a more detailed description of this technique).…”

A Liouville theorem on complete non-Kähler manifolds

“…Let s be a θ-invariant holomorphic section of a Higgs bundle (E, ∂ E , θ), i.e. there exists a holomorphic 1-form η on M such that θ(s) = η ⊗ s. When the base manifold (M, ω) is compact, following Kobayashi's techniques [9], one can obtain vanishing theorems for θ-invariant holomorphic sections on Higgs bundles or Higgs sheaves (see [1,3,13]). Now we consider the case that the base manifold is complete non-Kähler.…”

On Gieseker stability for Higgs sheaves

“…Higgs sheaves). In particular, there are Bochner's vanishing theorems for Higgs bundles [6]; and there are extensions of the Hitchin-Kobayashi correspondence to polystable reflexive Higgs sheaves [2] (a reflexive Higgs sheaf is polystable if and only if it has an admissible HYM-metric) and to semistable Higgs bundles [3,14]; in this case, the differential geometric counterpart is the notion of approximate Hermitian-Yang-Mills metric (from now on abbreviated apHYM-metric). More recently [7], the notion of T -stability (also called Bogomolov stability) has been studied in the context of Higgs sheaves over compact complex manifolds.…”