On vanishing theorems for Higgs bundles

Abstract

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

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.…”

$$T$$ T -stability for Higgs sheaves over compact complex manifolds

“…As it is well known, several results on holomorphic vector bundles and coherent sheaves can be extended to Higgs bundles and Higgs sheaves. In particular, Vanishing theorems for Higgs bundles have been recently studied in [7], and Simpson proved in [12] a Hitchin-Kobayashi correspondence for Higgs bundles over compact Kähler manifolds, i.e., an equivalence between the notion of Mumford-Takemoto polystability and the existence of Hermitian-Yang-Mills metrics (henceforth usually abbreviated HY M -metric). Now, Bruzzo and Graña Otero [4] proved that if a Higgs bundle admits an approximate Hermitian-Yang-Mills (henceforth abbreviated apHY M -metric), it is necessarily semistable in the sense of Mumford-Takemoto.…”