Higgs Bundles and Holomorphic Forms

For a complex manifold X which has a holomorphic form ̟ of odd degree k, we endow E = ⊕ p≥a Λ (p,0)(X) with a Higgs bundle structure θ given by θ(Z)(φ) := {i(Z)̟} ∧ φ. The properties such as curvature and stability of these and other Higgs bundles are examined. We prove (Theorem 2, section 2, for k > 1) E and additional classes of Higgs subbundles of E do not admit Higgs-Hermitian-Yang-Mills metric in any one of the cases: i. deg(X) < 0, ii. deg(X) = 0 and a ≤ n − k + 1, or iii. a ≤ n − k + 1 and k > n 2 + 1. We give examples of (noncompact) Kähler manifolds with the above Higgs structure which admit Higgs-Hermitian-Yang-Mills metrics. We also examine vanishing theorems for (p, q)−forms with values in Higgs bundles.