Compatible Almost Complex Structures on Quaternion Kähler Manifolds

Abstract

Let (M4n,g,Q) be a quaternion Kahler manifold with reduced scalar curvature ν = K/4n(n + 2). Suppose J is an almost complex structure which is compatible with the quaternionic structure Q and let θ = − δ F ∘ J be the Lee form of J. We prove the following local results: (1) if J is conformally symplectic, then it is parallel and ν = 0; (2) if J is cosymplectic, then ν ≤ 0 with equality if and only if J is parallel; (3) if J is integrable, then dθ is Q-Hermitian and harmonic; and (4) any closed self-dual 2-form ω = f(g ∘ J) ∈ Λ2+ = g ∘ Q ⊂ Λ2 is parallel. In Section 5, extending previous results of Salamon [24], we describe a correspondence among conformally balanced J, Killing vector fields X and self-dual 2-forms μ satisfying the twistor equation.