Compatible Almost Complex Structures on Quaternion Kähler Manifolds

Abstract

Let (M⁴ⁿ,g,Q) be a quaternion Kähler 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.

When M⁴ⁿ is compact our main global results are the following: (1) if ν > 0, then there exists no compatible almost complex structure J; (2) if the first Chern class c₁(T^(1,0) _J M) = 0, then ν = 0; (3) if ν = 0 a compatible complex structure J is parallel; and (4) if ν ≠ 0, then no compatible complex structure J exists. The last two results have been proved in [23] by twistor methods.