Compatible almost complex structures on quaternion Kähler manifolds (Q1268108)

Let \((M,g,Q)\) be a quaternionic Kähler manifold of scalar curvature \(K\). The authors study almost complex structures \(J\) on \(M\) which are compatible with the quaternionic structure \(Q\). Let \(\omega= g\circ J\) be the corresponding 2-form. Besides many useful formulas, the following local results are obtained: (1) If \(d\omega= \alpha\wedge \omega\) for some 1-form \(\alpha\), then \(J\) is parallel and \(K=0\); (2) if \(\delta\omega=0\), then \(K\leq 0\) with equality iff \(J\) is parallel; (3) if \(J\) is integrable, then \(d(\delta\omega\circ J)\) is a \(Q\)-Hermitian harmonic 2-form; (4) any closed section of \(\Lambda_+^2:= g\circ Q\subset \Lambda^2\) is parallel. The main global results for compact quaternionic Kähler manifolds are: (1) if \(K>0\), then there exists no compatible almost complex structure; (2) if \(K\neq 0\), then there exists no compatible complex structure; (3) if \(K=0\), then a compatible complex structure is necessarily parallel; (4) if the first Chern class of a compatible almost complex structure vanishes, then \(K=0\). Parts (2) and (3) were first obtained by the third author using twistor methods. Finally, extending results of S. Salamon, a correspondence is described between Killing vector fields, sections of \(\Lambda_+^2\) satisfying the so-called twistor equation, and a class of compatible complex structures on \(M\).