Special structures on four-manifolds
Received: 15 June 1992
Mathematics Subject Classification: 53C15
Abstract In this note the formalism of spinors is used to analyse properties af almost complex structures on manifolds in dimension four. The integrability property of an almost complex structure is, in the presence of a compatible metric, exactly complementary to the condition that the almost complex structure give rise to a symplectic form. This fact is particularly evident when one studies the situation of a 4-manifold which possesses two anti-commuting almost complex structures. The most striking instance of this is when the manifold has a hyperkahler metric, and our remarks serve to place these metrics in a more general setting. Consequences of the existence of a complex structure for the Riemann curvature tensor are pursued in the last section. A 4-manifold has an abundance of orthogonal complex structures if and anly if it is self-dual, which means that half of its conformal Weyl tensor vanishes. Thus the material below is closely connected with the more general theory of self-duality.