Anti-Self-Dual Metrics and Kahler Geometry

into the rank-3 bundles of self-dual and anti-self-dual 2-forms, respectively defined as the ±l-eigenspaces of the Hodge star operator * : /\ -> / \ ; this just reflects the fact that the adjoint representation of 50(4) on the skew (4 x 4)-matrices is the sum of two 3-dimensional representations, as indicated by the Lie algebra isomorphism so(4) = so(3)©so(3). The decomposition (1) is conformally invariant, in the sense that it is unchanged if g is replaced by ug for any positive function u; but reversing the orientation of M interchanges the bundles / \ . The central importance of (1) stems from the fact that curvature tensors are bundle-valued 2-forms and thus, on a Riemannian 4-manifold, can be broken up into self-dual and anti-self-dual parts. For the Riemannian curvature of our metric g, however, one can go even further; using the metric to reinterpret the curvature tensor as the curvature operator endomorphism 7£ of the bundle of 2-forms, it is apparent that 7 £ : / \ ®/\~—>/\ © A~ m a y ^ e considered as consisting of more primitive pieces