Walker 4-manifolds with proper almost complex structures

It is shown that a Walker 4-manifold, endowed with a canonical neutral metric depending on three arbitrary functions, admits a specific almost complex structure (called proper) and an associated opposite almost complex structure. We study when these two almost complex structures are integrable and when the corresponding Kähler forms are symplectic. The conditions for the canonical neutral metric to be Kähler imply that the three arbitrary functions in the metric are all harmonic with respect to two coordinate variables, and we obtain a useful method of constructing indefinite Kähler 4-manifolds. Petean’s example of a nonflat indefinite Kähler–Einstein 4-manifold is a special case of this construction.