Neutral hyperkähler structures on primary Kodaira surfaces (Q1961898)

A manifold \(M\) is a neutral almost hyper-Hermitian manifold (or a paraquaternionic Hermitian manifold), if there are three endomorphisms \((I,J,K)\) on the tangent bundle of \(M\) and a pseudo-Riemannian metric \(g\) satisfying: \(J^2= K^2=- I^2= Id\) and \(g(IX,IY)= -g(JX, JY)= -g(KX,KY)= g(X,Y)\), for tangent vectors \(X\) and \(Y\) (\(\dim M= 4n\) and the sinature of \(g\) is \((2n,2n)\), see \textit{N. Blăzić} [Publ. Inst. Math. (Beograd), 60, 101-107 (1996; MR 97h:53034)]). The manifold \(M\) is called a neutral hyper-Kähler manifold if \(I\), \(J\) and \(K\) are integrable and the fundamental forms \(\Omega_I(X, Y)= g(IX, Y)\), \(\Omega_J(X, Y)= g(JX, Y)\) and \(\Omega_K(X, Y)= g(KX, Y)\) are closed. In this paper, the author studies neutral hyper-Kähler structures on compact four-dimensional manifolds, which are called neutral hyper-Kähler surfaces. As the author points out, according to the Enriques-Kodaira classification for compact complex surfaces, for any compact neutral hyper-Kähler surface \((M,g,I,J,K)\) the underlying complex surface \((M,I)\) must be biholomorphic to one of the following: (a) a complex torus (which has the standard flat neutral hyper-Kähler structure), (b) a K3 surface (which admit no symplectic structures compatible with the opposite orientation), and (c) a primary Kodaira surface. In this paper, neutral hyper-Kähler structures on a primary Kodaira surface are investigated. Thus, the author gives a characteriation of neutral hyper-Kähler structures, in terms of a partial differential equation for the Kähler potentials, and proves that any primary Kodaira surface admits neutral hyper-Kähler structures, whose compatible neutral metrics can be chosen to be flat or nonflat.