Systems of symplectic forms on four-manifolds (Q2864590)

The paper under review records an investigation of Riemannian 4-dimensional manifolds that admit an orthonormal system of 5 symplectic forms. For a natural hyper-Kähler structure on these manifolds, one proves the existence of tri-holomorphic Killing vector fields and one describes the local structures of the Riemannian manifolds under consideration (Theorem 1.1). Among other results, one describes the almost-Kähler 4-dimensional manifolds for which the dimension of the holonomy algebra of their corresponding Hermitian connection has dimension at most 1 (Theorems 5.5 and 5.6).