On the geometry of generalized almost quaternionic manifolds of vertical type

We study generalized almost quaternionic manifolds of vertical type. Examples of this type of manifolds are given. It is proved that on a generalized almost quaternionic manifold there always exists an almost $\alpha$-quaternionic connection, which in the main bundle induces a metric connection. The criterion of the auto-duality of the projected vertical $2$-form on an almost $\alpha$-quaternion manifold is obtained. The components of the structural endomorphism on the space of the $G$-structure are obtained. The answer to the question is obtained: when does the Riemann-Christoffel endomorphism preserve the Kähler module of a variety. It is proved that the Riemann-Christoffel Hermitian endomorphism of an almost $\alpha$-quaternionic variety of vertical type preserves the Kähler module of a variety if and only if the structural sheaf of this variety is Einstein. Hence, as a consequence, we obtain that a four-dimensional manifold with a Riemannian or neutral pseudo-Riemannian metric is an Einstein manifold if and only if its module of auto-dual forms is invariant with respect to the Riemann-Christoffel endomorphism. The resulting corollary shows that the previous result is a broad generalization of the Atiyah-Hitchin-Singer theorem, which gives the Einstein criterion for 4-dimensional Riemannian manifolds in terms of auto-dual forms, since the result generalizes this theorem to the case of a neutral pseudo-Riemannian metric. On the other hand, this result is closely related to the well-known result of Berger, who clarifies it in the special case of quaternionic-Kähler manifolds: if a variety $M$ is quaternionic-Koehler, then its Riemann connectivity (and not just the Riemann-Christoffel operator) preserves the Koehler modulus of the variety. In this case, $M$ is an Einstein manifold.