A NEW CONSTRUCTION OF HOMOGENEOUS QUATERNIONIC MANIFOLDS AND RELATED GEOMETRIC STRUCTURES

Let V = ℝ^p,q be the pseudo-Euclidean vector space of signature (p, q), p ≥ 3 and W a module over the even Clifford algebra Cℓ⁰(V). A homogeneous quaternionic manifold (M, Q) is constructed for any 𝔰𝔭𝔦𝔫(V)-equivariant linear map II: ∧²W → V. If the skew symmetric vector valued bilinear form II is nondegenerate then (M, Q) is endowed with a canonical pseudo-Riemannian metric g such that (M, Q, g) is a homogeneous quaternionic pseudo-Kähler manifold. If the metric g is positive definite, i.e. a Riemannian metric, then the quaternionic Kähler manifold (M, Q, g) is shown to admit a simply transitive solvable group of automorphisms. In this special case (p = 3) we recover all the known homogeneous quaternionic Kähler manifolds of negative scalar curvature (Alekseevsky spaces) in a unified and direct way. If p > 3 then M does not admit any transitive action of a solvable Lie group and we obtain new families of quaternionic pseudo-Kähler manifolds. Then it is shown that for q = 0 the noncompact quaternionic manifold (M, Q) can be endowed with a Riemannian metric h such that (M, Q, h) is a homogeneous quaternionic Hermitian manifold, which does not admit any transitive solvable group of isometries if p > 3.

The twistor bundle Z → M and the canonical SO(3)-principal bundle S → M associated to the quaternionic manifold (M, Q) are shown to be homogeneous under the automorphism group of the base. More specifically, the twistor space is a homogeneous complex manifold carrying an invariant holomorphic distribution of complex codimension one, which is a complex contact structure if and only if II is nondegenerate. Moreover, an equivariant open holomorphic immersion into a homogeneous complex manifold of complex algebraic group is constructed.

Finally, the construction is shown to have a natural mirror in the category of supermanifolds. In fact, for any 𝔰𝔭𝔦𝔫(V)-equivariant linear map II : ∨²W → V a homogeneous quaternionic supermanifold (M, Q) is constructed and, moreover, a homogeneous quaternionic pseudo-Kähler supermanifold (M, Q, g) if the symmetric vector valued bilinear form II is nondegenerate.