A class of quaternionic manifolds admitting a compatible complex structure (Q1267761)

Let \((M,H,D)\) be a quaternionic manifold, that is, a manifold \(M\) with a torsion-free connection \(D\), which preserves a quaternionic structure \(H \subset \text{End }TM \). It is called a quaternionic Hermitian-Weyl manifold if a conformal metric \([b]\) is given which is \(H\)-Hermitian (\(b(J\cdot, \cdot) + b(\cdot , J \cdot) =0\), for all \(J \in H \)) and \(D\)-parallel \((Db = \omega \otimes b\), \(b \in [b])\). Assume that a metric \(b \in [b] \) is fixed such that \(\nabla \omega =0\), \(b^{-1}(\omega, \omega)=1 \), where \(\nabla\) is the Levi-Civita connection of \(b\). Then \(B = b^{-1}\omega \) is a Killing vector field. A quaternionic Hermitian-Weyl manifold \(M\) is called regular if \(B\) generates a free action of the group \(S^1\) on \(M\). Then the orbit space \(P =M/S^1\) is a manifold with induced metric \(g\) and naturally defined local 3-Sasakian structure. Theorem. Let \((M,H,D,[b])\) be a regular quaternionic Hermitian-Weyl manifold. Then it admits a \(D\)-parallel complex structure \(J\) compatible with \(H\) (i.e., \( J \in \Gamma (H) \)). This complex structure together with some metric \(b \in [b]\) defines a local Kähler structure on \(M\). Some relations between Betti numbers of compact regular quaternionic Hermitian-Weyl manifolds are also established.