Complex quaternionic manifolds and c-projective structures

We discuss complex quaternionic manifolds, i.e., those that have holonomy $GL(n,\mathbb{H})U(1)$, which naturally arise via quaternionic Feix–Kaledin construction. We show that for a fixed c-projective class, any real analytic connection with type $(1,1)$ curvature induces, via quaternionic Feix–Kaledin construction, an $S^1$-invariant connection with holonomy contained in $GL(n,\mathbb{H})U(1)$. As an application, we characterize in this setting the distinguished $U^{\ast }(2n):=SL(n,\mathbb{H})U(1)$ connection studied in Battaglia \cite{Bat} and Hitchin \cite{Hit3}.