Hermitian structures on a class of quaternionic Kähler manifolds

Any quaternionic Kähler manifold $(\bar{N},g_{\bar{N}},\mathcal{Q})$ equipped with a Killing vector field $X$ with nowhere vanishing quaternionic moment map carries an integrable almost complex structure $J_1$ that is a section of the quaternionic structure $\mathcal{Q}$. Using the HK/QK correspondence, we study properties of the almost Hermitian structure $(g_{\bar{N}},{\tilde{J}}_1)$ obtained by changing the sign of $J_1$ on the distribution spanned by $X$ and $J_{1}X$. In particular, we derive necessary and sufficient conditions for its integrability and for it being conformally Kähler. We show that for a large class of quaternionic Kähler manifolds containing the one-loop deformed c-map spaces, the structure ${\tilde{J}}_1$ is integrable. We do also show that the integrability of ${\tilde{J}}_1$ implies that $(g_{\bar{N}},{\tilde{J}}_1)$ is conformally Kähler in dimension four, but not in higher dimensions. In the special case of the one-loop deformation of the quaternionic Kähler symmetric spaces dual to the complex Grassmannians of two-planes we construct a third canonical Hermitian structure $(g_{\bar{N}},{\widehat{J}}_1)$. Finally, we give a complete local classification of quaternionic Kähler four-folds for which ${\tilde{J}}_1$ is integrable and show that these are either locally symmetric or carry a cohomogeneity $1$ isometric action generated by one of the Lie algebras $\mathfrak{o}(2)⋉{\mathfrak{h}\mathfrak{e}\mathfrak{i}\mathfrak{s}}_3(\mathbb{R})$, $\mathfrak{u}(2)$, or $\mathfrak{u}(1,1)$.