Hermitian structures on a class of quaternionic K\"ahler manifolds

TL;DR

This paper investigates the integrability and conformal Kähler properties of almost Hermitian structures on quaternionic Kähler manifolds with Killing vector fields, providing classifications and new structures in specific cases.

Contribution

It introduces new integrable Hermitian structures on quaternionic Kähler manifolds using the HK/QK correspondence and classifies four-dimensional cases with these properties.

Findings

The structure  is integrable on many quaternionic Ke4hler manifolds including one-loop deformed c-map spaces.

Integrability of  implies conformally Ke4hler in four dimensions, but not in higher dimensions.

Complete local classification of four-dimensional quaternionic Ke4hler manifolds with integrable .

Abstract

Any quaternionic K\"ahler 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 $J_1X$. In particular, we derive necessary and sufficient conditions for its integrability and for it being conformally K\"ahler. We show that for a large class of quaternionic K\"ahler 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 K\"ahler in dimension four, but not in higher dimensions. In the special case of the one-loop deformation of the quaternionic K\"ahler symmetric spaces dual to the complex Grassmannians of two-planes we construct a third canonical Hermitian structure $(g_{\bar N},\hat J_1)$. Finally, we give a complete local classification of quaternionic K\"ahler 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 $\mathfrak{o}(2)\ltimes\mathfrak{heis}_3(\mathbb R)$, $\mathfrak{u}(2)$, or $\mathfrak{u}(1,1)$.