Hermitian structures on a class of quaternionic Kähler manifolds

Any quaternionic Kähler manifold

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

upper Q $\mathcal Q$ Q

. Using the HK/QK correspondence, we study properties of the almost Hermitian structure

left parenthesis gN bar comma upper J tilde 1 right parenthesis $(g_{\bar N},\tilde J_1)$ ( g N ¯ , J ~ 1 )

obtained by changing the sign of J ₁ on the distribution spanned by X and

upper J 1 upper X $J_1X$ 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

upper J tilde 1 $\tilde J_1$ J ~ 1

is integrable. We do also show that the integrability of

upper J tilde 1 $\tilde J_1$ J ~ 1

implies that

left parenthesis gN bar comma upper J tilde 1 right parenthesis $(g_{\bar N},\tilde J_1)$ ( g N ¯ , J ~ 1 )

is conformally Kähler in dimension 4, 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

left parenthesis gN bar comma upper J caret 1 right parenthesis $(g_{\bar N},\hat J_1)$ ( g N ¯ , J ^ 1 )

. Finally, we give a complete local classification of quaternionic Kähler fourfolds for which

upper J tilde 1 $\tilde J_1$ 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

fraktur o left parenthesis 2 right parenthesis less than imesheis 3 left parenthesis upper R right parenthesis $\mathfrak{o}(2) \lt imes\mathfrak{heis}_3(\mathbb R)$ 𝔬 ( 2 ) < i m e s h e i s 3 ( R )

,

fraktur u left parenthesis 2 right parenthesis $\mathfrak{u}(2)$ 𝔲 ( 2 )

, or

fraktur u left parenthesis 1 comma 1 right parenthesis $\mathfrak{u}(1,1)$ 𝔲 ( 1 , 1 )

.