Hermitian structures on a class of quaternionic Kähler manifolds

Source abstract

Any quaternionic Kähler manifold left parenthesis upper N bar comma gN bar comma upper Q right parenthesis $(\bar N,g_{\bar N},\mathcal Q)$ ( N ¯ , g N ¯ , 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 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 ) .