Quaternionic complex manifolds and fixed-point sets of $S^1$-actions

In this paper, we study fixed-point sets of $S^1$-actions and compatible complex structures on quaternionic manifolds. We obtain an equation involving the first Chern classes of the fixed-point set and of a quaternionically flat manifold with compatible complex structure of closed type. In addition, if the first Chern class of the fixed-point set is not trivial, then the quaternionic manifold does not admit hypercomplex structures containing given compatible complex structure on any open set containing the fixed-point set. Moreover, we determine the connected components of the fixed-point set arising from quaternionic $S^1$-actions on the quaternionic projective space. We apply these results to Pontecorvo’s example ${\mathrm{S}\mathrm{O}}^{\ast }(2n+2)\mathrm{/}{\mathrm{S}\mathrm{O}}^{\ast }(2n)\times {\mathrm{S}\mathrm{O}}^{\ast }(2)$.