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

Abstract

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 $\mathrm{SO}^{\ast}(2n+2)/\mathrm{SO}^{\ast}(2n) \times \mathrm{SO}^{\ast}(2)$.