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

TL;DR

This paper investigates fixed-point sets of $S^{1}$-actions on quaternionic manifolds, deriving relations involving Chern classes, and explores conditions under which certain complex structures can exist, with applications to quaternionic projective spaces.

Contribution

It introduces new equations relating fixed-point sets and Chern classes, and characterizes the existence of hypercomplex structures under these conditions.

Findings

Derived an equation involving first Chern classes of fixed-point sets.

Showed that non-trivial Chern classes prevent certain hypercomplex structures.

Determined fixed-point components for quaternionic $S^{1}$-actions on quaternionic projective space.

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