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

Kazuyuki Hasegawa

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

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

Abstract

In this paper, we study fixed-point sets of -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 -actions on the quaternionic projective space. We apply these results to Pontecorvo's example .
Paper Structure (14 sections, 46 theorems, 159 equations)

This paper contains 14 sections, 46 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. A fixed-point set of a quaternionic $S^{1}$-action
  3. A quaternionic manifold and its complex submanifold
  4. A fixed-point set
  5. The twistor function and a quaternionic complex manifold
  6. The twistor operator
  7. The Swann bundle
  8. A quaternionic complex manifold
  9. A compatible complex structure of closed type
  10. An obstruction to the existence of hypercomplex structures
  11. Curvatures of the $\mu$-connection
  12. Proof of the main theorem
  13. The quaternionic projective space--Pontecorvo's example--
  14. The complex two-plane Grassmann manifold

Key Result

Lemma 2.1

Let $\nabla^{1}$ and $\nabla^{2}$ be quaternionic connections on $(M,Q)$. Then there exists a 1-form $\xi$ on $M$ such that for all $X$, $Y \in \Gamma(T M)$, where $S^{\xi}$ is defined by Conversely, for a given quaternionic connection $\nabla^{1}$, the connection $\nabla^{2}$ given by the equation above is also a quaternionic connection.

Theorems & Definitions (97)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Remark 2.6
  • ...and 87 more