Hypercomplex structures arising from twistor spaces

Abstract

A hyperkähler manifold is defined as a Riemannian manifold endowed with three covariantly constant complex structures that are quaternionically related. A twistor space is characterized as a holomorphic fiber bundle $p:\mathcal{Z}\to {\mathbb{CP}}^1$ possesses properties such as a family of holomorphic sections whose normal bundle is ${⨁}^{2n}\mathcal{O}(1)$, a holomorphic section of ${\Lambda }^2(N\mathcal{Z})\otimes p^{\ast }(\mathcal{O}(2))$ that defines a symplectic form on each fiber, and a compatible real structure. According to the Hitchin-Karlhede-Lindström-Roček theorem (Comm. Math. Phys., 108(4):535-589, 1987), there exists a hyperkähler metric on the parameter space $M$ for the real sections of $\mathcal{Z}$. Utilizing the Kodaira-Spencer deformation theory, we facilitate the construction of a hypercomplex structure on $M$, predicated upon more relaxed presuppositions concerning $\mathcal{Z}$. This effort enriches our understanding of the classical theorem by Hitchin-Karlhede-Lindström-Roček.