Hypercomplex structures arising from twistor spaces

Abstract

A hyperk\"ahler 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 $p: \mathcal{Z} \rightarrow \mathbb{CP}^1$ possesses properties such as a family of holomorphic sections whose normal bundle is $\bigoplus^{2n}\mathcal{O}(1)$, a holomorphic section of $\Lambda^2(N\mathcal{Z})\otimes p^*(\mathcal{O}(2))$ that defines a symplectic form on each fiber, and a compatible real structure. According to the Hitchin-Karlhede-Lindstr\"om-Ro\v{c}ek theorem (Comm. Math. Phys., 108(4):535-589, 1987), there exists a hyperk\"ahler 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-Lindstr\"om-Ro\v{c}ek.