Hypercomplex structures arising from twistor spaces

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: Z → CP¹ possesses properties such as a family of holomorphic sections whose normal bundle is ²ⁿO(1), a holomorphic section of Λ²(NZ)⊗ p^*(O(2)) that defines a symplectic form on each fiber, and a compatible real structure. According to the Hitchin-Karlhede-Lindstr\"om-Ro{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 Z. Utilizing the Kodaira-Spencer deformation theory, we facilitate the construction of a hypercomplex structure on M, predicated upon more relaxed presuppositions concerning Z. This effort enriches our understanding of the classical theorem by Hitchin-Karlhede-Lindstr\"om-Ro{c}ek.