标题： 从扭量空间中产生的超复结构 显示英文标题

标题： Hypercomplex structures arising from twistor spaces

摘要： 超凯勒流形被定义为带有三个以四元数方式相关联的协变常复结构的黎曼流形。 扭量空间被表征为一个全纯纤维丛$p: \mathcal{Z} \rightarrow \mathbb{CP}^1$，它具有如下的性质：一族全纯截面，其法丛为$\bigoplus^{2n}\mathcal{O}(1)$，一个全纯截面$\Lambda^2(N\mathcal{Z})\otimes p^*(\mathcal{O}(2))$，该截面在每个纤维上定义了一个辛形式，并且有一个相容的实结构。 根据Hitchin-Karlhede-Lindström-Roček定理（Comm. Math. Phys., 108(4):535-589, 1987），参数空间$M$上存在一个超凯勒度量，该参数空间对应于$\mathcal{Z}$的实截面。 利用Kodaira-Spencer形变理论，我们能够在更宽松的关于$\mathcal{Z}$的假定基础上，促进在$M$上构造一个超复结构。 这一努力加深了我们对Hitchin-Karlhede-Lindström-Roček经典定理的理解。 显示英文摘要

摘要： 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.