Year: 2024
Type: preprint
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 $p: \mathcal{Z} \rightarrow \mathbb{CP}^1$ possesses properties such as a family of holomorphic sections whose normal bundle is $\bigoplus^{2n}\mathcal{O}(1)$, a holomorphic section of $Λ^2(N\mathcal{Z})\otimes p^*(\mathcal{O}(2))$ that defines a symplectic form on each fiber, and a compatible real structure. According to the Hitchin-Karlhede-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.
Source: arXiv (Cornell University)
Authors: Wang, Shuo, Xu, Bin
Institution: University of Science and Technology of China
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4392089939