Principal SO(2n,ℂ)-Bundle Fixed Points over a Compact Riemann Surface

• Date: 2024
• Main Author: Antón-Sancho, Álvaro
• Format: Article
• Language: English
• Published: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна Національної академії наук України 2024
• Subjects: головне розшарування; ортогональна група; простiр модулiв; рiманова поверхня; автоморфiзм
• Online Access: https://jmag.ilt.kharkiv.ua/index.php/jmag/article/view/1063
• Journal Title: Journal of Mathematical Physics, Analysis, Geometry

Let $X$ be a compact connected Riemann surface of genus $g\geq 2$ equipped with a holomorphic involution $\sigma_X,$ and let $G$ be a semisimple complex Lie group which admits an outer involution $\sigma$. A principal $(G,\sigma_X,\sigma)$-bundle over $X$ is a pair $(E,\rho),$ where $E$ is a principal $G$-bundle over $X$ and $\rho:E\rightarrow\sigma_X^*(\sigma(E))$ is an isomorphism such that $(\sigma_X^*\rho)\circ\rho:E\rightarrow E$ is an automorphism of $E$ which acts as the product by an element of the center of $G$. In this paper, principal $(G,\sigma_X,\sigma)$-bundles over $X$ are introduced and the study is particularized to the case of $G=\textrm{SO}(2n,\mathbb{C})$. It is shown that the stability and polystability conditions for a principal $(\textrm{SO}(2n,\mathbb{C}),\sigma_X,\sigma)$-bundle coincide with those of the corresponding principal $\textrm{SO}(2n,\mathbb{C})$-bundle. Finally, the explicit form that a principal $(\textrm{SO}(2n,\mathbb{C}),\sigma_X,\sigma)$-bundle takes is provided, and the stability of these principal $(\textrm{SO}(2n,\mathbb{C}),\sigma_X,\sigma)$-bundles is discussed. Mathematical Subject Classification 2020: 14D20, 14H10, 14H60