Logarithmic connections on principal bundles over a Riemann surface
Logarithmic connections on principal bundles over a Riemann surface
复制标题
黎曼曲面上主丛上的对数连接
DOI：
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发表时间：
2017
期刊：
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通讯作者：
Arideep Saha
中科院分区：
文献类型：
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作者：
I. Biswas;A. Dan;Arjun Paul;Arideep Saha
设$E_G$是紧连通黎曼曲面$X$上的全纯主$G$-束，其中$G$是连通的约化复仿射代数群。修复一个有限子集$D \子集X$，并为D$中的每个$ X \修复$w_x \in \text{ad}(E_G)_x$。设$T$是$E_G$的全纯自同构群中的一个极大环面。给出了在$E_G$奇异/ $D$上的$T$不变对数连接的存在性的充分必要条件，使得D$中每个$x $的余项为$w_x$。在假设每个$w_x$是$T$刚性的情况下，我们还给出了$E_G$奇异/ D$上每个$x $上的余数为$w_x$的对数连接存在的充分必要条件。
Let $E_G$ be a holomorphic principal $G$-bundle on a compact connected Riemann surface $X$, where $G$ is a connected reductive complex affine algebraic group. Fix a finite subset $D \subset X$, and for each $x\in D$ fix $w_x \in \text{ad}(E_G)_x$. Let $T$ be a maximal torus in the group of all holomorphic automorphisms of $E_G$. We give a necessary and sufficient condition for the existence of a $T$-invariant logarithmic connection on $E_G$ singular over $D$ such that the residue over each $x \in D$ is $w_x$. We also give a necessary and sufficient condition for the existence of a logarithmic connection on $E_G$ singular over $D$ such that the residue over each $x \in D$ is $w_x$, under the assumption that each $w_x$ is $T$-rigid.