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arXiv 1608.00780math.AG
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不规则连接的同构变形与丛的稳定性

Isomonodromic deformations of irregular connections and stability of bundles

Indranil Biswas, Viktoria Heu, Jacques Hurtubise

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摘要：

设$G$是定义在$\mathbb C$上的约化仿射代数群，$\nabla_0$是光滑复曲线$X_0$上的全纯$G$ -丛$E_0$上的亚纯$G$ -联络，其极轨迹为$P_0 \subset X_0$ .我们假设$\nabla_0$是不可约的，在这个意义上，它不通过$G$的某些真抛物子群的因子。考虑$(E_0\to X_0, \nabla_0, P_0)$的泛同构变形$(E_t\to X_t, \nabla_t, P_t)_{t\in \mathcal{T}}$，其中$\mathcal{T}$是我们所描述的某个框架Teichmüller空间的某个商.证明了如果$X_0$的亏格$g$满足$g\geq 2$，则对于一般参数$t\in \mathcal{T}$，$G$ -丛$E_t\to X_t$是稳定的.对于$g\geq 1$，我们能够证明对于一般参数$t\in \mathcal{T}$，$G$ -丛$E_t\to X_t$是半稳定的.

英文摘要：

Let $G$ be a reductive affine algebraic group defined over $\mathbb C$, and let $\nabla_0$ be a meromorphic $G$-connection on a holomorphic $G$-bundle $E_0$, over a smooth complex curve $X_0$, with polar locus $P_0 \subset X_0$. We assume that $\nabla_0$ is irreducible in the sense that it does not factor through some proper parabolic subgroup of $G$. We consider the universal isomonodromic deformation $(E_t\to X_t, \nabla_t, P_t)_{t\in \mathcal{T}}$ of $(E_0\to X_0, \nabla_0, P_0)$, where $\mathcal{T}$ is a certain quotient of a certain framed Teichmüller space we describe. We show that if the genus $g$ of $X_0$ satisfies $g\geq 2$, then for a general parameter $t\in \mathcal{T}$, the $G$-bundle $E_t\to X_t$ is stable. For $g\geq 1$, we are able to show that for a general parameter $t\in \mathcal{T}$, the $G$-bundle $E_t\to X_t$ is semistable.

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