arXiv 2102.006242021-02math.AGmath.DGmath.SG

Riemann surfaces上丛的一个标准联络和θ丛上的Quillen联络

A canonical connection on bundles on Riemann surfaces and Quillen connection on the theta bundle

Indranil Biswas, Jacques Hurtubise

备注：: Final version

摘要：

研究了紧致黎曼曲面$X$上联络的模空间的辛几何和微分几何性质。在$X$上固定一个θ特征$K^{1/2}_X$;它在秩为$r$的$X$上的稳定向量丛的模空间${\mathcal M}$上定义一个θ因子。给定一个位于θ因子外的向量丛$E \in {\mathcal M}$，我们在$E$上构造了一个全纯依赖于$E$的自然全纯联络.利用这个全纯连接，我们在以下两个之间构造了一个规范的全纯同构：\begin{enumerate} \item the moduli space $\mathcal C$ of pairs $(E, D)$, where $E\in {\mathcal M}$ and $D$ is a holomorphic connection on $E$, and \item the space ${\rm Conn}(Θ)$ given by the sheaf of holomorphic connections on the line bundle on $\mathcal M$ associated to the theta divisor. \end{enumerate}上述$\mathcal C$和${\rm Conn}(Θ)$之间的同构是保辛结构的，并且当$X$在全纯黎曼曲面族上运行时，它全纯移动。

英文摘要：

We investigate the symplectic geometric and differential geometric aspects of the moduli space of connections on a compact Riemann surface $X$. Fix a theta characteristic $K^{1/2}_X$ on $X$; it defines a theta divisor on the moduli space ${\mathcal M}$ of stable vector bundles on $X$ of rank $r$ degree zero. Given a vector bundle $E \in {\mathcal M}$ lying outside the theta divisor, we construct a natural holomorphic connection on $E$ that depends holomorphically on $E$. Using this holomorphic connection, we construct a canonical holomorphic isomorphism between the following two: \begin{enumerate} \item the moduli space $\mathcal C$ of pairs $(E, D)$, where $E\in {\mathcal M}$ and $D$ is a holomorphic connection on $E$, and \item the space ${\rm Conn}(Θ)$ given by the sheaf of holomorphic connections on the line bundle on $\mathcal M$ associated to the theta divisor. \end{enumerate} The above isomorphism between $\mathcal C$ and ${\rm Conn}(Θ)$ is symplectic structure preserving, and it moves holomorphically as $X$ runs over a holomorphic family of Riemann surfaces.

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