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arXiv 2303.09701math.AG
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曲线上向量丛的模空间及其性质

Moduli spaces of vector bundles on a curve and opers

Indranil Biswas, Jacques Hurtubise, Vladimir Roubtsov

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摘要：

设$X$是亏格为$g$的紧致连通Riemann曲面，其中$g\, \geq\,2$为亏格，$ξ$是$X$上的全纯线丛，其中$ξ^{\otimes 2}\,=\, {\mathcal O}_X$为亏格.将θ特性$\mathbb L$固定在$X$上。设${\mathcal M}_X(r,ξ)$是秩为$r$的$X$上的稳定向量丛$E$的模空间，使得$\bigwedge^r E\,=\, ξ$和$H^0(X,\, E\otimes{\mathbb L})\,=\, 0$。考虑${\mathcal M}_X(r,ξ)$与$E\, \longmapsto\, E^*$给出的对合的商。我们从这个商到$X$上的${\rm SL}(r,{\mathbb C})$算子的模空间构造了一个代数态射。由于$\dim {\mathcal M}_X(r,ξ)$与${\rm SL}(r,{\mathbb C})$算子的模空间的维数一致，自然会问这个映射的内射性和满射性。

英文摘要：

Let $X$ be a compact connected Riemann surface of genus $g$, with $g\, \geq\,2$, and let $ξ$ be a holomorphic line bundle on $X$ with $ξ^{\otimes 2}\,=\, {\mathcal O}_X$. Fix a theta characteristic $\mathbb L$ on $X$. Let ${\mathcal M}_X(r,ξ)$ be the moduli space of stable vector bundles $E$ on $X$ of rank $r$ such that $\bigwedge^r E\,=\, ξ$ and $H^0(X,\, E\otimes{\mathbb L})\,=\, 0$. Consider the quotient of ${\mathcal M}_X(r,ξ)$ by the involution given by $E\, \longmapsto\, E^*$. We construct an algebraic morphism from this quotient to the moduli space of ${\rm SL}(r,{\mathbb C})$ opers on $X$. Since $\dim {\mathcal M}_X(r,ξ)$ coincides with the dimension of the moduli space of ${\rm SL}(r,{\mathbb C})$ opers, it is natural to ask about the injectivity and surjectivity of this map.

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