Monodromy of rank 2 twisted Hitchin systems and real character varieties
Monodromy of rank 2 twisted Hitchin systems and real character varieties
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DOI：
10.1090/tran/7144
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发表时间：
2015-06
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通讯作者：
David Baraglia;L. Schaposnik
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作者：
David Baraglia;L. Schaposnik
我们引入了一种计算Hitchin映射单一性的新方法，并利用它完全确定了$L$-twisted $G$-Higgs束的模空间的单一性，对于群$G = GL(2,\mathbb{C})$, $SL(2,\mathbb{C})$和$PSL(2,\mathbb{C})$。我们还确定了正则轨迹的扭曲Chern类，它阻碍了$L$的模空间的一部分的存在-秩$2$和阶$deg(L)+1$的扭曲希格斯束。通过计算具有$\mathbb{Z}_2$-系数的单作用的轨道，我们统一得到了实数群$G = GL(2,\mathbb{R})$, $SL(2,\mathbb{R})$, $PGL(2,\mathbb{R})$, $PSL(2,\mathbb{R})$的字符变体的分量个数，以及具有最大Toledo不变量的$Sp(4,\mathbb{R})$-字符变体的分量个数。我们还使用我们对$GL(2,\mathbb{R})$的结果来计算$SO(2,2)$ Hitchin映射的单一性，并确定$SO(2,2)$字符变化的组成部分。
We introduce a new approach for computing the monodromy of the Hitchin map and use this to completely determine the monodromy for the moduli spaces of $L$-twisted $G$-Higgs bundles, for the groups $G = GL(2,\mathbb{C})$, $SL(2,\mathbb{C})$ and $PSL(2,\mathbb{C})$. We also determine the twisted Chern class of the regular locus, which obstructs the existence of a section of the moduli space of $L$-twisted Higgs bundles of rank $2$ and degree $deg(L)+1$. By counting orbits of the monodromy action with $\mathbb{Z}_2$-coefficients, we obtain in a unified manner the number of components of the character varieties for the real groups $G = GL(2,\mathbb{R})$, $SL(2,\mathbb{R})$, $PGL(2,\mathbb{R})$, $PSL(2,\mathbb{R})$, as well as the number of components of the $Sp(4,\mathbb{R})$-character variety with maximal Toledo invariant. We also use our results for $GL(2,\mathbb{R})$ to compute the monodromy of the $SO(2,2)$ Hitchin map and determine the components of the $SO(2,2)$ character variety.