In this paper, we address the issue of quaternionic Toledo invariant to study the character variety of two dimensional complex hyperbolic uniform lattices into $SU(n,2)$. We construct four distinct representations to prove that the character variety contains at least four distinct components. We also address the existence of holomorphic horizontal lift to various period domains of $SU(n,2)$.

中文翻译：

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通过对称正方形在 SU(2,1) 中的格子的托莱多不变量

在本文中，我们解决了四元数托莱多不变量的问题，以研究二维复双曲均匀格子的特征变化为 $SU(n,2)$。我们构建了四个不同的表示来证明字符种类至少包含四个不同的组件。我们还解决了 $SU(n,2)$ 的各个周期域的全纯水平提升的存在。