Studying complex manifolds by using groups $G_{n}^{k}$ and $\Gamma_{n}^{k}$

In the present paper, we study several complex manifolds by using the following idea. First, we construct a certain moduli space and study the fundamental group of this space. This fundamental group is naturally mapped to the groups $G_{n}^{k}$ and $\Gamma_{n}^{k}$. This is the step towards "complexification" of the $G_{n}^{k}$ and $\Gamma_{n}^{k}$ approach first developed in \cite{2019arXiv190508049M}.