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

TL;DR

This paper explores complex manifolds by constructing moduli spaces and analyzing their fundamental groups, which are related to the groups $G_{n}^{k}$ and $ Gamma_{n}^{k}$, advancing the complexification of these algebraic structures.

Contribution

It introduces a new approach to study complex manifolds through the fundamental groups of moduli spaces linked to $G_{n}^{k}$ and $ Gamma_{n}^{k}$, extending previous work on their complexification.

Findings

Established a connection between moduli space fundamental groups and $G_{n}^{k}$, $ Gamma_{n}^{k}$ groups.

Developed a framework for the complexification of these groups.

Provided insights into the topology of complex manifolds via algebraic group actions.

Abstract

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}.