# Symmetries of complex flat manifolds

## TL;DR

This paper investigates the automorphism groups of flat Kähler manifolds, providing methods for their calculation and classification, and exploring the complex analogues of Bieberbach theorems with illustrative examples.

## Contribution

It introduces new approaches to compute automorphism groups and classifies flat Kähler manifolds up to biholomorphism, extending Bieberbach theorems to the complex setting.

## Key findings

- Automorphism groups can be computed via affine transformations and complex torus automorphisms.
- Classification of flat Kähler manifolds depends on their automorphism groups.
- Finiteness of automorphism groups is influenced by factors beyond the fundamental group.

## Abstract

In this article we show how to calculate the group of automorphisms of flat K\"ahler manifolds. Moreover we are interested in the problem of classification of such manifolds up to biholomorphism. We consider these problems from two points of view. The first one treats the automorphism group as a subgroup of the group of affine transformations, while in the second one we analyze it using automorphisms of complex tori. This leads us to the analogues of the Bieberbach theorems in the complex case. We end with some examples, which in particular show that in general the finiteness of the automorphism group depends not only on the fundamental group of a flat manifold.