# The geometric classification of Leibniz algebras

## TL;DR

This paper classifies all rigid four-dimensional Leibniz algebras over complex numbers, identifies irreducible components in their algebraic variety, and examines conjectures related to their structure, providing a comprehensive geometric understanding.

## Contribution

It provides a complete geometric classification of four-dimensional Leibniz algebras, including the validation and refutation of key conjectures in the field.

## Key findings

- Identified all rigid algebras in $\
- ,
- ,

## Abstract

We describe all rigid algebras and all irreducible components in the variety of four dimensional Leibniz algebras $\mathfrak{Leib}_4$ over $\mathbb{C}.$ In particular, we prove that the Grunewald--O'Halloran conjecture is not valid and the Vergne conjecture is valid for $\mathfrak{Leib}_4.$