Examples of varieties of structures

TL;DR

This paper explores various algebraic varieties related to complexes, structures, and deformation theory, including subcomplexes, maps, Grassmannians, determinantal varieties, and differential graded Lie algebras, with computed dimensions and properties.

Contribution

It introduces and analyzes several examples of algebraic varieties relevant to deformation theory, providing explicit dimension calculations and properties.

Findings

Computed dimensions of the introduced varieties

Analyzed properties of these algebraic structures

Provided examples relevant to moduli spaces in deformation theory

Abstract

In this article we will introduce, among others, the variety of subcomplexes and the variety of maps between complexes of given rank. Also, varieties of $\mathfrak{g}$-structure like $\mathfrak{g}$-Grassmannian, $\mathfrak{g}$-determinantal varieties and finally the variety $\mathcal{DGLA}(E)$ of differential graded Lie algebra structures on $E$. We will compute the dimensions of these varieties and also some relevant properties. The motivation of this article is to give examples of moduli spaces relevant to deformation theory.