Examples of varieties of structures

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.