  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.
