  We give necessary conditions for the existence of degenerations between two
complex Lie superalgebras of dimension $(m,n)$. As an application, we study the
variety $\mathcal{LS}^{(2,2)}$ of complex Lie superalgebras of dimension
$(2,2)$. First we give the algebraic classification and then obtain that
$\mathcal{LS}^{(2,2)}$ is the union of seven irreducible components, three of
which are the Zariski closures of rigid Lie superalgebras. As byproduct, we
obtain an example of a nilpotent rigid Lie superalgebra, in contrast to the
classical case where no example is known.
