I know that the Lie bracket for $(1,1)$ vector field corresponds to the obstruction map in deformation theory. I was wondering if the wedge product of the deformation vector with itself has any meaning.
More precisely, let $X$ be a smooth Calabi-Yau fourfold and let $v \in H^1(T_X)$. I can identify $v$ with the germ of an infinitesimal deformation of $X$. Is there any deformation-theoretic meaning to the equation:
$$ v \wedge v = 0, $$
in $H^2(\bigwedge^2 T_X)$?
Many thanks!
