  We study equivariant bi-vector fields on a toric variety. We prove that, on a
smooth toric variety of dimension $n$, the locus where the rank of an
equivariant bi-vector field is $\leq 2k$ is not empty and has at least a
component of dimension $\geq 2k+1$, for all integers $k> 0$ such that $2k < n$.
The same is true also for $k=0$, if the toric variety is smooth and compact.
While for the non compact case, the locus in question has to be assumed to be
non empty.
