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We study equivariant bi-vector fields on a toric variety. We prove that, on asmooth toric variety of dimension $n$, the locus where the rank of anequivariant bi-vector field is $\leq 2k$ is not empty and has at least acomponent 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 benon empty.

On degeneracy loci of equivariant bi-vector fields on a smooth toric variety