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

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 ≤ 2k is not empty and has at least a component of dimension ≥ 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.