  In this paper, we study the interplay between modules and sub-objects in
holomorphic Poisson geometry. In particular, we define a new notion of
"residue" for a Poisson module, analogous to the Poincar\'e residue of a
meromorphic volume form. Of particular interest is the interaction between the
residues of the canonical line bundle of a Poisson manifold and its degeneracy
loci---where the rank of the Poisson structure drops. As an application, we
provide new evidence in favour of Bondal's conjecture that the rank \leq 2k
locus of a Fano Poisson manifold always has dimension \geq 2k+1. In particular,
we show that the conjecture holds for Fano fourfolds. We also apply our
techniques to a family of Poisson structures defined by Fe\u{\i}gin and
Odesski\u{\i}, where the degeneracy loci are given by the secant varieties of
elliptic normal curves.
