Let X(\Sigma) be a smooth projective toric variety for a complex torus T_\C.
In this paper, a real T_\C-invariant Poisson structure \Pi_\Sigma is
constructed on the complex manifold X(\Sigma), the symplectic leaves of which
are the T_\C-orbits in X(\Sigma). It is shown that each leaf admits a
Hamiltonian action by a sub-torus of the compact torus T\subset T_\C. However,
the global action of T_\C on (X(\Sigma),\Pi_\Sigma) is Poisson but not
Hamiltonian. The main result of the paper is a lower bound for the first
Poisson cohomology of these structures. For the simplest case, X(\Sigma)=\CP^1,
the Poisson cohomology is computed using a Mayer-Vietoris argument and known
results on planar quadratic Poisson structures and in the example the bound is
optimal. The paper concludes with the example of \CP^n, where the modular
vector field with respect to a particular Delzant Liouville form admits a
curious formula in terms of Delzant moment data. This formula enables one to
compute the zero locus of this modular vector field and relate it to the
Euclidean geometry of the moment simplex.