Toric Poisson structures

Let $T_\mathbb C$ be a complex algebraic torus and let $X(\Sigma)$ be a complete nonsingular toric variety for $T_\mathbb C$. In this paper, a real $T_\mathbb C$-invariant Poisson structure $\Pi_\Sigma$ is constructed on the complex manifold $X(\Sigma)$, the symplectic leaves of which are the $T_\mathbb C$-orbits in $X(\Sigma)$. It is shown that each leaf admits an effective Hamiltonian action by a subtorus of the compact torus $T\subset T_\mathbb C$. However, the global action of $T_\mathbb C$ 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, $X(\Sigma)=\mathbb C\mathrm P^1$, the Poisson cohomology is computed using a Mayer–Vietoris argument and known results on planar quadratic Poisson structures. In this example, the bound is optimal. The paper concludes by studying the interaction of $\Pi_\Sigma$ with the symplectic structure on $\mathbb C\mathrm P^n$, 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.