Toric Poisson Structures

TL;DR

This paper constructs and analyzes a class of Poisson structures on smooth projective toric varieties, exploring their symplectic leaves, Hamiltonian actions, and Poisson cohomology, with explicit computations for projective lines and spaces.

Contribution

It introduces a real invariant Poisson structure on toric varieties, establishes bounds for their Poisson cohomology, and connects modular vector fields to Delzant moment data.

Findings

Poisson structures have symplectic leaves as torus orbits.

Lower bounds for Poisson cohomology are established.

Explicit cohomology computations for rac{CP^1}{ ext{and} } rac{CP^n}.

Abstract

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.