Non-exactness of toric Poisson structures

TL;DR

This paper proves that certain toric Poisson structures on projective toric varieties are inherently non-exact, based on a geometric criterion related to their symplectic leaves and invariance properties.

Contribution

It establishes a general criterion for non-exactness of Poisson structures with finite symplectic leaves and applies it to toric varieties, showing their structures are not exact.

Findings

Toric Poisson structures invariant under torus action are not exact.

A geometric criterion for non-exactness of Poisson structures with finite symplectic leaves.

Application of the criterion to projective toric varieties confirms non-exactness.

Abstract

We prove that a Poisson structure on a projective toric variety which is invariant by the torus action and whose symplectic leaves are the torus orbits is not exact. This is deduced from a geometric criterion for non-exactness of Poisson structures with a finite number of symplectic leaves.