Toric Poisson Structures

Abstract

Let T C be a complex algebraic torus and let X(Σ) be a smooth projective T C -variety. In this paper, a real T C -invariant Poisson structure Π Σ is constructed on the complex manifold X(Σ), the symplectic leaves of which are the T C -orbits in X(Σ). It is shown that each leaf admits a Hamiltonian action by a sub-torus of the compact torus T ⊂ T C . However, the global action of T C on (X(Σ), Π Σ ) 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(Σ) = 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.

On Toric Poisson Structures of Type (1,1) and their Cohomology

“…(z,z) = (ζz, ζz) and π = −2izz∂ z ∧ ∂z is a smooth toric Poisson structure. Note that π = −2izz∂ z ∧ ∂z is real (fixed by conjugation) and of type (1,1). In terms of real variables, z = x + iy and ∂ z = 1 2 (∂ x − i∂ y ) while ∂z = 1 2 (∂ x + i∂ y ) and one can check that…”

“…Holomorphic Poisson structures are necessarily bi-vector fields of type (2,0). No bi-vector field of type (1,1) can be holomorphic, nor can such a field occur as the real projection of holomorphic bi-vector field.…”

“…In [1], examples of toric Poisson structures of type (1, 1) on smooth toric varieties generalizing this example were constructed via a quotient construction. A smooth toric variety M determines and is determined by a simplicial fan Σ in t, the real Lie algebra of the torus T , which is simplicial with respect to the coweight lattice Λ ⊂ t of T C .…”

“…The double slash indicates that one does not divide C d by the action of N C but rather a dense open T d C -stable subset U Σ , determined from Σ, on which N C acts freely. The product Poisson structure π ⊕ · · · ⊕ π on C d , using π from (1.1), restricts to a real toric Poisson structure on U Σ of type (1,1) which is generically non-degenerate and, via the quotient map U Σ → C d //N C , coinduces a real toric Poisson structure of type (1, 1) on M which is generically non-degenerate. Each open cone in Σ determines a distinguished holomorphic coordinate chart C n on M in which the action of T C T d C /N C is equivalent to the action of T n C on C n .…”

“…, z n ), the Poisson structure on M has the form B pq Z p ∧ Z q making it clear that it is the image of an element of t C ∧ t C under the natural map induced infinitesimally by the holomorphic action of T C on M . In fact, since t C is abelian and the map from t C into sections of T 1,0 is a homomorphism of Lie algebras, it is clear that any element of t C ∧ t C maps to a toric Poisson structure on M of type (1,1). In terms of the coordinates (z 1 , .…”