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On Toric Poisson Structures of Type $(1,1)$ and their Cohomology
Symmetry, integrability and geometry: methods and applications, Tome 13 (2017) Cet article a éte moissonné depuis la source Math-Net.Ru

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We classify real Poisson structures on complex toric manifolds of type $(1,1)$ and initiate an investigation of their Poisson cohomology. For smooth toric varieties, such structures are necessarily algebraic and are homogeneous quadratic in each of the distinguished holomorphic coordinate charts determined by the open cones of the associated simplicial fan. As an approximation to the smooth cohomology problem in each ${\mathbb C}^n$ chart, we consider the Poisson differential on the complex of polynomial multi-vector fields. For the algebraic problem, we compute $H^0$ and $H^1$ under the assumption that the Poisson structure is generically non-degenerate. The paper concludes with numerical investigations of the higher degree cohomology groups of $({\mathbb C}^2,\pi_B)$ for various $B$.
Keywords: toric; Poisson structures; group-valued momentum map; Poisson cohomology. 
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     author = {Arlo Caine and Berit Nilsen Givens},
     title = {On {Toric} {Poisson} {Structures} of {Type} $(1,1)$ and their {Cohomology}},
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     url = {http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a22/}
}
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Arlo Caine; Berit Nilsen Givens. On Toric Poisson Structures of Type $(1,1)$ and their Cohomology. Symmetry, integrability and geometry: methods and applications, Tome 13 (2017). http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a22/

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