
SIGMA 13 (2017), 023, 16 pages arXiv:1610.09445
https://doi.org/10.3842/SIGMA.2017.023
Contribution to the Special Issue “Gone Fishing”
On Toric Poisson Structures of Type $(1,1)$ and their Cohomology
Arlo Caine and Berit Nilsen Givens
California State Polytechnic University Pomona, 3801 W. Temple Ave., Pomona, CA, 91768, USA
Received October 29, 2016, in final form March 28, 2017; Published online April 06, 2017
Abstract
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 multivector fields. For the algebraic problem, we compute $H^0$ and $H^1$ under the assumption that the Poisson structure is generically nondegenerate. The paper concludes with numerical investigations of the higher degree cohomology groups of $({\mathbb C}^2,\pi_B)$ for various $B$.
Key words:
toric; Poisson structures; groupvalued momentum map; Poisson cohomology.
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