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

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2017
• Main Authors: Caine, A., Givens, B.N.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2017
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/148600
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: On Toric Poisson Structures of Type (1,1) and their Cohomology / A. Caine, B.N. Givens // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 8 назв. — англ.

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 Cn chart, we consider the Poisson differential on the complex of polynomial multi-vector fields. For the algebraic problem, we compute H⁰ and H¹ under the assumption that the Poisson structure is generically non-degenerate. The paper concludes with numerical investigations of the higher degree cohomology groups of (C²,πB) for various B.