Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 023, 16 pages      arXiv:1610.09445
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

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 $B$.

Key words: toric; Poisson structures; group-valued momentum map; Poisson cohomology.

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  1. Caine A., Toric Poisson structures, Mosc. Math. J. 11 (2011), 205-229, arXiv:0910.0229.
  2. Goto R., Unobstructed deformations of generalized complex structures induced by $C^\infty$ logarithmic symplectic structures and logarithmic Poisson structures, in Geometry and Topology of Manifolds, Springer Proc. Math. Stat., Vol. 154, Springer, Tokyo, 2016, 159-183, arXiv:1501.03398.
  3. Liu Z.J., Xu P., On quadratic Poisson structures, Lett. Math. Phys. 26 (1992), 33-42.
  4. Lu J.-H., Momentum mappings and reduction of Poisson actions, in Symplectic Geometry, Groupoids, and Integrable Systems (Berkeley, CA, 1989), Math. Sci. Res. Inst. Publ., Vol. 20, Springer, New York, 1991, 209-226.
  5. Monnier P., Formal Poisson cohomology of quadratic Poisson structures, Lett. Math. Phys. 59 (2002), 253-267.
  6. Monnier P., Poisson cohomology in dimension two, Israel J. Math. 129 (2002), 189-207.
  7. Nakanishi N., Poisson cohomology of plane quadratic Poisson structures, Publ. Res. Inst. Math. Sci. 33 (1997), 73-89.
  8. Vaisman I., Lectures on the geometry of Poisson manifolds, Progress in Mathematics, Vol. 118, Birkhäuser Verlag, Basel, 1994.

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