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

Abstract

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 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 (C 2 , π B ) for various B.

Holonomic Poisson Manifolds and Deformations of Elliptic Algebras

“…For example the "general position" and "Pnormal" conditions of [58,59] will guarantee holonomicity. On the other hand [10,Example 3.7] gives an example for X = C 4 where the Lichnerowicz cohomology is infinite-dimensional; the failure of holonomicity can be explained by the fact that the modular foliation has infinitely many zero-dimensional leaves.…”

“…This condition always holds when (X, π) is a projective log symplectic surface, as follows from the classification [41,Section 7] or by examining what happens when we blow up the possible minimal models. In this case, the map ∧ • T X → j * Ω • U splits the sequence (10), giving…”