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Monday, November 28, 2016

**Abstract:** We show that the real Poisson structures on complex toric manifolds $(M,T_{\mathbb C})$ which are invariant under the complex torus action are in bijection with the hermitian forms on $\mathfrak t_{\mathbb C}$. In holomorphic coordinates in which the torus action is equivalent to the product action of $\mathbb T_{\mathbb C}^n$ on $\mathbb C^n$, the components of such tensors are homogeneous quadratic functions of the coordinates and their conjugates with coefficients given by the matrix $[B_{pq}]$ representing the associated hermitian form $B$. We investigate the dependence of the Poisson cohomology on the associated form, focusing on examples where $B$ is integral. In the algebraic category, the computation of the Poisson cohomology on such a chart $\mathbb C^n$ admits a complete solution using representation theory. We prove that, with respect to the wedge product, the cohomology is a module over the exterior algebra of $\mathfrak t_\mathbb C$ with generators which are monomial decomposable multi-vector fields corresponding to solutions of Diophantine inequalities determined by the form $B$. This is joint work Berit Givens of Cal Poly Pomona.