关于$(1,1)$型环面Poisson结构及其上同调
On Toric Poisson Structures of Type $(1,1)$ and their Cohomology
摘要：
我们对复复环面流形上的$(1,1)$型实Poisson结构进行了分类，并研究了它们的Poisson上同调。对于光滑复曲面簇，这样的结构必然是代数的，并且在由相关单纯扇的开锥确定的每个杰出的全纯坐标图中是齐次二次的。作为每个${\mathbb C}^n$图中光滑上同调问题的一个近似，我们考虑多项式多向量场复形上的泊松微分。对于代数问题，我们计算$H^0$和$H^1$的假设下，泊松结构一般是非退化的。本文最后数值研究了不同$B$的$({\mathbb C}^2,π_B)$的高次上同调群。
英文摘要：
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 $({\mathbb C}^2,π_B)$ for various $B$.
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