Holomorphic Poisson structures

量子化空间的思想是用一个非对易函数环来代替函数的交换环，这个非对易环应该被实现(“量化”)为某个Hilbert空间上的线性算子代数。“替换‘’意味着在同一函数空间上找到一个参数为h的非交换乘法(普朗克常数的抽象)，当h=0时，它给出了函数的普通乘法，即经典极限。H中的一阶项(“准经典”极限)定义了一种称为泊松结构的数学结构。它可以独立地用微分几何定义，事实上，康采维奇早在10多年前就证明了一个强大的定理，该定理说，至少形式上(作为h的展开)可以从泊松结构返回到量子化，但很少有泊松结构屈服于显式的非对易变形。另一方面，理论物理学家Sklyanin多年前利用椭圆函数定义了向量空间上的非对易代数结构，并由此引出射影空间上的全纯Poisson结构。因此，我们有一个关于非对易几何和泊松几何的一般原理，但很少有例子，也很少理解允许显式量子化的泊松流形的世界有多宽或多窄。多年来，泊松几何一直在国际上被研究，但提出的问题似乎与代数几何没有很好的互动，这也是本提案主要关注的。旨在加深我们对全纯Poisson流形及其可能的非对易变形之间的关系的理解。

The idea of quantizing a space is to replace the commutative ring of functions by a non-commutative one which is supposed to be realized ("quantized") as an algebra of linear operators on some Hilbert space. "Replacement'' means finding a non-commutative multiplication on the same space of functions with a parameter h (an abstraction of Planck's constant) which when h=0 gives the ordinary multiplication of functions, the classical limit. The term to first order in h (the "quasi-classical" limit) defines a mathematical structure called a Poisson structure. It can be defined independently in differential geometric terms and in fact Kontsevich over 10 years ago proved a powerful theorem which said that at least formally (as an expansion in h) one could go from the Poisson structure back to a quantization, yet very few Poisson structures have yielded to explicit noncommutative deformations. On the other hand non-commutative algebra structures on vector spaces were defined many years ago by the theoretical physicist Sklyanin using elliptic functions, and these induce holomorphic Poisson structures on projective space. We thus have a general principle relating non-commutative geometry and Poisson geometry, but few examples and little understanding of how wide or narrow is the world of Poisson manifolds which admit explicit quantizations. Poisson geometry has been pursued for many years at an international level, but the questions that were posed seemed not to interact well with algebraic geometry, which is what this proposal is mainly concerned with. It is intended to advance our understanding of the relationship between holomorphic Poisson manifolds and their possible non-commutative deformations.