We survey the theory of Poisson traces (or zeroth Poisson homology) developed by the authors in a series of recent papers. The goal is to understand this subtle invariant of (singular) Poisson varieties, conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations. The main technique is the study of a canonical D-module on the variety. In the case the variety has finitely many symplectic leaves (such as for symplectic singularities and Hamiltonian reductions of symplectic vector spaces by reductive groups), the D-module is holonomic, and hence, the space of Poisson traces is finite-dimensional. As an application, there are finitely many irreducible finite-dimensional representations of every quantization of the variety. Conjecturally, the D-module is the pushforward of the canonical D-module under every symplectic resolution of singularities, which implies that the space of Poisson traces is dual to the top cohomology of the resolution. We explain many examples where the conjecture is proved, such as symmetric powers of du Val singularities and symplectic surfaces and Slodowy slices in the nilpotent cone of a semisimple Lie algebra. We compute the D-module in the case of surfaces with isolated singularities and show it is not always semisimple. We also explain generalizations to arbitrary Lie algebras of vector fields, connections to the Bernstein–Sato polynomial, relations to two-variable special polynomials such as Kostka polynomials and Tutte polynomials, and a conjectural relationship with deformations of symplectic resolutions. In the appendix we give a brief recollection of the theory of D-modules on singular varieties that we require.

中文翻译：

---

泊松迹线、D 模和辛分辨率

我们调查了作者在最近的一系列论文中提出的泊松迹（或零泊松同调）理论。目标是理解（奇异）泊松簇的这种微妙的不变量，它是有限维的条件，它与辛分辨率的几何和拓扑的关系，以及它在量化中的应用。主要技术是对品种的规范 D 模块的研究。在该变体具有有限多个辛叶的情况下（例如对于辛奇点和通过约简群对辛向量空间进行哈密顿约简），D 模是完整的，因此，泊松迹的空间是有限维的。作为一种应用，对于变体的每个量化，都有有限多个不可约的有限维表示。推测，D模是正则D模在奇点的每个辛分辨率下的推进，这意味着泊松迹的空间与分辨率的顶上同调是对偶的。我们解释了许多证明猜想的例子，例如 du Val 奇点和辛曲面的对称幂以及半单李代数的幂零锥中的 Slodowy 切片。我们在具有孤立奇点的表面的情况下计算 D 模，并表明它并不总是半简单的。我们还解释了向量场的任意李代数的推广、与 Bernstein-Sato 多项式的连接、与诸如 Kostka 多项式和 Tutte 多项式等二元特殊多项式的关系，以及与辛分辨率变形的猜想关系。