This paper is intended both as an introduction to the algebraic geometry of holomorphic Poisson brackets, and as a survey of results on the classification of projective Poisson manifolds that have been obtained in the past 20 years. It is based on the lecture series delivered by the author at the Poisson 2016 Summer School in Geneva. The paper begins with a detailed treatment of Poisson surfaces, including adjunction, ruled surfaces and blowups, and leading to a statement of the full birational classification. We then describe several constructions of Poisson threefolds, outlining the classification in the regular case, and the case of rank-one Fano threefolds (such as projective space). Following a brief introduction to the notion of Poisson subspaces, we discuss Bondal’s conjecture on the dimensions of degeneracy loci on Poisson Fano manifolds. We close with a discussion of log symplectic manifolds with simple normal crossings degeneracy divisor, including a new proof of the classification in the case of rank-one Fano manifolds.

中文翻译：

---

射影泊松簇的构造和分类

本文旨在介绍全纯泊松括号的代数几何，以及对过去 20 年来获得的射影泊松流形分类结果的调查。它基于作者在日内瓦 Poisson 2016 暑期学校讲授的系列讲座。该论文首先详细处理了泊松曲面，包括附加曲面、直纹曲面和膨胀曲面，并得出了完整的双有理分类的声明。然后，我们描述了泊松三重的几种构造，概述了常规情况下的分类，以及一阶法诺三重的情况（例如投影空间）。在简要介绍 Poisson 子空间的概念之后，我们讨论 Bondal 关于 Poisson Fano 流形上简并轨迹维数的猜想。