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Constructions and classifications of projective Poisson varieties. (English) Zbl 1390.53093

This paper is an introduction to the geometry of holomorphic Poisson structures. It grew out of the lecture series delivered by the author at the “Poisson 2016” summer school in Geneva. This work is also a review of all the results obtained in the past twenty years on the classification of projective Poisson manifolds. First, the author starts with a detailed discussion of Poisson surfaces. Then he studies many types of Poisson structures on threefolds. An introduction to log-symplectic manifolds is then given, with many explicit examples. A new proof of the classification in the case of rank-one Fano manifolds is also obtained.

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
14J45 Fano varieties
17B63 Poisson algebras

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