Abstract
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.
Highlights
This paper is an introduction to the geometry of holomorphic Poisson structures, i.e. Poisson brackets on the ring of holomorphic or algebraic functions on a complex manifold or algebraic variety
This paper serves a second purpose: it is an overview of results on the classification of projective Poisson manifolds that have been obtained by several authors over the past couple of decades, with some added context for the results and the occasional new proof
We will focus on the related problems of construction and classification: how do we produce holomorphic Poisson structures on compact complex manifolds, and how do we know when we have found them all?
Summary
This paper is an introduction to the geometry of holomorphic Poisson structures, i.e. Poisson brackets on the ring of holomorphic or algebraic functions on a complex manifold or algebraic variety (and sometimes on more singular objects, such as schemes and analytic spaces). It grew out of a mini-course delivered by the author at the “Poisson 2016” summer school in Geneva. The first (albeit minor) difference comes already in the definition: while a Poisson bracket on a C∞ manifold is defined by a Poisson bracket on the ring of global smooth functions, this definition is no longer appropriate in the holomorphic setting.
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